Groups, Algorithms, Geometries & Applications A · 2019. 11. 19. · Groups, Algorithms, Geometries...

AG Algorithmik & Symbolisches Rechnen

Groups, Algorithms,Geometries &Applications AUniversität StuttgartSoSe 19

Prof.Frederik

Witt

Table of contents

1. Rings, modules and morphisms

2. Gröbner bases

3. Affine and projective varieties

For concrete computations we use SINGULAR, a C-based programming languageand computer algebra system which is freely available at

https://www.singular.uni-kl.de/

The website also provides further documentation and examples, see

https://www.singular.uni-kl.de/index.php/singular-manual.html

In these notes, input to SINGULAR will be preceeded by >; output of SINGULAR willbe in red:

>5*4;20


2. Gröbner bases


1. Definition.(i) A ring (A,+, ·) consists of an abelian group (A,+) with neutral element 0,

and an associative multiplication · : A× A→ A such that for all a, b, c ∈ A,

a · (b + c) = a · b + a · c and (b + c) · a = b · a + b · c.

Furthermore, in this course a ring will always be commutative and withidentity, i.e., multiplication · is commutative, and there is a neutral element 1for ·. A subgroup of a ring A is a subring if it contains 1 and is closed undermultiplication. A unit u ∈ A \ {0} is an element such that there exists u′ ∈ Awith uu′ = 1. A ring whose nonzero elements are units is a field.

(ii) A ring morphism ϕ : A→ B is a group morphism such that ϕ(1A) = 1B andϕ(a · b) = ϕ(a) · ϕ(b). A ring isomorphism is a bijective ring morphism (itsinverse is then also a ring morphism).

In the sequel, A and k will always denote a ring and a field respectively.

2. Examples. Z, its quotients Zm = Z/mZ or fields such as Q, R or C. On theother hand, N = {m ∈ Z | m ≥ 0} is not a ring (it is not even a group).

3. Definition.(i) Let A be a ring. A set M, together with two operations + : M ×M → M

(addition) and · : A×M → M (scalar multiplication) is called an A–module if(M,+) is an abelian group, and for all a, b ∈ A, m, n ∈ M

. (a + b) ·m = a ·m + b ·m, a · (m + n) = a ·m + a · n

. (ab) ·m = a · (b ·m)

. 1 ·m = m

(ii) Let M, N be A–modules. A map ϕ : M → N is called A–module morphism orA-linear if, for all a ∈ A and m, n ∈ M

ϕ(am) = aϕ(m), ϕ(m + n) = ϕ(m) + ϕ(n).

If N = M, then ϕ is called an endomorphism.(iii) HomA(M,N) denotes the set of all A–module morphisms M → N.(iv) A bijective A–linear map ϕ : M → N is called an isomorphism (ϕ−1 being

automatically A-linear). In this case M and N are isomorphic.

4. Example. If A is a ring, then the direct sum (as abelian groups)An = A⊕ . . .⊕ A with the usual scalar action is an A-module.

5. Polynomials over A. A monomial in the variables x1, . . . , xn is a formalexpression

xα := xα11 . . . xαn

n , α = (α1, . . . , αn) ∈ Nn.

The set of monomials Mon(x1, . . . , xn) = {xα | α ∈ Nn} forms a semi-group withneutral element 1 = x0

1 . . . x0n via

xα11 . . . xαn

n · xβ11 . . . xβn

n = xα1+β11 . . . xαn+βn

n .

The degree of xα is the sum |α| = α1 + . . .+ αn. A polynomial over A in thevariables x1, . . . , xn is a finite linear combination of monomials xα over A:

f =∑

α=(α1,...,αn)

aαxα11 . . . xαn

n , aα ∈ A.

The support of f is the set supp f = {xα | aα 6= 0}. For f 6= 0 the degree of f is themaximum deg(f ) of all |α|, xα ∈ supp f ; by convention, we put deg 0 = −1.

A[x1, . . . , xn] = set of polynomials in n variables.

6. Ring structure on the polynomials. A term is a polynomial of the form axα.The ring structure on A[x1, . . . , xn] is defined as follows:∑

aαxα +∑

bαxα :=∑

(aα + bα)xα(∑aαxα

)·(∑

bαxα)

:=∑γ

( ∑α+β=γ

aαbβ)xγ .

If we identify 1 ∈ A with 1 = x0, then A ⊂ A[x1, . . . , xn] as a subring (with theobvious definitions). The elements of A ⊂ A[x1, . . . , xn] will be called constantpolynomials, and A is the base ring. In particular, this gives A[x1, . . . , xn] thestructure of an A-module for which Mon(x1, . . . , xn) provides a basis.

7. Definition. Let A be a ring. An A-algebra is a ring B which contains A as asubring. Let B and C be two A-algebras. A ring morphism ϕ : B → C is called anA-algebra morphism if ϕ(a) = a for all a ∈ A, i.e., ϕ|A = IdA. More generally, ifϕ : A→ B and ψ : A→ C are two ring maps, then a ring map α : B → C is anA–algebra morphism if α ◦ ϕ = ψ, that is, α is an A-algebra morphism with respectto the A-algebra structure on B and C induced by ϕ(a) · b and ψ(a) · b.

8. Remark. In fact, if B is an A-algebra via the ring morphism ϕ : A→ B, ϕ(a) · bturns B into an A-module. Hence any A-algebra is also an A-module (the conversebeing false), for instance A[x1, . . . , xn]. Similarly, any abelian group (which is aZ-algebra) is a Z-module. Moreover, an A-algebra morphism is an A-linear map.

9. Example. Let k be a field. Then we have a natural ring morphism ϕ : Z→ kdefined by 1 ∈ Z 7→ 1 ∈ k , i.e., k is a ϕ(Z)-algebra. Note that the kernel kerϕ iseither trivial, or there exists a unique positive prime p ∈ Z which divides anyelement in kerϕ (cf. Remark 22). We call p the characteristic of k and writep = char k . For instance, charQ = 0 while charZp = p.

10. Proposition [EH, Thm. 1.1]. Let B be an A-algebra, and let α1, . . . , αn ∈ B.Then there exists a unique A-algebra morphism ϕ : A[x1, . . . , xn]→ B determinedby ϕ(xi) = αi , the so-called substitution morphism.

11. Proposition [EH, Thm. 1.3]. There is a natural A-algebra isomorphismA[x1, . . . , xn−1, xn] ∼= A[x1, . . . , xn−1][xn].

Definition of polynomial base rings 1 [GP, 1.1.9]> ring A1=0,x,dp; //defines the ring Q[x ]: characteristic 0, x thevariable, dp= degree reverse lexicographical ordering on Mon(x1, . . . , xn)(see below)> poly f=2/3x3+5x2+1/7x+2/5;> poly g=1/3x4+2/9x+2/3;> f*g;2/9x7+5/3x6+1/21x5+38/135x4+14/9x3+212/63x2+58/315x+4/15

Definition of polynomial base rings 2 [GP, 1.1.9]> ring A2=(7,a,b),(x,y,z),dp; //defines the ring Z7(a,b)[x , y , z] withdegree reverse lexicographical ordering. It has characteristic 7,invertible parameters a, b, and variables x, y, z> poly f=2ax3y2z+(1/b)*x2z+xyz+2;> poly g=3xy4z2+2xyz+3bx;> f*g;(-a)*x4y6z3+3/(b)*x3y4z3+3*x2y5z3+(-3a)*x4y3z2+(-ab)*x4y2z-xy4z2+2/(b)*x3yz2+2*x2y2z2+3*x3z+(3b)*x2yz-3*xyz+(-b)*x

Definition of algebra morphisms [GP, 1.1.10]> ring A=0,(a,b,c),dp; //defines the ring Q[a,b, c] with degree reverselexicographical ordering> poly f=a+b+ab+c3+5;> ring B=0,(x,y,z),dp; //new basering B; if we do not specifyotherwise, all computations now take place in B> map F=A,x+y,x-y,z; //map F : A → B defined by a 7→ x + y, b 7→ x − y,c 7→ z> poly g=F(f); //apply F to f> g;z3+x2-y2+2x+5

IMAP and FETCH [GP, 1.1.10]> ring C=0,(x,y,c,b,a,z),dp; //defines the ring Q[x , y , z,a,b, c]> imap(A,f); //“identity map” from A → C preserving the names of thevariables, but displaying f with respect to the new orderingc3+ba+b+a+5> fetch(A,f); //fetch preserves order of variablesc3+xy+x+y+5

12. Definition.(i) A subset I ⊂ A is an ideal if it is an abelian subgroup and a ∈ A, f ∈ I implies

a · f ∈ I, that is, I is an A-module. The ideal generated I = (fλ)λ∈Λ by thesubset {fλ | λ ∈ Λ} of A is the set of finite linear combinations of the fλ, i.e.

I = ({∑finite

ai fi | i ∈ Λ, ai ∈ A}).

(this is indeed an ideal). If Λ is a finite set, then we call I finitely generated.(ii) The null ideal is the ideal (0) = {0} generated by 0. If an ideal is generated

by one element it is called a principal ideal.(iii) If I and J are two ideals of A, then I + J and I · J are the ideals generated by{a + b | a ∈ I, b ∈ J} and {a · b | a ∈ I, b ∈ J}, respectively.

13. Example. Let ϕ : A→ B be a ring morphism,and J ⊂ B be an ideal. Then itspreimage or contraction ϕ−1(J) is an ideal. In particular, kerϕ = ϕ−1((0)) is anideal. ϕ is injective, if kerϕ = (0). On the other hand, if I is an ideal in A, then ϕ(I)is merely a subring of B, but not an ideal in general. The extension of I by ϕ isthe ideal generated by ϕ(I). It is written ϕ(I) · B.

Injectivity of ring morphisms [GP, 1.3.3]> ring A=0,(a,b,c),dp; //defines A = Q[a,b, c]> ring B=0,(x,y,z),dp; //defines B = Q[x , y , z] which is the base ring> ideal I=x,y,x2-y3; //defines the ideal I = (x , y , x2− y3) ⊂ B> I;I[1]=xI[2]=yI[3]=-y3+x2> map phi=A,I; //defines φ : A→ B, φ(a) = x; φ(b) = y, φ(c) = x2 − y2

> LIB "algebra.lib"; //loads additional library// ** loaded /usr/bin/../share/singular/LIB/algebra.lib...> is_injective(phi,A);0 //φ : A→ B is not injective> ideal J=x,x+y,z-x2+y3; //defines the ideal J ⊂ B> map psi=A,J; //ψ : A→ B> is_injective(psi,A);1 //ψ : A→ B is injective

Computing the kernel of ring morphisms [GP, 1.3.3]> alg_kernel(phi,A,"ker"); //defines the ideal ker ⊂ A and computesgeneratorsb3-a2+c> setring A; //Since B is the basering we need first to set A as basering if we want to work with ideals in A> ker; //print generators of the ideal ker ⊂ Aker[1]=b3-a2+c> setring B;> alg_kernel(phi,A); //short version of alg_kernelb3-a2+c> alg_kernel(psi,A);0> ideal Z; //defines the zero ideal in B> Z;Z[1]=0

Surjectivity of ring morphisms [GP, 1.3.3]> setring A; //we can now compute ideals in A> preimage(B,phi,Z); //computes the preimage of φ, i.e., generators ofthe ideal φ−1(Z ) = kerφ ⊂ A_[1]=b3-a2+c> ideal P=preimage(B,phi,Z); //defines the ideal P ⊂ A as the preimageφ−1(0)> P;P[1]=b3-a2+c> setring B; //sets B as base ring, we can now deal with ψ : A→ B> psi;psi[1]=xpsi[2]=x+ypsi[3]=y3-x2+z> is_surjective(psi,A);1 //ψ : A→ B is surjective> is_bijective(psi,A); //ψ : A→ B is surjective1 //ψ : A→ B is injective

Remark on the base rings in SINGULAR

If we are working with several rings and work with an ideal I inside some ring A,the latter must be the base ring; if necessary, use the command

setring A.

The command basering prints the current base ring. If we are given generators ofan ideal I ⊂ B we can define a ring morphism φ : A→ B via the command

map phi=A,I

provided B is the base ring – morphisms are regarded as target space objects.This command maps the variables of the domain ring to the generators of theideal. If there are less generators of the ideal than variables of the domain ring,then the remaining variables are mapped to 0. If there are more generators of theideal than variables in the domain ring, the addtional generators are ignored. Sowe always specify the map and the domain ring, the target ring being the basering. As another example in the code above,

is_bijective(phi,A)

tests if a given morphism φ from A to the base ring is bijective.

14. Remark [EH, Exercise 1.5]. If I = (f1, . . . , fr ) and J = (g1, . . . ,gs) are twofinitely generated ideals, then so are I + J and I · J with

I + J = (f1, . . . , fr ,g1, . . . ,gs), I · J = (figj | i = 1, . . . , r , j = 1, . . . , s).

15. Definition and Proposition. Let I ⊂ A be an ideal. Then a ∼ b if and only ifa− b ∈ I defines an equivalence relation on A with equivalence class a. Thequotient ring A/I is the ring on the coset space {a | a ∈ A} with the followingoperations:

. a + b := a + b

. a · b := a · bIn particular, the projection or quotient map πI : A→ A/I, πI(a) = a becomes asurjective ring A-algebra morphism. Moreover, the scalar multiplicationa · b := a · b turns A/I into an A-module.

16. Example. In Q[x , y ]/(x2, xy) we have x2 = 0 and x · y = 0. For instance,5x2y + 3x4 + 1 + y · 2x + xy + 7y3 = 1 + y · 2x + 7y3 = 2x + 7y3 + 7y4.

17. Comparison Lemma A↔ A/I.(i) A ring morphism ϕ : A→ B induces an isomorphism A/kerϕ ∼= Im B. In

particular, B is a ring isomorphic to A/I, I ideal of A, if ϕ is surjective.(ii) The map J 7→ πI(J) induces a bijection

{ideals in A containing I} ←→ {ideals in A/I}

whose inverse assigns to J ⊂ A/I the ideal π−1I (J) in A.

18. Definition.(i) An element a ∈ A is a zero divisor if there exists b ∈ A \ {0} such that

ab = 0.(ii) A ring A which has only 0 as a zero divisor is called an integral domain.(iii) A principal ideal ring is a ring for which any ideal is principal; it is a

principal ideal domain if it is in addition an integral domain.

19. Example. Z is an integral domain. Further, it is easy to see that if A is anintegral domain, then so is the polynomial ring A[x ]. By recursion, any polynomialring A[x1, . . . , xn] is integral. On the other hand, quotients of integral domains areusually not integral. Consider, for instance, the ideal I = (f · g). Then f , g 6= 0, butfg = 0.

20. Definition. A proper ideal I ( A is prime if f · g ∈ I implies f ∈ I or g ∈ I, andmaximal, if I ⊂ I′ ( A implies I = I′.

21. Lemma [GP, 1.3.11].(i) I ⊂ A is prime if and only if A/I is an integral domain.(ii) I ⊂ A is maximal if and only if A/I is a field. In particular, a maximal ideal is

prime.

22. Remark.(i) If ϕ : A→ B is a ring morphism, and J ⊂ B a prime ideal, then so is its

contraction ϕ−1(J). On the other hand contractions of maximal ideals are notnecessarily maximal (consider the inclusion Z→ Q and (0) ⊂ Q).

(ii) Consider the canonical Z-algebra structure of a field k (cf. Example 9). Theimage of ϕ : Z→ k is an integral ring and is isomorphic with Z/kerϕ, so thatkerϕ is a prime ideal. It is therefore equal to the trivial ideal or the principalideal of a prime in Z.

23. Lemma [GP, 1.3.12].(i) Let P, I, J ⊂ A be ideals with P prime. If I 6⊂ P, I · J ⊂ P, then J ⊂ P.(ii) Let P, I1, . . . , In ⊂ A be ideals with P prime and

⋂ni=1 Ii ⊂ P (resp. =). Then

Ij ⊂ P (resp. =) for some j.(iii) (Prime avoidance) Let P1, . . . , Pn, I ⊂ A be ideals with Pi prime and

I ⊂⋃n

i=1 Pi . Then I ⊂ Pj for some j.

Definition of quotient rings and testing for equality [GP, 1.3.13]> ring A=0,(x,y,z),dp;> ideal I=x2+y2-z5,z-x-y2;> I;I[1]=-z5+x2+y2I[2]=-y2-x+z> std(I); //computes a particularly nice generating system (“Gröbner”or “standard base”) of I – we will explain this in greater detaillater_[1]=y2+x-z_[2]=z5-x2+x-z> qring Q=std(I); //defines the quotient ring Q = A/I where I must bespecified via a Gröbner base, and sets Q as current base ring againwhose variables are still denoted x, y and z> poly f=z2+y2;> poly g=z2+2x-2z-3z5+3x2+6y2; //elements of Q> reduce(f-g,std(0)); //computes the remainder of the division of f − gby the Gröbner base of (0); this is zero if and only if f = g0

24. Definition. A monomial ideal of k [x1, . . . , xn] is an ideal generated bymonomials.

25. Proposition [EH, 1.7]. For an ideal I ⊂ k [x1, . . . , xn] are equivalent:(i) I is monomial.(ii) For any f ∈ I one has supp f ⊂ I.

26. Corollary [EH, 1.8]. Let I ⊂ k [x1, . . . , xn] be a monomial ideal, and G a set ofmonomials in I. Then G is a set of generators of I if and only if for each monomialv ∈ I there exists u ∈ G such that u|v.

27. Remark. On the subset of monomials Mon(x1, . . . , xn) ⊂ k [x1, . . . , xn] we havethe following natural partial order: α ≤nat β, α, β ∈ Nn, if and only if αi ≤ βi for alli = 1, . . . ,n. Equivalently, α ≤nat β if and only if xα|xβ, i.e., xα divides xβ.

28. Theorem (Dickson’s Lemma) [EH, 1.9], [GP, 1.2.6]. Let∅ 6= M ⊂ Mon(x1, . . . , xn). Then there is a finite subset B ⊂ M such that for allxα ∈ M there exists xβ ∈ B with xβ ≤nat xα.

29. Corollary (Hilbert’s Basissatz for monomial ideals) [EH, 1.10]. LetI ⊂ k [x1, . . . , xn] be a monomial ideal. Then any subset of generators of I has afinite subset generating I.

30. Definition. Let I be a monomial ideal. A subset G of monomial generators isminimal if any proper subset of G does not generate I any longer.

31. Proposition [EH, 1.11]. Any monomial ideal I has a unique minimal set ofmonomial generators, the so-called minimal basis which is denoted by G(I).

32. Remark. If I and J are monomial ideals, then so are I + J and I · J. We have

G(I + J) ⊂ G(I) ∪G(J) and G(I · J) ⊂ G(I)G(J)

for their minimal bases.

33. Corollary [EH, 1.12]. Each ascending sequence of monomial idealsI1 ⊂ I2 ⊂ I3 ⊂ . . . in k [x1, . . . , xn] eventually becomes stationary, that is, there existsa k ∈ N such that Ik = Ik+1 = Ik+2 = . . ..

34. Definition. A ring A is called Noetherian if any ascending sequence of idealseventually becomes stationary.

35. Example.(i) Every principal ideal ring such as Z or k [x ] is Noetherian.(ii) Hilbert’s celebrated basis theorem states that k [x1, . . . , xn] is Noetherian (see

below). More generally, if A is Noetherian, then so is A[x ].

We pass to modules next and consider further examples of modules andmorphisms between them.

36. Lemma and Definition [GP, 2.1.4]. We can define an A-module structure onHomA(M,N) by

(ϕ+ ψ)(m) := ϕ(m) + ψ(m)

(aϕ)(m) := a · ϕ(m).

In particular, M∨ := HomA(M,A) is an A-module, the so-called dual module.

37. Remark [GP, 2.1.5]. If M, N and L are A-modules, and ϕ : M → N is A-linear,then

Φ : HomA(N,L)→ HomA(M,L), Φ(λ) := λ ◦ ϕ

andΨ : HomA(L,M)→ HomA(L,N), Ψ(λ) := ϕ ◦ λ

are A-linear for the natural A-module structure on Hom.

Matrix operations 1 [GP, 2.1.6]> ring A = 0,(x,y,z),dp;> matrix M[2][3] = 1, x+y, z2, x, 0, xyz; //2x3 matrix. Note that forSINGULAR matrices have polynomial entries, hence we need to specifya base ring> matrix N[3][3] = 1,2,3,4,5,6,7,8,9; //3x3 matrix> M; //lists all entries of MM[1,1]=1M[1,2]=x+yM[1,3]=z2M[2,1]=xM[2,2]=0M[2,3]=xyz> print(N); //displays N as usual if the entries are small1,2,3,4,5,6,7,8,9

Matrix operations 2 [GP, 2.1.6]> print(M+M); //addition of matrices2, 2x+2y,2z2,2x,0,2xyz> print(x*N); //scalar multiplicationx, 2x,3x,4x,5x,6x,7x,8x,9x> print(M*N); //multiplication of matrices7z2+4x+4y+1,8z2+5x+5y+2,9z2+6x+6y+3,7xyz+x, 8xyz+2x, 9xyz+3x> M[2,3]; //access to single entryxyz> M[2,3]=37; //change single entry> print(M);1,x+y,z2,x,0, 37

Matrix libraries 1 [GP, 2.1.6]> LIB "matrix.lib"; LIB "inout.lib"; //libraries for matrix operations> print(power(N,3)); //exponentiation of matrices468, 576, 684,1062,1305,15481656,2034,2412> matrix K = concat(M,N); //concatenation of matrices; number of rowsof result matrix is max(nrows(A1),nrows(A2),...)> print(K);1,x+y,z2,1,2,3,x,0, 37,4,5,6,0,0, 0, 7,8,9

Matrix libraries 2 [GP, 2.1.6]> ideal(M); //converts matrix to ideal by taking the entries asgenerators_[1]=1_[2]=x+y_[3]=z2_[4]=x_[5]=0_[6]=37> print(unitmat(5)); //5x5 unit matrix1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,

38. Definition.(i) Let M be an A-module. An abelian subgroup N of M is an A-submodule of M

if it is closed under scalar multiplication of A, i.e., for all a ∈ A, n ∈ N we havea · n ∈ N.

(ii) If N ⊂ M is an A-submodule we define the quotient module M/N as the set ofequivalence classes m = m + n for m ∈ M, n ∈ N. This is again an A-module(cf. the case of A/I discussed above). Indeed, m + m′ = m + m′ anda ·m = a ·m are well-defined operations on M/N turning it into an A-module.

39. Example. Let ϕ : M → N be an A-module morphism. The kernelkerϕ := {m ∈ M | ϕ(m) = 0} and the imageimϕ := {n ∈ N | n = ϕ(m) for some m ∈ M} are A-submodules of M and Nrespectively. The cokernel cokerϕ = N/imϕ is a quotient module.

Submodules of An [GP, 2.1.10]> ring A=0,(x,y,z),dp;> module M=[xy-1,z2+3,xyz],[y4,x3,z2]; //submodule in A3

> M; //gives the generating vectors with respect to the canonicalbasis gen(i) = ei of A3

M[1]=xyz*gen(3)+xy*gen(1)+z2*gen(2)+3*gen(2)-gen(1)M[2]=y4*gen(1)+x3*gen(2)+z2*gen(3)> ideal I=x2+y2+z2;> qring Q=std(I); //creates quotient ring A/I> module M=fetch(A,M); //maps M from A-modules to Q = A/I-modules> vector s1=[x2,y3,z];> vector s2=[0,x2-y2,z];> poly f=xyz;> module N=s1,f*s2; //alternative definition of a module (over thecurrent base ring Q)> N;N[1]=y3*gen(2)+x2*gen(1)+z*gen(3)N[2]=x3yz*gen(2)-xy3z*gen(2)+xyz2*gen(3)

Kernel and image of module morphisms [GP, 2.1.13]> ring A=0,(x,y,z),(c,dp); //(c,dp) specifies how to order monomialsof the module, see below. It also implies that vectors are representedcomponentwise> matrix M[2][3]=x,xy,z,x2,xyz,yz;> module Ker=syz(M); //syz computes the kernel of M (in factthe first syzygy, see below); it is a generalization of Gaussianelimination from fields to rings> Ker;Ker[1]=[y2z-yz2,xz-yz,-x2y+xyz]> vector k=[y2z-yz2,xz-yz,-x2y+xyz];> M*k;_[1,1]=0_[2,1]=0> module Im=M[1],M[2],M[3];> Im;Im[1]=[x,x2]Im[2]=[xy,xyz]Im[3]=[z,yz]

40. Proposition [GP, 2.1.16]. Let ϕ : M → N be an A-module morphism. Then

M/kerϕ ∼= imϕ.

In analogy to the cokernel one also calls M/kerϕ the coimage of ϕ. Theisomorphism then reads coimϕ ∼= imϕ. In particular, given A-modules N ⊂ M ⊂ L,we have

(L/N)/(M/N) ∼= L/M.

41. Definition.(i) Let M be an A-module with submodules Mλ. λ ∈ Λ. Then the sum of the Mλ

is defined by∑λ∈Λ

Mλ := {∑λ∈Λ

mλ | mλ ∈ Mλ, mλ 6= 0 only for finitely many λ} ⊂ M

and their intersection by⋂λ∈Λ Mλ ⊂ M

(ii) Let I ⊂ A be an ideal and M an A-module. We define IM by

IM := {∑finite

aimi | ai ∈ I, mi ∈ M}.

(iii) The direct sum is the module (with the obvious operations)⊕λ∈Λ

Mλ = {(mλ)λ∈Λ | mλ ∈ Mλ, mλ 6= 0 for only finitely many λ}

The direct product is the module (with the obvious operations)∏λ∈Λ

Mλ = {(mλ)λ∈Λ | mλ ∈ Mλ}.

For a finite index set these notions coincide. We write M1 ⊕ . . .⊕Mn.(iv) A module M which is isomorphic with the direct sum

⊕λ∈Λ A is called a free

module (e.g., A[x1, . . . , xn]). By convention, M = (0) if Λ = ∅. The cardinality ofΛ is the rank of M. A subset S of M is called a basis, if every m ∈ M is a finitelinear combination of elements in S in a unique way.

(v) An A-module M is finitely generated if there exists m1, . . . ,mr ∈ M such thatany m ∈ M is a linear combination of the generators mi . We writeM = 〈m1, . . . ,mr 〉. A module is cyclic if it is generated by one element (e.g., aprincipal ideal).

(vi) The torsion submodule of an A-module M is defined by

Tors(M) := {m ∈ M | am = 0 for some non zerodivisor a ∈ A}.

M is torsionfree if Tors(M) = 0, and a torsion module if Tors(M) = M.(vii) Let N, P ⊂ M be submodules. The ideal quotient N : P is defined by

N : P := {a ∈ A | aP ⊂ N}.

If N = 0 then Ann(P) := (0 : P) is called the annihilator of P.

42. Remark. The ring of polynomials A[x ] is isomorphic to the direct sum⊕∞

i=0 A,where the sequence (a0,a1, . . .) is sent to the polynomial

∑∞i=0 aix i . Since only

finitely many ai are different from zero we get a well-defined polynomial. On theother hand, the direct product

∏∞i=0 A yields the ring AJxK of formal power series.

Operation on modules 1 [GP, 2.1.20]> ring A=0,(x,y,z),(c,dp);> module M=[xy,xz],[x,x];> module N=[y2,z2],[x,x];> M+N; //sum of two ideals_[1]=[xy,xz]_[2]=[x,x]_[3]=[y2,z2]> intersect(M,N); //intersection of two modules_[1]=[x,x]_[2]=[xy2,xz2]> quotient(M,N); //quotient M : N as submodules of An

_[1]=x> quotient(N,M); //quotient N : M as submodules of An

_[1]=y+z

Operation on modules 2 [GP, 2.1.20]> qring Q=std(x5);> module M=fetch(A,M);> module Null; //the trivial module> M;M[1]=[xy,xz]M[2]=[x,x]> Null;Null[1]=0> quotient(Null,M); //the annihilator of M_[1]=x4

43. Remark. The sum of submodules of an A–module, the product of an idealwith an A–module, the direct sum and the direct product of A–modules are againA–modules. The module quotient of two submodules of an A–module is an ideal inA. The quotient of a submodule by an ideal is a submodule of M. The torsionmodule Tors(M) is a submodule of M.

44. Proposition [GP, 2.1.21]. Let M be an A-module and N1, N2 ⊂ M besubmodules, then

(N1 + N2)/N1∼= N2/(N1 ∩ N2).

45. Lemma [GP, 2.1.22]. Let M be an A-module. M is finitely generated if andonly if M ∼= An/L for some n ∈ N and a submodule L ⊂ An. Equivalently, thereexists a surjective homomorphism ϕ : An � M.

46. Definition. Let M be an A-module. Then M is called Noetherian if everyA-submodule of M is finitely generated. In particular, A is a Noetherian ring if andonly if it is Noetherian as an A-module over itself.

47. Lemma [GP, 2.1.28].(i) Submodules and quotient modules of Noetherian modules are Noetherian.(ii) Let N ⊂ M be A–modules, then M is Noetherian if and only if N and M/N are

Noetherian.(iii) Let M be an A–module, then the following properties are equivalent:

. M is Noetherian.

. Every ascending chain of submodules M1 ⊂ M2 ⊂ . . . ⊂ Mk ⊂ . . . becomesstationary.

. Every non–empty set of submodules of M has a maximal element with respectto inclusion.

The same holds for Noetherian rings and ideals instead of Noetherian modulesand submodules.

48. Proposition [GP, 2.1.29]. Let A be a Noetherian ring and M be a finitelygenerated A-module, then M is a Noetherian A-module.

49. Lemma (Nakayama) [GP, 2.1.30]. Let A be a ring and I be an ideal containedin the Jacobson ideal of A, i.e., the intersection of all maximal ideals of A. If M isa finitely generated A-module and N a submodule of M such that M = IM + N,then M = N. In particular, if M = IM, then M = 0.

50. Corollary [GP, 2.1.31]. Let (A,m) be a local ring, that is, m is the onlymaximal ideal. Further, let M be a finitely generated A-module. If m1, . . . ,mn ∈ Minduce generators of the A/m-vector space M/mM, then they also generate M.


2. Gröbner bases


The advantage of working with monomial ideals lied in the existence of a canonicalgenerating set, namely the minimal basis G(I). Formalising this idea gives rise toGröbner bases.

51. Definition. A monomial ordering is a total ordering < on Mon(x1, . . . , xn)such that

xα < xβ =⇒ xγxα < xγxβ

for all α, β and γ ∈ Nn. A monomial ordering < is called global resp. local ifxα > 1 resp. xα < 1 for all α ∈ Nn \ {0}.

52. Remark.(i) If we consider 0 as a monomial we set 0 < xα for all α.(ii) All orderings we will consider in this course are global in view of the important

characterisation below (Lemma 56). Still, local orderings have importantapplications in geometry. One should think of these as a device ofimplementing negative exponents, that is, of inverting monomials, leading tothe localisation of a polynomial ring. This is an important idea in commutativealgebra and algebraic geometry, whence the name of local ordering. See forinstance [GP, Chapter 1.4] for an elementary introduction.

53. Examples of global orderings.(i) lexicographical ordering >lp:

xα >lp xβ ⇔ α1 = β1, . . . , αi−1 = βi−1, αi > βi

(ii) Degree reverse lexicographical ordering >dp:

xα >dp xβ ⇔deg xα > deg xβ or if deg xα = deg xβ

then αn = βn, . . . , αi+1 = βi+1, αi < βi

(iii) Degree lexicographical ordering >Dp:

xα >Dp xβ ⇔deg xα > deg xβ or if deg xα = deg xβ

then α1 = β1, . . . , αi−1 = βi−1, αi > βi

54. Remark. Let > be a general ordering. Define >′ by xα >′ xβ if and only ifxα < xβ. Then > is global if and only if >′ is local.

Global monomial orderings> ring A=0,(x,y,z),lp; //A = Q[x , y , z] with lexicographical ordering> poly f=x3yz+y5+z4+x3+xy2;> f;x3yz+x3+xy2+y5+z4> ring B=0,(x,y,z),dp; //B = Q[x , y , z] with degree reverse lexicogra-phical ordering> poly g=imap(A,f);> g;y5+x3yz+z4+x3+xy2> ring C=0,(x,y,z),Dp; //C = Q[x , y , z] with degree lexicographical or-dering> poly h=imap(A,f);> f;x3yz+y5+z4+x3+xy2

Once we have fixed an ordering, we can give a definite representation of apolynomial 0 6= f ∈ k [x1, . . . , xn] as a sum of non-zero terms

f = aα1xα1 + . . .+ aαr xαr , xα1 > . . . > xαr .

55. Definition.(i) The leading monomial of f is LM(f ) = xα1 .(ii) The leading exponent of f is LE(f ) = α1.(iii) The leading term of f is LT(f ) = aα1xα1 .(iv) The leading coefficient of f is LC(f ) = aα1 ∈ k .(v) The tail of f is tail(f ) = f − LT(f ) = aα2xα2 + . . .+ aαr xαr .

By convention we set LM(0) = 0.

From now on we will fix some ordering > on Mon(x1, . . . , xn) which will be globalunless mentioned otherwise (we shall write sometimes global ordering foremphasis).

Leading data of a polynomial> ring A=0,(x,y,z),lp; //A = Q[x , y , z] with lexicographical ordering> poly f=y4z3+2x2y2z2+3x5+4z4+5y2;> f;3x5+2x2y2z2+y4z3+5y2+4z4 //displays f according to the lexicographicalordering> leadmonom(f); //leading monomialx5> lead(f); //leading term3x5> leadexp(f); //leading exponent5,0,0> leadcoef(f); //leading coefficient3

56. Lemma [GP, 1.2.5]. Are equivalent:(i) > is a well ordering, i.e., every ∅ 6= S ⊂ Mon(x1, . . . , xn) has a least element

xβ (this means that for all xα ∈ S we have α = β or xα > xβ).(ii) xi > 1 for i = 1, . . . ,n.(iii) xα > 1 for all α 6= (0, . . . ,0), that is, > is global.(iv) xα ≥nat xβ and α 6= β implies xα > xβ.

The well-ordering property immediately yields the

57. Corollary. Let > be an ordering on Mon(x1, . . . , xn). If u1, u2, . . . is a sequenceof monomials with u1 ≥ u2 ≥ . . ., then there exists an integer m such that ui = umfor all i ≥ m.

58. Lemma [Exercise]. Let f1, . . . , fr ∈ k [x1, . . . , xn] be nonzero polynomials. Then(i) LM(f1 · f2 · . . . · fr ) = LM(f1) · . . . · LM(fr ).(ii) LM(f1 + . . .+ fr ) ≤ max

(LM(f1), . . . ,LM(fr )

). Equality holds if and only if∑

LC(fi) 6= 0, where the sum is taken over those i whose leading monomialsLM(fi) are maximal on the right hand side.

59. Definition. Let G ⊂ k [x1, . . . , xn] be a nonempty subset. The leading ideal ofG is the monomial ideal

LM(G) := (LM(f ) | 0 6= f ∈ G).

For G = (0) we set LM((0)) = (0).

60. Remark. Let I ⊂ k [x1, . . . , xn] be an ideal with generating subset G. Themonomial ideal LM(I) is generated by the leading monomials of all polynomials inI, and LM(G) is in general properly contained in LM(I). For instance, considerI = (f ,g) where f = x1x2 − x3x4, g = −x2

2 + x1x3. For >=>dp we haveLM(f ) = x1x2 and LM(g) = x2

2 . Now h = x21 x3 − x2x3x4 = x2f + x1g ∈ I, but

LM(I) 3 LM(h) = x21 x3 6∈ (x1x2, x2

2 ).

61. Definition. Let I ⊂ k [x1, . . . , xn] be a nontrivial ideal. A Gröbner basis of I is afinite subset G = {g1, . . . ,gm} of I with LM(I) = LM(G) = (LM(g1), . . . ,LM(gm)).

For instance, the generator g of a principal ideal I = (g) provides a Gröbner basis,for LM(g) = (LM(g)) by (i) of Lemma 58. In general, we know that the monomialideal LM(I) is finitely generated by Corollary 29 which yields existence of aGröbner basis for any ideal.

62. Theorem [EH, 2.8]. Let I ⊂ k [x1, . . . , xn] be a nontrivial ideal, and{g1, . . . ,gm} be a Gröbner basis. Then I = (g1, . . . ,gm).

Theoretical application 1: Hilbert’s basis theorem63. Corollary (Hilbert’s basis theorem) [EH, 2.9 & 10]. k [x1, . . . , xn] is aNoetherian ring, that is, each ideal is finitely generated. Equivalently, everyascending chain of ideals I1 ⊂ I2 ⊂ . . . becomes stationary, i.e., there exists a ksuch that Ij = Ik for all j ≥ k.

Next we address the question of computing Gröbner bases explicitely. This willbe based on a general division algorithm for multivariate polynomials.

64. Definition. Let G ⊂ k [x1, . . . , xn] be any subset.(i) G is called interreduced if 0 6∈ G and if LM(g) 6 | LM(f ) for any two distinct

elements f , g ∈ G. An interreduced Gröbner basis is also called minimal.(ii) f ∈ k [x1, . . . , xn] is reduced with respect to G if for each g ∈ G, g 6= f ,

LM(g) does not divide any monomial in supp f .(iii) G is reduced if G is interreduced and if for each g ∈ G, LC(g) = 1 and tail(g)

is reduced with respect to G.

65. Remark. Any finite set G can be transformed to a minimal one G′ with(G) = (G′), i.e., they generate the same ideal. Indeed, if f and g ∈ G such thatLM(g) = aLM(f ), then replace g by g − af .

66. Definition. Let

G = {G = (g1, . . . ,gs) | gi ∈ k [x1, . . . , xn], s ∈ N}be the set of finite s-tuples of elements in k [x1, . . . , xn]. A function

NF : k [x1, . . . , xn]× G → k [x1, . . . , xn], (f ,G) 7→ NF(f |G)

is called a normal form on k [x1, . . . , xn] if for all G ∈ G and f ∈ k [x1, . . . , xn] thefollowing properties hold:

. NF(0|G) = 0

. NF(f |G) 6= 0⇒ LM(NF(f |G)) 6∈ LM(G)

. If G = (g1, . . . ,gs), then f =∑s

i=1 qigi + NF(f |G) for qi ∈ k [x1, . . . , xn] withLM(

∑si=1 qigi) ≥ LM(qigi) for any i with qigi 6= 0, that is,

∑si=1 qigi is a

standard representation of f −NF(f |G). In particular, f = NF(f |G) ink [x1, . . . , xn]/(G).

Note that for a polynomial p ∈ k [x1, . . . , xn], p =∑s

i=1 qigi is a standardrepresentation if no cancellation of leading terms occurs on the right hand side,that is, LM(p) ≥ LM(qigi) whenever qigi 6= 0, and LM(p) = LM(qigi) for at leastone i .

67. Remark. Put differently, the existence of a normal form is a division theoremwhere f is “divided” by an s-tuple G = (g1, . . . ,gs) ∈ G with main part

∑aigi and

remainder NF(f |G) such that LM(f ) ≥ LM(∑

aigi) and LM(f ) ≥ LM(NF(f |G))(exercise). Note, however, that the result may depend on the ordering of G!

68. Definition. We call NF a reduced normal form if for all f and G, NF(f |G) isreduced with respect to G.

The normal form has particularly nice properties if taken with respect to a Gröbnerbasis.

69. Lemma [GP, 1.6.7]. Let I ⊂ k [x1, . . . , xn] be a nontrivial ideal, G be a Gröbnerbasis of I, and NF be a normal form on k [x1, . . . , xn].

(i) For any f ∈ k [x1, . . . , xn] we have f ∈ I if and only if NF(f |G) = 0.

(ii) If NF is a further reduced normal form, then NF(f | G) = NG(f | G) for all f . Inparticular, since changing the order in G would produce a new normal form,NF(· | G) does not depend on the ordering of G.

Normal form 1 [GP, 1.6.13]> ring A=0,(x,y,z),dp;> poly f=x2yz+xy2z+y2z+z3+xy;> ideal I=xy+y2-1,xy;> I;I[1]=xy+y2-1I[2]=xy> reduce(f,I); //computes redNFB(f |(xy + y2− 1, xy))// ** I is no standard basis //Warning: result possibly depends onordering of the generatorsy2z+z3> I=xy,xy+y2-1; //Indeed...> I;I[1]=xyI[2]=xy+y2-1> reduce(f,I); //computes redNFB(f |(xy , xy + y2− 1))// ** I is no standard basisy2z+z3-y2+1

Normal form 2 [GP, 1.6.13]> ideal G=std(I); //computes a Gröbner basis of I> G;G[1]=xG[2]=y2-1> reduce(f,G);z3+z> G=y2-1,x;> G;G[1]=y2-1G[2]=x> reduce(f,G);// ** G is no standard basis //For SINGULAR an ideal comes with anordered set of generators, so G is not considered as a Gröbner basez3+z> reduce(f,G,1); //optional value “1” suppresses tail reduction, hencecomputes NFB(f |G)z3+xy+z

70. Definition. Let f , g ∈ k [x1, . . . , xn] with LM(f ) = xα, LM(g) = xβ andlcm(xα, xβ) = xγ . Then the s-polynomial of f and g is defined as

spoly(f ,g) := xγ−αf − LC(f )

LC(g)· xγ−βg

If LM(g)|LM(f ), we can write more simply spoly(f ,g) = f − LT(f )LT(g) · g.

71. Division algorithm NFB(f |G) [GP, Algorithm 1.6.10].

input : f ∈ k [x1, . . . , xn], G ∈ Goutput: NFB(f |G) ∈ k [x1, . . . , xn], a normal form of f with respect to G

1 h := f ;2 while (h 6= 0) and (Gh := {g ∈ G | LM(g)|LM(h)} 6= ∅) do3 choose any g ∈ Gh;4 h := spoly(h,g);5 end6 return h

(Note: B stands for Bruno Buchberger (austrian mathematician))

72. Remark. During the i-th while loop we compute an s-polynomial

hi = hi−1 −migi , LM(hi−1) > LM(hi),

where mi is a term with LT(migi) = LT(hi=1), gi ∈ G. In particular, setting h0 := f ,we have g1 = f −m1g1, h2 = h1 −m2g2 = f −m1g1 −m2g2 etc. This process endsafter a finite number m of steps, and we finally get

hm = f −m∑

i=1

migi ,

where hm = NF(f |G). Thus, keeping track of the coefficients mi , the Buchbergeralgorithm compute both the normal form and a standard representation off −NF(f |G). In particular, we can write f as a linear combination f =

∑migi if

f ∈ I = (G).

We can easily modify the division algorithm in order to get a reduced normal form:

73. Reduced normal form algorithm redNFB(f |G) [GP, Algorithm 1.6.11].

input : f ∈ k [x1, . . . , xn], G ∈ Goutput: redNFB(f |G) ∈ k [x1, . . . , xn], a reduced normal form of f with respect to G

1 g := 0;2 h := f ;3 while (h 6= 0) do4 h := NFB(h|G);5 if (h 6= 0) then6 g := g + LT(h);7 h := tail(h);8 end9 end

10 return g/LC(g)

74. Explicit computation of NFB and redNFB using the algorithm [GP, 1.6.12].Consider >dp on Mon(x , y , z), and f = x3 + y2 + 2z2 + x + y + 1, G = {x , y}.

(i) Computation of NFB(f |G) = 2z2 + x + y + 1. We initialise

h := f = x3 + y2 + 2z2 + x + y + 1.Then we have for the while-loops:

. LM(h) = x3, Gh = {x}h1 = spoly(f , x) = y2 + 2z2 + x + y + 1

. LM(h1) = y2, Gh1 = {y}h2 = spoly(h1, y) = 2z2 + x + y + 1

. LM(h2) = 2z2, Gh2 = ∅.(ii) Computation of redNFB(f |G) = z2 + 1/2. We initialise

g := 0, h := f = x3 + y2 + 2z2 + x + y + 1.Then we have for the while-loops:

. h1 = NFB(h|G) = 2z2 + x + y + 1g1 = LT(h1) = 2z2, h2 = tail(h1) = x + y + 1

. h3 = NFB(h2|G) = 1g2 = g1 + LT(h3) = 2z2 + 1, h4 = tail(h3) = 0.

75. The Buchberger algorithm STD(G) [GP, Algorithm 1.7.1]. Next we want tocompute Gröbner bases for a given ordering > on Mon(x1, . . . , xn) and normalform NF.

input : G ∈ Goutput: S = STD(G) a Gröbner basis of I = (G)

1 S := G;2 P := {(f ,g) | f , g ∈ S, f 6= g};3 while (P 6= ∅) do4 choose (f ,g) ∈ P;5 P := P \ {(f ,g)};6 h := NF(spoly(f ,g)|S);7 if (h 6= 0) then8 P := P ∪ {(h, f ) | f ∈ S};9 S := S ∪ {h};

10 end11 end12 return S

76. Theorem (Buchberger’s criterion) [GP, 1.7.3], [GP, 2.5.9], [CLS, 2.6.6]. LetI ⊂ k [x1, . . . , xn] be an ideal, and let G = {g1, . . . ,gs} ⊂ I. Are equivalent:

(i) G is a Gröbner basis of I.(ii) NF(f | G) = 0 for all f ∈ I.(iii) (G) = I and NF(spoly(gi ,gj) | G) = 0 for i, j = 1, . . . , s.

77. Remark. From a given Gröbner basis it is easy to construct a minimal andsubsequently a reduced one (this is actually what SINGULAR does via theSTD-command). Note that a reduced Gröbner basis of an ideal is unique (withrespect to the given monomial ordering) [GP, Exercise 1.6.1].

78. Example [GP, 1.7.4]. We now compute STD(G) explicitly using Buchberger’salgorithm with NFB. Consider >dp on Mon(x , y) and G = (x2 + y , xy + x). Weinitialise

S := S0 = (x2 + y , xy + x), P := P0 = {(x2 + y , xy + x)}.

Then we have for the while-loops:. P1 = ∅

h1 = NFB(spoly(x2 + y , xy + x) | S0) = NFB(−x2 + y2 | S0) = y2 + y 6= 0P1 = {(y2 + y , x2 + y), (y2 + y , xy + x)}, S1 = {x2 + y , xy + x , y2 + y}

. P2 = {(y2 + y , xy + x)}h2 = NFB(spoly(y2 + y , x2 + y) | S1) = NFB(−x2y + y3 | S1) = 0

. P2 = ∅h3 = NFB(−x2y + y3 | S1) = 0

The algorithm terminates and returns the Gröbner basisS = S1 = {x2 + y , xy + x , y2 + y} for I = (G).

Practical application 1: Ideal membership [GP, 1.8.1]Problem: Determine for a given ideal I whether f is in I or not.

input : I = (f1, . . . , fr ) ⊂ k [x1, . . . , xn], f ∈ k [x1, . . . , xn]output: true or false according to whether or not f ∈ I

1 G := STD(I);2 p := NF(f |G);3 if p = 0 then4 B := true;5 else6 B := false7 end8 return B

Ideal membership [GP, 1.8.1]> ring A=0,(x,y),dp;> ideal I=x10+x9y2,y8-x2y7;> ideal G=std(I);G[1]=x2y7-y8G[2]=x9y2+x10G[3]=x12y+xy11G[4]=x13-xy12G[5]=y14+xy12G[6]=xy13+y12> poly f=x2y7+y14;> reduce(f,G);-xy12+y8 // F is not in I> f=xy13+y12;> reduce(f,G);0 // F is in I

Testing ideal equality [GP, 1.8.1]> ring A=0,(x,y),dp;> ideal I=x10+x9y2,y8-x2y7;> ideal G=std(I);> ideal K=x2y7+y14,y14+xy12;> K;K[1]=y14+x2y7K[2]=y14+xy12> reduce(K,G); //reduces the generators of K with respect to G_[1]=-xy12+y8_[2]=0 //K not contained in I> poly f=(x10+x9y2)*(y8-x2y7);> K=f;> reduce(K,G);_[1]=0 //K contained in I

Practical application 2: Solving polynomial equations [GP, 1.8.7], [CLS,Kap. 3.1]Problem: f1, . . . , f` ∈ C[x1, . . . , xn] (or more generally, an algebraically closed fieldk , as for instance C), determine the set of points (a1, . . . ,an) ∈ Cn with

f1(a1, . . . ,an) = . . . = f`(a1, . . . ,an) = 0.

This depends on the ideal (f1, . . . , f`) generated by the fi and not on thegenerators! In general, if I = (f1, . . . , f`) ⊂ k [x1, . . . , xn] is an ideal, we set

V (I) := {(a1, . . . ,an) ∈ kn | f (a1, . . . ,an) = 0 for all f ∈ I} = V (f1, . . . , f`).

The problem is now to determine V (I).

79. Remark. As for linear equations it is in general impossible to write down thesolutions explicitly. The best we can do is to write down a basis of the solutionspace in the linear case, or a Gröbner base of I in the nonlinear case if we fixsome monomial ordering. However, V (I) will be a finite set of points if I is zerodimensional, see Remark 83 below. In this case we set on computing V (I)explicitly. The reason to work over C, or more generally, an algebraically closedfield, is that V (I) is not empty whenever 1 6∈ I, see Hilbert’s Nullstellensatz below.

Solving equations 1 [GP, 1.8.7]> ring A=0,(x,y,z),lp; //”solve”-command below uses Gröbner basis withrespect to LP-ordering> ideal I=x2+y+z-1,x+y2+z-1,x+y+z2-1;> ideal G=std(I);> G;G[1]=z6-4z4+4z3-z2G[2]=2yz2+z4-z2G[3]=y2-y-z2+zG[4]=x+y+z2-1> dim(G); //computes the dimension of the ideal (G) (see below); ifdim(G)=0, then the solution set is a finite collection of points0> LIB "solve.lib"; //library for solving polynomial equationsnumerically

Solving equations 2 [GP, 1.8.7]> def AC=solve(G,8,0,"nodisplay"); //solves the equations given by Gnumerically in the complex ring AC, ”0”: without multiplicities. SOL-VE returns a ring with complex coefficients, DEF defines objectswithout a specific type: they inherit their type from the firstassignment to them.// ’solve’ created a ring, in which a list SOL of numbers (the complexsolutions)// is stored.// To access the list of complex solutions, type (if the name R wasassigned// to the return value):setring R; SOL;> setring AC;> size(SOL); //gives the size of SOL, i.e., the number of elements inSOL, the solution set5

Solving equations 3 [GP, 1.8.7]> SOL; //displays the five solutions up to 8 digits[1]: [2]: [3]: [4]: [5]:[1]: [1]: [1]: [1]: [1]: //values of X

-2.4142136 0.41421356 0 1 0[2]: [2]: [2]: [2]: [2]: //values of Y

-2.4142136 0.41421356 0 0 1[3]: [3]: [3]: [3]: [3]: //values of Z

-2.4142136 0.41421356 1 0 0

An example with complex solutions> ring A=0,(x,y),lp;> ideal I=(xy-4,y2-x3+1);> ideal G=std(I);> dim(G);0> LIB "solve.lib";> def AC=solve(G,6,0,"nodisplay");// ’solve’ created a ring, in which a list SOL of numbers (the complexsolutions)// is stored.// To access the list of complex solutions, type (if the name R wasassigned// to the return value):setring R; SOL;> setring AC;> size(SOL);5

> SOL;[1]: [2]: [3]:[1]: [1]: [1]:1.80699 (-1.38823-i*1.08623) (-1.38823+i*1.08623)[2]: [2]: [2]:2.21363 (-1.78719+i*1.3984) (-1.78719-i*1.3984)[4]: [5]:[1]: [1]:(0.484732-i*1.61705) (0.484732+i*1.61705)[2]: [2]:(0.680372+i*2.26969) (0.680372-i*2.26969)

Theoretical application 2: Elimination and Extension

In the previous example, the generator G[1] = z6 − 4z4 + 4z3 − z2 was in I ∩ C[z]and thus an univariate polynomial. We formalise this idea for general (notnecessarily complete) fields as follows.

80. Definition. Given I = (f1, . . . , fs) ⊂ k [x1, . . . , xn], the `-th elimination ideal I`is the ideal of k [x`+1, . . . , xn] defined by

I` = I ∩ k [x`+1, . . . , xn].

In particular, I0 = I.

The `-th elimination ideal can be determined as follows.

81. The Elimination Theorem [CLS, 3.1.2]. Let I ⊂ k [x1, . . . , xn] be an ideal andlet G be a Gröbner basis of I with respect to >lp with x1 >lp x2 >lp . . . >lp xn. Then,for every 0 ≤ ` ≤ n − 1, the set

G` = G ∩ k [x`+1, . . . , xn]

is a Gröbner basis of the `-th elimination ideal I`.

82. Example. In the previous example, whereI = (x2 + y + z − 1, x + y2 + z − 1, x + y + z2 − 1) ⊂ C[x , y , z], we find

I0 = I ∩ C[x , y , z] = I

I1 = I ∩ C[y , z] = (G1) = (z6 − 4z4 + 4z3 − z2,2yz2 + z4 − z2, y2 − y − z2 + z)

I2 = I ∩ C[z] = (G2) = (z6 − 4z4 + 4z3 − z2).

83. Remark.(i) By [CLS, Corollary 9.5.4], we can define dim I, the dimension of the ideal I,

to be the largest integer r such that there exist variables xi1 , . . . , xir withI ∩ C[xi1 , . . . , xir ] = 0. Then the condition dim I = dim(G) = 0 garantuees thatIn−1 is not empty.

(ii) Instead of >lp (which is often inefficient) one can consider an eliminationordering on x1, . . . , xs. This is a monomial ordering > on k [x1, . . . , xn] suchthat LM(f ) ∈ k [xs+1, . . . , xn] implies f ∈ k [xs+1, . . . , xn] (>lp is actually anelimination ordering for any choice of variables). If G is a Gröbner basis for Iwith respect to >, then G′ = {g ∈ G | LM(g) ∈ k [xs+1, . . . , xn]} is a Gröbnerbasis for I ∩ k [xs+1, . . . , xn], see for instance [CLS, 1.8.3].

Assume now that we have a partial solution (a`+1, . . . ,an) ∈ V (I`). When does thisextend to a solution (a`, . . . ,an) ∈ V (I`−1) = V (G`−1)?

84. Extension Theorem [CLS, 3.1.3]. Let I = (f1, . . . , fs) ⊂ k [x1, . . . , xn] be anideal and let I1 be the first elimination ideal of I (with respect to >lp or anelimination order). For each 1 ≤ i ≤ s we write fi in the form

fi = ci(x2, . . . , xn)xNi1 + terms in which x1 has degree < Ni ,

with Ni ≥ 0 and ci ∈ k [x2, . . . , xn] nonzero. If we have a partial solution(a2, . . . ,an) ∈ V (I1) \ V (c1, . . . , cs), there exists a1 ∈ k with (a1,a2, . . . ,an) ∈ V (I).

85. Corollary [CLS, 3.1.4]. Let I = (f1, . . . , fs) ⊂ k [x1, . . . , xn] be an ideal andassume that for some i, fi is of the form

fi = cixNi1 + terms in which x1 has degree < Ni ,

with ci ∈ k nonzero and Ni > 0. If I1 is the first elimination ideal of I and(a2, . . . ,an) ∈ V (I1), there exists a1 ∈ k with (a1,a2, . . . ,an) ∈ V (I).

This is based on the following

86. Proposition [CLS, 3.5.2]. Let G = {g1, . . . ,gt} be a Gröbner basis ofI ⊂ k [x1, . . . , xn] for >lp with x1 > . . . > xn. For each 1 ≤ j ≤ t , let cj = cgj , so that

gj = cj(x2, . . . , xn)xNj1 + terms in which x1 has degree < Nj ,

where Nj ≥ 0 and cj ∈ k [x2, . . . , xn] is nonzero. Assume a = (a2, . . . ,an) ∈ V (I1) isa partial solution with the property that a 6∈ V (c1, . . . , ct ). Then

{f (x1,a) | f ∈ I} = (go(x1,a)) ⊂ k [x1],

where go ∈ G satisfies co(a) 6= 0 and go has minimal x1-degree among allelements gj ∈ G with cj(a) 6= 0 (the subscript letter “o” stands for “optimal”).Furthermore,

(i) deg(go(x1,a)) > 0.(ii) if go(a1,a) = 0 for a1 ∈ k, then (a1,a) ∈ V (I).

Thus, if a ∈ V (I1), the defining system g1 = . . . = gt = 0 reduces to go(x1,a) = 0when evaluated at a.

87. Example. Consider the ideal I = (x2y + xz + 1, xy − xz2 + z − 1) in Q[x , y , z].SINGULAR computes the Gröbner basis G of I with respect to >lp, x > y > z:

g1 = y2 − 2yz2 − yz + y + 2z4 − z3,

g2 = xz3 − xz2 − y + z2 + z − 1,

g3 = xy − xz2 + z − 1,

g4 = x2z2 + x + 1.

It follows that I1 = I ∩Q[y , z] = (g1) and

c1 = g1, c2 = z2(z − 1), c3 = y − z2, c4 = z2.

Since V (c1, . . . , c4) = {(0,0)}, we have for any partial solution (b, c) 6= (0,0) that{f (x ,b, c) | f ∈ f} = (go(x ,b, c)) ⊂ Q[x ]. For instance we have go = g4 for (1,1),and {f (x ,1,1)} = (x2 + x + 1). For (0, 1

2) we have g1(x ,0, 12) = 0,

g2(x ,0, 12) = −x/8− 1

4 , g3(x ,0, 12) = −x/4− 1

2 and g4(x ,0, 12) = x2/4 + x + 1. For

go we can take either g2 or g3. In particular, we can ignore the quadratic equationg4(x ,0, 1

2) = 0 when extending the partial solution (0, 12).

Next we will generalise Gröbner bases to modules. For this, we let from now onk [x] be shorthand for k [x1, . . . , xn]. We consider the free module

k [x]r =r⊕

i=1

k [x]ei ,

where ei = (0, . . . ,1, . . . ,0) denotes the i-th canonical basis vector of k [x]r .Monomials are now given by

xαei = (0, . . . ,0, xα,0, . . . ,0) ∈ k [x]r , α ∈ Nn \ {0}, i = 1, . . . ,n.

The support of f =∑

i,α∈Λiai,αxαei is supp f = {xαei | ai,α 6= 0}.

88. Definition. Let > be a monomial ordering on k [x]. A module monomialordering on k [x]r or module ordering for short is a total ordering >m on themonomials {xαei | α ∈ Nn \ {0}, i = 1, . . . , r} which satisfies

(i) xαei >m xβej =⇒ xα+γei >m xβ+γej

(ii) xα > xβ ⇒ xαei >m xβei

for all α, β and γ ∈ Nr , i , j = 1, . . . , r . In the sequel we often denote a moduleordering >m simply by > if there is no risk of confusion.

89. Remark. The second condition of Definition 88 implies that the moduleordering is well-ordered and thus global if and only if the ordering on k [x] iswell-ordered and thus global.

90. Example. If > is an ordering on k [x], then one can define a module ordering>m by giving either priority to the components, that is,

xαei >m xβej if and only if i < j or (i = j and xα > xβ).

This will be suggestively denoted by >m= (c, >). Or – this is SINGULAR’S default –we give priority to the monomials in k [x], that is,

xαei >m xβej if and only if xα > xβ or (xα = xβ and i < j).

This will be therefore denoted by >m= (>, c). In both cases, the restriction to eachcomponent >m |k [x]ei×k [x]ei

is just >.

91. Some definitions for module orderings. All the notions from the ring casecarry over to the module case. We have, for instance, for any vector f ∈ k [x]r \ {0}the notion of leading monomial, coefficient and term. Note that LM(f ) andLT(f ) are elements of k [x]r , while LC(f ) ∈ k . For a subset G of k [x]r we call

LM(G) := (LM(g) | g ∈ G \ {0}) ⊂ k [x]r

the leading submodule of G. Since we can identify the monomials of k [x]r withNn × E r ⊂ Nn × Nr = Nn+r , where E r = {e1, . . . ,er}, the natural partial orderingon Nn+r induces a natural partial ordering on the monomials of k [x]r , namely

xαei ≤nat xβej if and only if i = j and xα|xβ.

Equivalently, we say that xαei divides xβej and write xαei |xβej . Then for any set ofmonomials G ⊂ k [x]r and any monomial xαei , we have

xαei ∈ (G) ⇔ xαei is divisible by some element of G.

In particular, Dickson’s Lemma for Nn+r is equivalent to the statement that anymonomial submodule of k [x]r is finitely generated – in strict analogy to the case ofmonomial ideals.

92. Definition.(i) Let M ⊂ k [x]r be a submodule. A Gröbner basis of M is a finite subset G of

M with LM(M) = LM(G), that is, for any f ∈ M \ {0} there exists a g ∈ Gsuch that LM(g)|LM(f ). As for rings, a Gröbner basis generates the module.

(ii) G ⊂ k [x]r is called interreduced if 0 6∈ G and if LM(g) 6 | LM(f ) for any twodistinct elements f , g ∈ G. An interreduced Gröbner basis is also calledminimal.

(iii) f ∈ k [x]r is reduced with respect to G ⊂ k [x]r if no monomial of supp f iscontained in LM(G).

(iv) G ⊂ k [x]r is reduced if G is interreduced and if for each g ∈ G, LC(g) = 1and tail(g) is reduced with respect to G.

93. Definition. Let G = {G = (g1, . . . ,gs) | gi ∈ k [x]r , s ∈ N} be the set of finites-tuples of elements in k [x]r . A function

NF : k [x]r × G → k [x1, . . . , xn], (f ,G) 7→ NF(f |G)

is called a normal form on k [x]r if for all G ∈ G and f ∈ k [x]r the followingproperties hold:

. NF(0|G) = 0

. NF(f |G) 6= 0⇒ LM(NF(f |G)) 6∈ LM(G)

. If G = (g1, . . . ,gs), then there exists a standard representation

f −NF(f |G) =s∑

i=1

aigi , ai ∈ k [x], s ≥ 0.

In particular, f = NF(f |G) and LM(NF(f |G)) 6= 0 in k [x]r/(G).

Lemma 69 holds mutatis mutandis also for modules.

94. Definition. Let f , g ∈ k [x]r \ {0} with LM(f ) = xαei , LM(g) = xβej andlcm(xα, xβ) = xγ . Then we define the s-polynomial of f and g by

spoly(f ,g) :=

{xγ−αf − LC(f )

LC(g) · xγ−βg, i = j

0, i 6= j .

Then the Algorithms 71 (NFB) and 73 (redNFB) carry directly over to the modulecase.

95. Theorem [GP, 2.3.13]. Let M ⊂ k [x]r and G = {g1, . . . ,gs} ⊂ M. Areequivalent:

(i) G is a Gröbner basis of M.(ii) NF(f |G) = 0 for all f ∈ M.(iii) M = (G) and NF(spoly(gi ,gj)|G) = 0 for i, j = 1, . . . , s.

Computing the Gröbner basis of modules 1 [GP, 2.3.12]> ring A=0,(x,y,z),(c,dp); //monomial ordering DP on the modulemonomials in Q[x ,y ,z] with priority given to components (priority tomonomials is SINGULAR’s default)> module M=[x+1,y,1],[xy,z,z2]; //defines the submodule M of Q[x , y , z]3

> std(M);_[1]=[0,xy2-xz-z,-xz2+xy-z2]_[2]=[y,y2-z,-z2+y]_[3]=[x+1,y,1]> ring B=0,(x,y,z),dp; //switches to SINGULAR’s default priority(DP,C)> module M=fetch(A,M); //Considers M now as a submodule of Q[x ,y ,z] withrespect to (DP,C)> std(M);_[1]=x*gen(1)+y*gen(2)+gen(3)+gen(1)_[2]=y2*gen(2)-z2*gen(3)+y*gen(3)+y*gen(1)-z*gen(2)

Computing the Gröbner basis of modules 2 [GP, 2.3.12]> ring C=0,(x,y,z),lp;> module M=fetch(A,M);> std(M);_[1]=y2*gen(2)+y*gen(3)+y*gen(1)-z2*gen(3)-z*gen(2)_[2]=x*gen(1)+y*gen(2)+gen(3)+gen(1)> ring D=0,(x,y,z),(c,lp);> module M=fetch(A,M);> std(M);_[1]=[0,xy2-xz-z,xy-xz2-z2]_[2]=[y,y2-z,y-z2]_[3]=[x+1,y,1]

Normal form for modules [GP, 2.3.10]> ring A=0,(x,y,z),(c,dp);> module M=[x,y,1],[xy,z,z2];> vector f=[zx,y2+yz-z,y];> reduce(f,M);// ** M is no standard basis //warning that the reduction depends onthe given generators of M as we do not have a Gröbner basis[0,y2-z,y-z]> reduce(f,std(M));[0,0,z2-z]

Practical application 3: Computing the intersection with free modules [GP,2.8.2]Problem: Given f1, . . . , f` ∈ k [x]r which define M := (f1, . . . , f`) ⊂ k [x]r . Findgenerators of the submodule

M ′ := M ∩r⊕

i=s+1

k [x]ei .

We say that M ′ is obtained from f1, . . . , f` by eliminating e1, . . . ,es.

Solution: Compute a Gröbner basis G of M with respect to >m= (c, >). Then

G′ := G ∩r⊕

i=s+1

k [x]ei

is a Gröbner basis for M ′.

Theoretical application 3: Computing syzygies96. Definition. Let M be a k [x]-module. A syzygy or relation betweenf1, . . . , fk ∈ M is a k -tuple (q1, . . . ,qk ) ∈ k [x]k satisfying

k∑i=1

qi fi = 0.

The set of all syzygies between f1, . . . , fk is the kernel of the module morphism

ϕ : F1 :=k⊕

i=1

k [x]ei → M, ei 7→ fi

and thus a submodule of k [x]k . It is called the module of syzygies and writtensyz(f1, . . . , fk ). We therefore have

syz(f1, . . . , fk ) ⊂ k [x]kϕ� (f1, . . . , fk ).

Since k [x] is Noetherian and k [x]r is finitely generated, k [x]r is Noetherian, thussyz(f1, . . . , fk ) ⊂ k [x]k is also finitely generated.

97. Lemma [GP, 2.5.3]. Let (f1, . . . , fk ) ⊂ k [x]r be a M = k [x]-module. Considerthe canonical embedding k [x]r ⊂ k [x]r+k with projection π : k [x]r+k → k [x]k ontothe last k components. Let G = {g1, . . . ,gs} be a Gröbner basis ofF = (f1 + er+1, . . . , fk + er+k ) with respect to a module ordering (c, <) given byxαei > xβej if i < j or i = j and xα > xβ. If, possibly under relabeling,{g1, . . . ,g`} = G ∩

⊕r+ki=r+1 k [x]ei in k [x]r+k , then

syz(f1, . . . , fk ) = (π(g1), . . . , π(g`)).

98. Computing generators for syz(f1, . . . , fk ) [GP, Algorithm 2.5.4].input : f1, . . . , fk ∈ k [x]r

output: S = {s1, . . . , s`} ⊂ k [x]k such that (S) = syz(f1, . . . , fk ) ⊂ k [x]k

1 F := {f1 + er+1, . . . , fk + er+k} ⊂ k [x]r+k ;2 compute G = STD(F ) with respect to (c, >);3 {g1, . . . ,g`} = G ∩

⊕r+ki=r+1 k [x]ei with gi =

∑kj=1 aijer+j , i = 1, . . . , `;

4 si := (ai1, . . . ,aik ) = π(gi), i = 1, . . . , `;5 return S = {s1, . . . , s`}

Syzygies [GP, 2.5.5]> ring A=0,(x,y,z),(c,dp);> ideal M=xy,yz,xz;> module S=syz(M); //computes the module of syzygies> S; //syzygy module in Q[x ,y ,z]3, k=3, given in terms of a Gröbner basisS[1]=[0,x,-y]S[2]=[z,0,-y]> module F=[xy,1,0,0],[yz,0,1,0],[xz,0,0,1]; //computes the syzygy viathe algorithm; F⊂Q[x ,y ,z]1+3

> module G=std(F);> G; //the first two elements give the syzygiesG[1]=[0,0,x,-y] //s1=π(G[1])=[0,x ,−y ]

G[2]=[0,z,0,-y] //s2=π(G[2])=[z,0,−y ]

G[3]=[yz,0,1] // G[3]=yze1+1e3 etc, but G[3] 6∈G∩⊕4

i=2 k [x]ei

G[4]=[xz,0,0,1] //G[4] 6∈G∩⊕4

i=2 k [x]ei

G[5]=[xy,1] //G[5] 6∈G∩⊕4

i=2 k [x]ei

99. Remark [GP, 2.5.6]. Let I = (f1, . . . , fs) ⊂ k [x] be an ideal and letΣ = {m1, . . . ,mk}, mi ∈ (k [x]/I)r ∼= k [x]r/I · k [x]r . We want to compute syzk [x]/I(Σ)

where we consider (Σ) and the resulting syzygy module as k [x]/I-module. We cando this as follows. Let

Σ = {m1, . . . ,mk , f1e1, . . . , f1er , . . . , fse1, . . . , fser} ⊂ k [x]r ,

where mi ∈ k [x]r are representatives of mi . Note that(f1e1, . . . , f1er , . . . , fse1, . . . , fser ) = I · k [x]r . We then compute a Gröbner basis{s1, . . . , s`} of syz(Σ) with si = (si1, . . . , siN), N = k + rs. Ifsi = (si1, . . . , sik ) ∈ k [x]k , i = 1, . . . , `, then

(s1, . . . , sl) ⊂ k [x]kπI·k [x]r

� (Σ) ⊂ k [x]r/I · k [x]r

that is, syz(Σ) = (s1, . . . , sl). Projecting this to k [x]k/(I · k [x]k

)gives the desired

syzygy module syzk [x]/I(Σ).

Practical application 4: Module membership [GP, 2.8.1]

Problem: Given f , f1, . . . , fk ∈ k [x]r , decide whether or notf ∈ M := (f1, . . . , fk ) ⊂ k [x]r . If it is, find qi ∈ k [x] such that f =

∑qi fi .

Solution: Compute a Gröbner basis G = {g1, . . . ,gs} of M with respect to >m andchoose any normal form NF on k [x]r . Then

f ∈ M ⇐⇒ NF(f |G) = 0

since the proof of Lemma 69 applies here as well. Next choose a Gröbner basis{h1, . . . ,h`} of syz(f , f1, . . . , fk ) ⊂ k [x]k+1 with respect to the ordering (c, >).Relative to the standard basis (e0, . . . ,ek ) of k [x]k+1 we write hi = (ai0, . . . ,aik ).Since hi ∈ syz(f , f1, . . . , fk ) we have ai0f +

∑kj=1 aij fj = 0 for all j . On the other

hand, f =∑

bi fi for f ∈ M so that h = (1,−b1, . . . ,−bk ) ∈ syz(f , f1, . . . , fk ) = (G).Since LM(hi)|LM(h) = e0 for some i it follows that LM(hi) = ce0 with c ∈ k∗,whence f = −

∑j≥1 aij fj/c.

Module membership [GP, 2.8.1]> ring A=0,(x,y,z),(c,dp);> module M=[-z,-y,x+y,x],[yz+z2,yz+z2,-xy-y2-xz-z2,0];> vector f=[-xz-z2,-xz+z2,x2+xy-yz+z2,0];> reduce(f,std(M));[0,xy-xz+yz+z2,-xz-2yz+z2,-x2-xz] //V is not in M> vector w=[-x5yz-x5z2-z,-x5yz-x5z2-y,x6y+x5y2+x6z+x5z2+x+y,x];> reduce(w,std(M));0> syz(w+M); //express W as a linear combination of m1 and m2 viacomputing the syzygy module of the (w ,m1,m2)

_[1]=[1,-1,x5] //Gröbner basis of syz(w ,m1,m2)⊂Q[x ,y ,z]3 is {e1−e2+x5e3}> lift(M,w); //the command LIFT(M,N) expresses the generators of thesubmodule N⊂M in terms of the generators of M. Here, N=(w1) withw1=w._[1,1]=1_[2,1]=-x5 //w1=a11m1+a21m2

Practical application 5: Kernel of a module homomorphism [GP, Section2.8.7]Let A = k [x]/I for some ideal I ⊂ k [x], and let M ⊂ Am and N = (n1, . . . ,ns) ⊂ An

be submodules. We consider the A-module morphism ϕ : Am/M → An/N inducedby ϕ : Am → An via πN ◦ ϕ = ϕ ◦ πM . If ϕ(ei) = bi ∈ An, then ϕ is given by thematrix B = (b1, . . . ,bm) considered as an A-linear operator Am → An.

Problem: Compute representatives in Am of a system of generators of kerϕ.

Solution: ϕ(a) = 0 if and only if ϕ(a) ∈ N so we are looking for elements(a1, . . . ,an,a′1, . . . ,a

′s) with

∑aibi =

∑a′ini . Therefore, we compute a system of

generators {h1, . . . ,h`} of syz(b1, . . . ,bm,n1, . . . ,ns) ⊂ Am+s. Then hi ∈ Am whichis obtained by deleting the last s components of hi is the desired system, and itsprojection down to Am/M generates kerϕ.

Kernel of a module morphism 1 [GP, 2.8.9]> ring K=0,(x,y,z),(c,dp);> ideal I=x2y2-xyz2;> qring A=std(I); //quotient ring A=Q[x ,y ,z]/I= current basering> module N=[x2,xy,y2],[xy,xz,yz];> matrix B[3][2]=x,y,zx,zy,y2,z2; //B is a 3x2-matrix, B:A2→A3

> module T=B[1],B[2],N[1],N[2]; //submodule T of An+s=A2+2

> module S=syz(T); //syzygy module in A4

> module Ker; //defines variable of type ”module”> int i;> for(i=1;i<=size(S);i++){Ker=Ker+S[i][1..2];} //keeps only the firsttwo components

Kernel of a module morphism 2 [GP, 2.8.9]>Ker;Ker[1]=[xy2-yz2]Ker[2]=[y3z-x2z2-xyz2+y2z2-xz3+yz3,xy3-x2yz]Ker[3]=[x2yz-xz3]Ker[4]=[x3z+x2z2-xyz2-y2z2+xz3-yz3,x3y-xy2z]Ker[5]=[x3y-x2z2]> reduce(B*Ker,std(N)); //We test by reducing the image of B(Ker)⊂Am byN

_[1]=0_[2]=0_[3]=0_[4]=0_[5]=0

100. Solving linear equations over a ring [GP, 2.8.8]. The solution ofcomputing the kernel of module morphisms also enables us to compute solutionsto linear equations in general. Let I = (f1, . . . , fk ) ⊂ k [x] be an ideal, andA = k [x]/I. We consider the system of linear equations given by

a11Z1 + . . . + a1mZm = b1...

......

ar1Z1 + . . . + armZm = br

for aij , bi ∈ k [x]. Writing T = (aij), Z = (Zi)> and b = (bi)

> we wish to solve thematrix equation TZ = b. Unlike the linear systems considered in linear algebra,the entries of the matrix T now belong to a ring and no longer to a field. However,considering T as a map Am → Ar , the solution set is of the form z + kerT as forvector spaces. To get a special solution z we first test whether b is a member ofthe submodule Im (T ) ⊂ Ar . If it is, we compute a representation b =

∑riai with

ri ∈ A and ai = the columns of T . This was done in the Practical application 4. Thecomputation of the kernel is Practical application 5.

101. Definition. A sequence of A-modules with A-linear maps

L α // Mβ // N

is called a complex if imα ⊂ kerβ, i.e., β ◦ α = 0, and exact at M if imα = kerβ.A sequence

. . .Mk−1αk−1 // Mk

αk // Mk+1 . . .

is a complex respectively is exact if it is a complex respectively exact at every Mk ,i.e., αk ◦ αk−1 = 0 respectively imαk−1 = kerαk . A sequence of the form

0 // L α // Mβ // N // 0

is called a short exact sequence of A-modules.

102. Remark.

(i) 0 // Mϕ // N is exact⇔ kerϕ = {0} ⇔ ϕ is injective.

(ii) Mϕ // N // 0 is exact⇔ imϕ = N ⇔ ϕ is surjective.

(iii) 0 // Mϕ // N // 0 is exact at M and N if and only if ϕ is an

isomorphism.

103. Examples. If ϕ : M → N is an A-module morphism, then the naturalinclusions kerϕ→ M, imϕ→ N induce short exact sequences

0 // kerϕ ι // Mϕ // imϕ // 0

and0 // imϕ

ι // N // cokerϕ // 0.

Here are two special instances of this:. If N1,2 are submodules of M, then the inclusion map N1 ∩ N2 → N1 ⊕ N2,

n 7→ n ⊕ n, and the difference map N1 ⊕ N2 → N1 + N2, (n1,n2) 7→ n1 − n2give the short exact sequence

0 // N1 ∩ N2 // N1 ⊕ N2 // N1 + N2 // 0.

. If I ⊂ A is an ideal, and a ∈ A, then A/I maps onto A/(I + (a)). The kernel isgenerated by a ∈ A/I. It follows that

0 // A/(I : (a)) // A/I // A/(I + (a)) // 0

is exact. Here, the map A/(I : (a))→ A/I assigns to πI:(a)(b) the elementπI(a · b).

104. Remark.(i) Splitting of exact sequences: An exact sequence

M1α1 // M2

α2 // M3α3 // M4 gives rise to

M1α1 // M2 // imα2 = kerα3 // 0

and0 // kerα3 = imα2 // M3

α3 // M4 .

(ii) Glueing of exact sequences: The exact sequences

M1α1 // M2

α2 // imα2 = kerα3 // 0 and

0 // kerα3 = imα2 // M3α3 // M4 give rise to the exact sequence

M1α1 // M2

α2 // M3α3 // M4.

105. Example. If ϕ : M → N is an A-module morphism, then the exact sequence

0 // kerϕ // Mϕ // N // cokerϕ // 0

can be split into

0 // kerϕ // Mϕ // imϕ // 0

and0 // imϕ // N // cokerϕ // 0.

In particular, any exact sequence

0 // M1α1 // M2

α2 // . . .αn−1 // Mn

αn // 0

can be split up into short exact sequences

0 // kerαi = imαi−1 // Miαi // imαi = kerαi+1 // 0

which can be glued back to the original exact sequence.

106. Definition. A finitely generated A-module M is said to have a finitepresentation if there exists an A-linear map ϕ : Am → An such that M isisomorphic to cokerϕ, that is, we have an exact sequence

Am ϕ // An πimϕ // An/imϕ = cokerϕ ∼= M // 0.

ϕ is called the presentation morphism of M.

Presentation of submodules 1 [GP, 2.1.24]> ring A=0,(x,y,z),dp;> module N=[xy,0,yz],[0,xz,z2]; //defines the submodule N of A3=Q[x ,y ,z]3

> N;//2 generatorsN[1]=xy*gen(1)+yz*gen(3)N[2]=xz*gen(2)+z2*gen(3)> LIB "inout.lib"; //library for formatting output gives command SHOW

> show(N); //shows the generators as vectors// module, 2 generator(s)[xy,0,yz][0,xz,z2]> print(N); //the matrix corresponding to A2→N⊂A3, ei 7→N[i]

xy,0,0,xz,yz,z2

Presentation of submodules 2 [GP, 2.1.24]> show(N+x*N); //shows the generators of N+x∗N[xy,0,yz][0,xz,z2][x2y,0,xyz][0,x2z,xz2]> print(N+x*N); //the matrix corresponding to A4→N+x∗N⊂A3, ei 7→N[i]

xy,0, x2y,0,0, xz,0, x2z,yz,z2,xyz,xz2> module K1=syz(N);> K1;K1[1]=0 //computes the kernel of A2→N=(N[1],N[2]).> print(K1) //gives the presentation matrix of N which is (0), i.e.,N∼=A2

0,0

Presentation of submodules 3 [GP, 2.1.24]> module M=[xy,yz],[xz,z2]; //defines the submodule M⊂A2

> print(M);xy,xz,yz,z2> matrix p=M; //automatic type conversion module with correspondingmatrix A2 p−→M→0

> p;p[1,1]=xyp[1,2]=xzp[2,1]=yzp[2,2]=z2> module K2=syz(M); //computes the kernel of A2→M

> show(K2); //gives the generator of K 2

K2[1]=[-z,y]> print(K2); //presentation morphism A1

ϕ =

(−zy

)// K 2⊂A2

p=πK 2// M∼=cokerϕ // 0-z,y

107. Definition.(i) Let M be a finitely generated A-module. A free resolution of M is an exact

sequence

. . . // Fk+1ϕk+1 // Fk // . . . // F1

ϕ1 // F0ϕ0 // M // 0

with finitely generated, free A-modules Fk∼= Ark , k ≥ 0. This free resolution

has length n ∈ N if Fk = 0 for all k > n and n is minimal with this property.(ii) If (A,m) is a local ring, then a free resolution is minimal if ϕk (Fk ) ⊂ mFk−1 for

k ≥ 1. In this case we call bk (M) := rank(Fk ) = rk , k ≥ 0, the k-th Bettinumber of M.

108. Computing free resolutions RES(I,m) [GP, 2.5.7]. Since the sygyzymodules are finitely generated, we obtain a free resolution of given length m bysuccessively computing syzygies.

input : f1, . . . , fk ∈ k [x]r , 0 6= m ∈ Noutput: A list of matrices A1, . . . ,Am with Ai ∈ Mat(ri−1 × ri , k [x]), i = 1, . . . ,m,

r0 = r and r1 = k , with free resolution of M = k [x]r/(f1, . . . , fk ):

k [x]rmAm // k [x]rm−1 // . . . // k [x]r1

A1 // k [x]rπ(f1,...,fk )//

M // 01 i := 1;2 A1 := matrix(f1, . . . , fk ) ∈ Mat(r × k , k [x]);3 while (i < m) do4 i := i + 1;5 Ai := syz(Ai−1);6 end7 return A1, . . . ,Am

109. Theorem [GP, 2.4.11]. Let (A,m) be a local Noetherian ring and M be afinitely generated A-module. Then M has a minimal free resolution. Furthermore,the Betti numbers and the length (if defined) are independent of the minimalresolution and are thus invariants of M.

110. Remark. The proof also shows that the length of a general free resolution isalways equal or greater than the length of a minimal resolution.

111. Example. If (A,m) = (K ,0) is a field and M thus a K -vector space, then aminimal resolution of M has length 0 and b0(M) = dimK M.

112. Definition. Let f1, . . . , fk ∈ k [x]r =: F0 with module ordering >0, and letF1∼= k [x]k be the free k [x]-module containing syz(f1, . . . , fk ). The Schreyer

ordering is the module ordering >1 on F1 which is defined as follows:

xαεi >1 xβεj ⇐⇒ LM(xαfi) >0 LM(xβfj) or

LM(xαfi) = LM(xβfj) and i < j ,

where εi , i = 1, . . . , k , is the canonical basis of F1. Of course, >1 depends onf1, . . . , fk .

113. Notation. Let f1, . . . , fk ∈ k [x]r . For each i 6= j such that fi and fj have theirleading monomials in the same component, i.e., LM(fi) = xαi eν , LM(fj) = xαj eνand γ = lcm(αi , αj) we define

mji := xγ−αi ∈ k [x].

In particular, spoly(fi , fj) = mji fi − cimij fj/cj if ci,j = LC(fi,j). Finally, assume thatspoly(fi , fj) =

∑kν=1 a(ij)

ν fν , a(ij)ν ∈ k [x] is a standard representation (for instance,

NF(spoly(fi , fj)|(f1, . . . , fk )) = 0 for some normal form). Then for i < j we set

sij := mjiεi −ci

cjmijεj −

∑ν

a(ij)ν εν .

114. Lemma [GP, 2.5.8]. sij ∈ syz(f1, . . . , fk) and LM(sij) = mjiεi .

115. Theorem [GP, 2.5.9]. Let G = {f1, . . . , fk} be a set of generators ofM ⊂ k [x]r , and let NF be a given normal form. Let

Λ := {(i , j) | 1 ≤ i < j ≤ k , LM(fi) = xαeν , LM(fj) = xβeν for some ν}.

Assume that NF(spoly(fi , fj) | (f1, . . . , fk )) = 0 for all i , j = 1, . . . k. Then thefollowing statements hold:

(i) G is a Gröbner basis of M (Buchberger’s criterion).(ii) S := {sij | (i , j) ∈ Λ} is a Gröbner basis of syz(f1, . . . , fk ) with respect to the

Schreyer ordering.

Theoretical application 4: Hilbert’s syzygy theorem116. Lemma [GP, 2.5.13]. Let G = {g1, . . . ,gs} be a minimal Gröbner base of thesubmodule M of k [x]r such that LM(gi) ∈ {e1, . . . ,er} for all i . Let Λ denote the setof indices ` such that e` 6∈ {LM(g1), . . .LM(gs)}. Then

M =s⊕

i=1

k [x]gi , k [x]r/M ∼=⊕`∈Λ

k [x]e`.

117. Lemma [GP, 2.5.14]. Let G = {g1, . . . ,gs} be a Gröbner basis of M ⊂ k [x]r

such that (g1, . . . ,gs) is ordered in the following way: if i < j and LM(gi) = xαi eν ,LM(gj) = xαj eν , then αi ≥ αj lexicographically. Let sij denote the syzygies definedabove. Suppose that LM(g1), . . . ,LM(gs) do not depend on the variablesx1, . . . , xk . Then the LM(sij) taken with respect to the Schreyer ordering do notdepend on x1, . . . , xk+1.

118. Theorem (Hilbert’s Syzygy Theorem) [GP, 2.5.15]. Let > be any monomialordering on k [x]. Then any finitely generated k [x]-module M has a free resolution

0 // Fm // Fm−1 // . . . // F0 // M // 0

of length m ≤ n, where the Fi are free k [x]-modules.

Next we want to investigate Hilbert functions and polynomials.

119. Definition. A graded ring A is a ring with a decomposition A =⊕

ν∈N Aν asa direct sum of abelian groups such that AνAµ ⊂ Aν+µ for all ν, µ ∈ N. The Aν arecalled the homogeneous components of degree ν of A, and elements of Aν arecalled homogeneous. A graded k -algebra is a k -algebra which is a graded ringfor which Aν is a k -vector space for all ν ∈ N and A0 = k .

120. Examples.(i) Every ring A carries a trivial structure of a graded ring by setting A0 = A and

Aν = 0 for ν ≥ 1.(ii) If I ⊂ A is an ideal, then

GrI(A) =⊕ν≥0

GrI(A)ν :=⊕ν≥0

Iν/Iν+1

is a graded ring (where by convention, I0 = (1) = A). Multiplication is given asfollows: If a ∈ Iν/Iν+1 and b ∈ Iµ/Iµ+1, then a · b := a · b ∈ Iν+µ/Iνµ+1 (this isindeed well-defined).

(iii) Let A = k [x1, . . . , xn] and w = (w1, . . . ,wn) be an n-tuple of positive integers.We let wdeg(xα) =

∑wiαi which is additive in the αi . Then Aν = the k -vector

generated by monomials of wdeg = ν is a graded k -algebra. The elements ofAν are called quasihomogeneous or weighted homogeneous of degree ν.

121. Definition. Let A =⊕

ν∈N Aν be a graded ring. An A-module M is a gradedA-module if there is a direct sum decomposition M =

⊕µ∈Z Mµ into abelian

groups with AνMµ ⊂ Mµ+ν .

122. Examples.(i) Let A =

⊕ν∈N Aν be a graded k -algebra, and consider the free A-module

Am =⊕m

i=1 Aei . Let deg(ei) := νi ∈ Z, and Mν be the k -vector spacegenerated by fei with f ∈ Aν−νi . Then Am =

⊕ν∈Z Mν becomes a graded

A-module.(ii) If M =

⊕ν∈Z Mν is a graded A-module, we define for d ∈ Z the d-th shift of

M by M(d) :=⊕

ν∈Z M(d)ν , where M(d)ν := Mν+d . Then M(d) is a gradedA-module. In particular, A(d) is a graded A-module.

123. Lemma [Exercise]. Let M =⊕

ν∈Z Mν be a graded A-module and N ⊂ M bea submodule. Are equivalent:

(i) N is graded by the induced grading, i.e., N =⊕

ν∈Z Nν with Nν = Mν ∩ N.(ii) N is generated by homogeneous elements.(iii) Let m =

∑mν , mν ∈ M. Then m ∈ N if and only if mν ∈ N for all ν.

A submodule satisfying one und thus all of these conditions is called a graded orhomogeneous submodule if M. If M = A, where A is a graded ring, we call N agraded or homogeneous ideal.

124. Remark. Let M =⊕

ν∈Z Mν be a graded module and N ⊂ M be a gradedsubmodule. Then the quotient M/N is a graded module with(M/N)ν = Mν/Nν

∼= (Mν + N)/N. If M = A is a graded ring and N = I is ahomogeneous ideal, then A/I is a graded ring.


ν≥0 Aν be a graded ring, and M =⊕

ν∈Z Mν andN =

⊕ν∈Z Nν be two graded A-modules. An A-linear map ϕ : M → N is graded or

homogeneous of degree d if ϕ(Mν) ⊂ Nν+d . We call ϕ simply homogeneous if itis homogeneous of degree 0.

126. Examples. Under the assumptions of the previous definition, multiplicationwith f ∈ Ad defines a graded morphism ϕ = f · : M → M of degree d .

127. Remark. If ϕ : M → N is a homogeneous morphism, then kerϕ, imϕ andcokerϕ are graded A-submodules.

Graded rings and modules 1 [GP, 2.2.15]> ring A=0,(x,y,z),dp;> ideal I=y3-z2,x3-z;> homog(I); //returns 1 or 0 depending on whether or not I ishomogeneous0> qhweight(I); //gives the weights of the variables x ,y ,z for which I

becomes quasihomogeneous (if they exist)1,2,3> ring B=0,(x,y,z),wp(1,2,3); //Let ν1,...,νn be positive integers.Then wp(ν1,...,νn) is weighted reverse lexicographical ordering withdeg(xα)=ν1α1+···+νnαn.> ideal I=fetch(A,I); //redefines I as an ideal of B where thevariables x , y , z now have the weights 1, 2, 3 respectively> homog(I);1

Graded rings and modules 2 [GP, 2.2.15]> module M=[y3-z2,x3-z],[x3,1];> homog(M);1> attrib(M,"isHomog");0,3 //M is a homogeneous submodule of Q[x ,y ,z]2=Be1⊕Be2 if the grading isinduced by by the grading on B2 given by deg e1=0, deg e2=3

//For more on the handling of graded modules see SINGULAR’S library”gradedModules_lib”

128. Lemma [GP, 2.2.14]. Let A =⊕

ν≥0 Aν be a Noetherian graded k-algebraand let M =

⊕ν∈Z Mν be a finitely generated, graded A-module. Then

(i) there exists m ∈ Z such that Mν = (0) for ν < m;(ii) dimk Mν <∞ for all ν.


ν≥0 Aν be a Noetherian graded k -algebra, and letM =

⊕ν∈Z Mν be a finitely generated, graded A-module. The Hilbert function

HM : Z→ Z is defined byHM(ν) := dimk (Mν).

The Hilbert-Poincaré series on M is defined by

HPM(t) :=∑ν∈Z

HM(ν) · tν ∈ ZJtK[t−1].

Both the Hilbert function and the Hilbert-Poincaré series depend only on thegraded structure of M. In particular, we are free to consider M as anA/Ann(M)-module and we shall usually do so when carrying out concretecomputations.

130. Lemma [GP, 5.1.2]. Let A =⊕

ν≥0 Aν be a Noetherian graded k-algebra,and let M =

⊕ν∈Z Mν be a finitely generated graded A-module.

(i) Let N ⊂ M be a graded submodule, then

HM(ν) = HN(ν) + HM/N(ν)

for all ν. In particular, HPM(t) = HPN(t) + HPM/N(t).(ii) Let d be an integer, then

HM(d)(ν) = HM(ν + d)

for all ν. In particular, HPM(d)(t) = t−dHPM(t).(iii) Let d be a non-negative integer, let f ∈ Ad , and let ϕ : M(−d)→ M be

defined by ϕ(m) := f ·m. Then kerϕ and cokerϕ are graded A/(f )-moduleswith respect to the induced gradings and

HM(ν)−HM(ν − d) = Hcokerϕ(ν)−Hkerϕ(ν − d).

In particular, HPM(t)− tdHPM(t) = HPcokerϕ(t)− tdHPkerϕ(t).

131. Definition. Let A be a graded k -algebra and M be a finitely generated,graded A-module. A graded free resolution of M is a resolution

. . . // Fk+1ϕk+1 // Fk // . . . // F1

ϕ1 // F0ϕ0 // M // 0

such that every Fk is a finitely generated free A-module of the formA(−d1)⊕ . . .⊕ A(−dp), and the ϕk are homogeneous of degree 0.

132. Corollary [Exercise]. Let A =⊕

ν≥0 Aν be a Noetherian graded k-algebra,and let M =

⊕ν≥0 Mν be a finitely generated and positively graded A-module. If

0 // Fkϕk // Fk−1 // . . . // F1

ϕ1 // F0ϕ0 // M // 0

is a graded free resolution of M, then we have

HM(ν) = dimk Mν =k∑

j=0

(−1)j dimk (Fj)ν =k∑

j=0

(−1)jHFj (ν).

133. Theorem [GP, 5.1.3]. Let A be a Noetherian graded k-algebra, and let M bea finitely generated and positively graded A-module. Furthermore, assume that Ais generated by x1, . . . , xn ∈ A1 as a k-algebra, that is, A = k [x1, . . . , xn]/I for someideal I of k [x1, . . . , xn]. Then

HPM(t) =Q(t)

(1− t)n

for some Q(t) ∈ Z[t ].

Canceling all common factors of HPM(t) = Q(t)/(1− t)n yields a representation

HPM(t) =G(t)

(1− t)s , 0 ≤ s ≤ n, G(t) =d∑

k=0

gk tk ∈ Z[t ]

such that gd 6= 0 and G(1) 6= 0. This leads to the


ν≥0 Aν be a Noetherian graded k -algebra, and letM =

⊕ν≥0 Mν be a finitely generated and positively graded A-module.

(i) Q(t) and G(t) ∈ Z[t ] are called the first and second Hilbert series of M.(ii) Let d = deg G(t) be the degree of G, and let s be the order of the pôle of

HPM(t) at 1. For ν ≥ d , we let

PM(ν) :=d∑

k=0

gk ·(

s − 1 + ν − ks − 1

).

Then PM(ν) is a polynomial expression in ν and we can view PM formally as arational polynomial in ν (if treat ν as a variable, then the coefficients are nolonger necessarily in Z). PM ∈ Q[ν] is called the Hilbert polynomial of M (by

convention(

n−1

)= 0 for n ≥ 0 and = 1 for n = −1; the degree of the zero

polynomial is −1).

135. Corollary (Hilbert) [GP, 5.1.5]. We have deg PM = s − 1 and the polynomialexpression in ν of PM(ν) equals HM(ν) for ν ≥ d. In particular, the Hilbert functionis a polynomial in ν for ν big enough. Moreover, there exists ak ∈ Z such that

PM =s−1∑k=0

ak

(νk

)=

as−1

(s − 1)!· νs−1 + lower terms in ν,

where as−1 = G(1) > 0.

136. Example. Consider k [x1, . . . , xn] with its natural k -algebra grading. Then

Hk [x](ν) =

0, ν < 0(

ν + n − 1n − 1

), ν ≥ 0

and thereforeHPk [x](t) =

∞∑ν=0

(ν + n − 1

n − 1

)tν =

1(1− t)n .

It follows that G(t) = 1 and Pk [x](ν) = Hk [x](ν) for all ν ≥ 0.

Next we adress the question of computing the Hilbert-Poincaré series for modulesof the form k [x]/I. Again, we start with monomial ideals.

137. Lemma [GP, 5.2.2]. Let I ⊂ k [x] be a homogeneous ideal, and let f ∈ k [x]be a homogeneous polynomial of degree d, then

HPk [x]/I(t) = HPk [x]/I+(f )(t) + tdHPk [x]/I:(f )(t)

138. Example. Let I = (xz, yz) ⊂ k [x , y , z]. For f = z we have I + (f ) = (z) andI : (f ) = k [x , y ], whence

HPk [x ,y ,z]/(xz,yz)(t) = HPk [x ,y ](t) + tHPk [z](t) =−t2 + t + 1

(1− t)2

and thus

Pk [x ,y ,z]/(xz,yz)(ν) =

(ν + 1

1

)+

(ν1

)−(ν − 1

1

)= ν + 2

139. The algorithm MONOMIALHP(I) [GP, 5.2.4].input : I = (f1, . . . , fr ) ⊂ k [x], fi ∈ Mon(x1, . . . , xn)output: Q(t) ∈ Z[t ] such that Q(t)/(1− t)n is the Hilbert-Poincaré series of k [x]/I

1 compute G(I) = {xα1 , . . . , xαs} the minimal generating subset of monomials of I;2 if G(I) = {0} then3 return 14 end5 if G(I) = {1} then6 return 07 end8 if all elements of G(I) have degree 1 then9 return (1− t)s

10 end11 choose 1 ≤ i ≤ s such that deg(xαi ) > 1 and 1 ≤ k ≤ n such that xk |xαi ;12 return MONOMIALHP(I, xk ) + t ·MONOMIALHP(I : (xk ))

Monomial Hilbert-Poincaré series [GP, 5.2.5]> proc MonomialHP(ideal I).{. I=interred(I); //computes the minimal basis of I

. int s=size(I);

. if(I[1]==0){return(1);} //I=0?

. if(I[1]==1){return(0);} //I=k [x]?

. if(deg(I[s])==1){return((1-var(1))ˆs);} //otherwise deg I[s]>1

. int j=1;

. while(leadexp(I[s])[j]==0){j++;} //find first xj | deg I[s]

. return(MonomialHP(I+var(j))+var(1)*MonomialHP(quotient(I,var(j))));

.}> ring A=0,(t,x,y,z),dp;> ideal I=x5y2,x3,y3,xy4,xy7;> MonomialHP(I);t6-2t3+1 //=Q(t), divide by (1−t)n, n=3, to obtain HPk [x,y,z]/I

> intvec v=hilb(std(I),1); //alternatively, use the HILB(·,1)-command> v; //This is Q(t) since we used 1 as option (use 2 for G(t))1,0,0,-2,0,0,1,0 //interpretation: if v=(v0,...,vd ,0), then Q(t)=

∑di=0 vi t i

140. Theorem [GP, 5.2.6], [GP, 5.2.7]. Let > be any monomial ordering on k [x],and let I ⊂ k [x] be a homogeneous ideal. Then

HPk [x]/I(t) = HPk [x]/LM(I)(t),

where LM(I) is the leading ideal of I with respect to >. In particular, we have

dimk k [x]/I = dimk k [x]/LM(I).

141. The Algorithm HILBERTPOINCARE(I) [GP, 5.2.8]. We wish to compute theHilbert-Poincaré series of k [x]/I for a homogeneous ideal I.

input : I = (f1, . . . , fk ) ∈ k [x] a homogeneous ideal, x = (x1, . . . , xr )output: Q(t) ∈ Z[t ] such that Q(t)/(1− t)r is the Hilbert-Poincaré series of k [x]/I

1 compute STD(I);2 return MONOMIALHILBERTPOINCARE(LM(g1), . . . ,LM(gs))


2. Gröbner bases


In the sequel we fix an algebraically closed field k subsequently referred to asthe ground field. k is necessarily infinite; otherwise, f (x) = 1 +

∏a∈k (x − a)

would be a nontrivial polynomial without zeroes. In particular, we can identifypolynomials in k [x1, . . . , xn] with polynomial functions on kn.

142. Definition. An algebraic set of kn is the common zero locus of a finitenumber of polynomial equations, i.e., a set of the form

X = V(f1, . . . , fn) := {(a1, . . . ,an) ∈ kn | f1(a1, . . . ,an) = . . . = fn(a1, . . . ,an) = 0}.

143. Remark. In fact, X only depends on the ideal generated by f1, . . . , fn. Sincek [x1, . . . , xn] is Noetherian, the algebraic sets are exactly the sets

V(I) = {x ∈ kn | f (x) = 0 for all f ∈ I}for an ideal I ⊂ k [x1, . . . , xn]. Conversely, given an algebraic set X we define itsassociated ideal by

I(X ) = {f ∈ k [x1, . . . , xn] | f (x) = 0 for all x ∈ X}.We obviously have

Y ⊂ X ⇒ I(X ) ⊂ I(Y ) and I ⊂ J ⇒ V(J) ⊂ V(I).

144. Example. First we recall that

1 ∈ I ⇐⇒ I = k [x1, . . . , xn],

an observation we will use quite often. Consider then, for instance, I({a}) fora = (a1, . . . ,an) ∈ kn. We claim that this is a maximal ideal equal to(x1 − a1, . . . , xn − an). First, we note that I({a}) is the kernel of the evaluation map

eva : k [x1, . . . , xn]→ k , f 7→ f (a) = f (a1, . . . ,an),

whence I({a}) is maximal. Moreover, the inclusion (x1 − a1, . . . , xn − an) ⊂ I({a})is clear. Conversely, if f were in I({a}) \ (x1 − a1, . . . , xn − an), then on degreegrounds, NF(f |x1 − a1, . . . , xn − an) = b ∈ k , whence we have a standard formf − b =

∑qi(xi − ai), qi ∈ k [x1, . . . , xn]. It follows that b ∈ I, whence 1 ∈ I, a

contradiction.

When is V(I) nonempty?

145. Theorem (Weak Nullstellensatz) [CLS, 4.1.1]. V(I) 6= ∅ if and only if I is aproper ideal of k [x1, . . . , xn].

Clearly, we have X = V(I(X )) for any algebraic set X . What about I(V(I))?

146. Definition. The radical of I is the ideal√

I defined by√

I = {a ∈ k [x1, . . . ,an] | there exists n ∈ N such that an ∈ I}.

Note that I ⊂√

I and√√

I =√

I. I is called reduced or radical if√

I = I.

147. Theorem (Hilbert’s Nullstellensatz) [CLS, 4.2.6], [CLS, 4.5.11].

I(V(I)) =√

I

In particular, V(·) and I(·) set up a bijection between algebraic subsets of kn andthe set of radical ideals in k [x1, . . . , xn]. For instance, points a = (a1, . . . ,an) ∈ kn

correspond to maximal ideals m ⊂ k [x1, . . . , xn].

Computation of the radicalThere are algorithms using Gröbner bases to compute the radical of an ideal.However, this requires some further technical tools so we content ourselves withan example using the built-in command RADICAL of SINGULAR.

> ring A=0,(x,y,z),dp;> LIB "primdec.lib"; //loads library “primdec.lib” containing thecommand RADICAL

// ** loaded /usr/bin/../share/singular/LIB/primdec.lib...> ideal I=xyz,x2,y4+y5;> ideal RI=radical(I);>RI;RI[1]=xRI[2]=y2+y

Practical application 6: Radical membership problem [GP, Section 1.8.6]Problem: Let I = (f1, . . . , fk ) be an ideal of k [x] and let > be a monomial orderingon Mon(x1, . . . , xn). Given f ∈ k [x] we want to determine whether or not f ∈

√I.

Solution: Recall Rabinovich’s trick we used for the proof of Hilbert’sNullstellensatz 147. This is the statement that

f ∈√

I ⇐⇒ 1 ∈ I := (f1, . . . , fk ,1− tf ) ⊂ k [x, t ]

for some additional new variable t . But this is the membership problem 90 whichwe already solved.

Radical membership problem [GP, 1.8.9]> ring A=0,(x,y,z),dp;> ideal I=x5,xy3,y7,z3+xyz;> poly f=x+y+z;> ring B=0,(t,x,y,z),dp;> ideal I=imap(A,I);> poly f=imap(A,f);> ideal II=I,1-t*f;> std(II);_[1]=1> LIB"primdec.lib"; //for a cross check, compute

√I directly

> setring A;> radical(I);_[1]=z_[2]=y_[3]=x

Practical application 7: Computing sodukos [DP, Section 3]

Source: W. Decker and G. Pfister, A First Course in Computational Algebraic Geometry, Lecture notes, p. 104

Problem: Suppose we are given a well-posed Sudoku, that is, we are given acollection of numbers in {1, . . . ,9} so that the Sudoku has a unique solution.

Solution: We represent the 9 · 9 = 81 cells by the variables x1, . . . , x81. Then theentry ai in the i-th cell of a completed Sudoku satisfies ai ∈ {1, . . . ,9} if and only ifai is a root of Fi(x) = Fi(xi) =

∏9k=1(xi − k) ∈ Q[xi ].

Let 1 ≤ i < j ≤ 81. If a = (a1, . . . ,a81) ∈ V (xi − xj), that is, ai = aj , thenFi(ai) = Fj(aj), whence Fi(xi)− Fj(xj) vanishes on V (xi − xj) and is thus divisibleby xi − xj . Consequently, the polynomials

Gij(x) = Gij(xi , xj) =Fi(xi)− Fj(xj)

xi − xj∈ Q[xi , xj ], i < j .

are well-defined. The condition that neither a row, nor a column, nor a distin-guished 3× 3-block in a completed Sudoku has repeated entries can beimplemented as follows. Set

E ={(i , j) | 1 ≤ i < j ≤ 81, and the i th and j th cell are in the samerow, column, or distinguished 3× 3-block}.

Let I ⊂ Q[x1, . . . , x81] (Q = algebraic closure of Q) be the ideal which is generatedby the 891 polynomials Fi , i = 1, . . . ,81, and Gij , (i , j) ∈ E .

148. Proposition [DP, 3.1]. Let a = (a1, . . . ,a81) ∈ Q81. Then a ∈ V(I) ⊂ Q81 ifand only if ai ∈ {1, . . . ,9}, for i = 1, . . . ,81, and ai 6= aj , for (i , j) ∈ E.

Put differently, a completed Sudoku is nothing but a point in V(I) ⊂ Q81 ∩Q81, orequivalently, a maximal ideal in Q[x1, . . . , x81] containing

√I.

149. Proposition [DP, 3.2]. Let S be a well–posed Sudoku with preassignednumbers {ai}i∈L, for some subset L ⊂ {1, . . . ,81}. We assign to S the idealIS = I + ({xi − ai}i∈L). Then IS is the maximal ideal corresponding to thecompleted sudoko (a1, . . . ,a81), that is, IS = {x1 − a1, . . . , x81 − a81}. We candetermine the ai by computing the reduced Gröbner basis of IS with respect to anymonomial ordering.

3 4 1 7 5 9 2 8 68 7 9 1 6 2 3 4 56 5 2 4 3 8 7 9 12 1 7 3 4 5 9 6 84 8 5 2 9 6 1 3 79 3 6 8 7 1 4 5 25 9 3 6 2 7 8 1 41 2 4 5 8 3 6 7 97 6 8 9 1 4 5 2 3

150. Definition.(i) Let X be some set and T a subset of the power set of X . Then T is said to

define a topology on X if the following conditions hold:. ∅, X ∈ T. F1, . . . ,Fs ∈ T , then

⋃si=1 Fi ∈ T

. If Fi ∈ T , I any indexing set, then⋂

i∈I Fi ∈ T .

The elements of T are called closed sets; their complements X \ F , Fclosed, are called open sets. X together with the topology T is called atopological space. If Y ⊂ X , TY = {F ∩ Y | F ∈ T } defines a topology on Y ,the so-called subspace topology.

(ii) A map f : X → Y between topological spaces is called continuous, if foreach closed subset G of Y , f−1(G) is a closed subset of X .

151. Remark. The set theoretic de Morgan rules imply that one can equally welldefine a topology by its open sets, and a function f : X → Y is continuous if anypreimage under f of an open set of Y is open in X . For instance, the usualEuclidean topology on Rn are the open sets which are arbitrary unions of openballs. Functions f : Rm → Rn which are continuous with respect to this topologyare precisely the functions which satisfy f (xn)→ f (x) for any sequence xn → x .

152. Proposition [CLS, 4.3.4, 4.3.15]. For ideals {Iλ}λ∈Λ and J in k [x1, . . . , ], wehave

V(∑λ

Iλ) =⋂λ

V(Iλ) and V(I ∩ J) = V(I) ∪ V(J).

In particular, the set of algebraic sets defines a topology on kn, the so-calledZariski topology. We therefore get a correspondence compatible with the latticestructure between closed sets of kn and radical ideals of k [x1, . . . , xn].

153. Example.(i) For X = k endowed with the Zariski topology the algebraic sets other than ∅

and k are the zeroes of a polynomial for k [x ] is principal. In particular, theyare finite. It follows that proper nonempty subsets of k are Zariski open in ifand only if their complement is finite. In particular, nonempty open subsetsare dense, and the Zariski topology is not Hausdorff on k .

(ii) For any f ∈ k [x1, . . . , xn] define the so-called basic open set by kn \ V(f ). It iseasy to see that the basic open sets form a base for the Zariski topology, i.e.,every open set is a union of basic open sets.

Operations on ideals [GP, Section 1.8.7]It is easy to compute the intersection of two ideals I and J of k [x1, . . . , xn] viaelimination orderings. SINGULAR has the built-in command INTERSECT:

> ring A=0,(x,y,z),dp;> ideal I=x,y;> ideal J=y2,z;> intersect(I,J);_[1]=yz_[2]=xz_[3]=y2

154. Definition. A closed set is called irreducible if it is not the union of twostrictly smaller closed subsets. An affine variety is an irreducible closed set.

155. Proposition [CLS, 4.5.3]. An algebraic set X is an affine variety if and only ifI(X ) is prime. In particular, there is a one-to-one correspondence between affinevarieties in kn and prime ideals of k [x1, . . . , xn].

156. Examples. Any point in kn corresponds to a maximal ideal (weakNullstellensatz). Hence they are irreducible; this, of course, can be checkeddirectly. On the other hand, kn with its Zariski topology is irreducible, but this isdifficult to show by using the definition unless n = 1 (do you see why this case isspecial?). The twisted cubic curve C = {(t , t2, t3) | t ∈ k} =V(y − x2) ∩ V(z − x3) in k3 is irreducible as I(C) = (y − x2, z − x3) is prime.

157. Definition. The affine variety kn is called affine space and written Ank or

simply An.

The intersection of V (y − x2) and V (z − x3)

Source: W. Decker and G. Pfister, A First Course in Computational Algebraic Geometry, Lecture notes, p. 18

158. Proposition [CLS, 4.5.13]. An algebraic set X is irreducible if and only if forevery closed subset Y of X , the difference X \ Y is Zariski dense in X. Inparticular, an open subset of an irreducible algebraic set is always dense.

159. Definition. A topological space X is Noetherian if any descending chain ofclosed sets becomes stationary, that is, for X ⊃ X0 ⊃ X1 ⊃ X2 ⊃ . . ., Xi closed inX , there exists N such that XN = XN+1 = XN+2 = . . ..

160. Proposition [CLS, 4.6.1]. Any algebraic set of An is Noetherian with respectto the induced subspace topology.

161. Proposition [CLS, 4.6.4]. Every algebraic set X can be written as a finiteunion

X = X1 ∪ . . . ∪ Xk

of affine varieties Xi with Xi 6⊆ Xj if i 6= j . The set {X1, . . . ,Xk} is uniquelydetermined. Its elements Xi are called the components of X.

162. Remark.(i) If X = V(I) is an algebraic set with irreducible components Xi = V(Pi), then

the Pi are the minimal primes containing I, that is, there is no prime ideal Qcontaining I and strictly contained in one of the Pi . If I =

√I is a radical ideal,

then the decomposition into irreducibles V(I) =⋃

i Xi corresponds to thedecomposition I =

√I =

⋂P where the union is taken over all prime ideals P

with I ⊂ P and P is prime and minimal, cf. the problem class.(ii) Let X , Y ⊂ An

k be two closed subsets with Y ⊂ X . Then X \ Y is Zariskidense in X if and only if Y contains no irreducible component of X .

Computing the decomposition 1> LIB "primdec.lib";> ring A=0,(x,y,z),dp;> ideal I=xy; ideal J=y-x2,z-x3;> radical(I);_[1]=xy //The ideal I is radical> primdecGTZ(I); //The ideals [i][1] are the prime ideals of theirreducible components. The ideals [i][2] are only relevant if I is notradical.[1]:[1]:_[1]=y[2]:_[1]=y[2]:[1]:_[1]=x[2]:_[1]=x

Computing the decomposition 2> reduce(radical(J),std(J));_[1]=0_[2]=0 //The ideal J is radical.> primdecGTZ(J);[1]:[1]:_[1]=y3-z2_[2]=-y2+xz_[3]=xy-z_[4]=x2-y[2]:_[1]=y3-z2_[2]=-y2+xz_[3]=xy-z_[4]=x2-y> reduce(-y2+xz,std(J));0> reduce(y3-z2,std(J));0 //The ideal J is even prime.

163. Definition. Let X ⊂ An1 and Y ⊂ An2 be two algebraic sets. A map

F : X → Y

will be called a morphism if there exists n2 polynomials f1, . . . , fn2 ∈ k [x1, . . . , xn1 ]such that

F (p) =(f1(p1, . . . ,pn1), . . . , fn2(p1, . . . ,pn1)

)for all points p = (p1, . . . ,pn1) ∈ X . We denote by Hom(X ,Y ) the set of morphismsbetween X and Y . If Y = A1 ∼= k then we call F a regular function.

164. Proposition [CLS, 5.1.2]. Two polynomials f and g ∈ k [x1, . . . , xn] define thesame regular function on an algebraic set X if and only if f − g ∈ I(X ).

165. Definition. If X ⊂ An is an algebraic set, we call

A(X ) := k [x1, . . . , xn]/I(X )

the coordinate ring of X .

166. Remark.(i) The coordinate ring of an affine variety is a finitely generated k -algebra which

is an integral domain if X is an affine variety. Such ring is also called an affinering for every affine ring is isomorphic with the coordinate ring of an affinevariety, cf. Proposition 170. In view of Proposition 164 we can regard A(X ) asthe ring of regular functions on X .

(ii) Fix a monomial ordering on k [x1, . . . , xn]. For an ideal I let the complementof monomials be the set C(I) := {α ∈ Nn | xα 6∈ I}. Then the reduced normalform identifies the coordinate ring A(X ) = k [x1, . . . , xn]/I(X ) as a k-vectorspace with

⊕α 6∈C(I(X)) k · xα, cf. also [CLS, 5.3.1,4] and the problem class.

167. Definition. Let X ⊂ An and Y ⊂ Am be affine varieties. We say that X and Yare isomorphic if there exist polynomial mappings α : X → Y and β : Y → X suchthat α ◦ β = IdY and β ◦ α = IdX .

168. Theorem [CLS, 5.4.8, 5.4.9]. If X and Y are affine varieties, then there is anatural isomorphism Hom(X ,Y ) ∼= Homk (A(Y ),A(X )) by assigning F : X → Y tothe k-algebra morphism F ∗ : A(Y )→ A(X ), F ∗G = G ◦ F. In particular, two affinevarieties are isomorphic if and only if their coordinate rings are isomorphic.

169. Example. The map F ∗ : k [x , y , z]→ k [t ] induced by F (x) = t , F (y) = t2 andF (z) = t3 (cf. Proposition 10) induces actually an isomorphismk [x , y , z]/(y − x2, z − x3) corresponding to the morphism A1 → C, t 7→ (t , t2, t3).Therefore, the twisted cubic is isomorphic with A1. On the other hand, t 7→ (t2, t3)induces a morphism A1 → X = {y3 − x2 = 0} ⊂ A2 which does not induce anisomorphism k [x , y ]/(y3 − x2)→ k [t ] (can you see why?Algebraically/Geometrically?).

170. Corollary (category of affine varieties vs. category of finitely generatedk -algebras which are integral domains). The assignement X → A(X ) extendsto a contravariant functor between the category of affine k-varieties + morphismsand the category of finitely generated, integral k-algebras + k-algebra morphisms.In particular, this defines an equivalence of categories.

171. The projective space. The second natural space for doing geometry is theprojective space Pn

k (or simply Pn if we do not wish to emphasise the ground field).This is the set of lines in kn+1 through the origin (=one dimensional subspaces ofkn+1). A line ` ∈ Pn

k is determined by any of its points 0 6= (x0, . . . , xn) ∈ `. Twopoints x , y ∈ kn+1 \ {0} determine the same line if they differ by a nonzero scalar,i.e., x = λy for λ ∈ k∗. Hence we get a map

π : kn+1 \ {0}, (x0, . . . , xn) 7→ span((x0, . . . , xn))k

leading to the bijection Pn ∼= kn+1 \ {0}/k∗. Intuitively, it is clear that Pnk should be

an n-dimensional space. The line ` determined by the point (x0, . . . , xn) will bedenoted by [x0 : . . . : xn] – these are the homogeneous coordinates of `. Thesegive rise to the subsets Ui = {[x0 : . . . : xn] | (x0, . . . , xn) ∈ kn+1 \ {0}, xi 6= 0}which we identify with kn via the bijections

Ui → kn, (x0, x1, . . . , xn) 7→(x0

xi,x1

xi, . . . ,

xn

xi

)with xi/xi omitted. This yields two important ways of understanding Pn:

. Pnk =

⋃ni=0 Ui , the standard covering of Pn.

. Pnk = kn ∪ Pn−1

k with kn ∼= Ui , Pn−1k = Uc

i , the cellular decomposition.

172. The cellular decomposition.

Source: D. Cox, J. Little and D. O’Shea, Ideals, algorithms and varieties, pp. 390 and 391

173. Geometrical motivation of projective space. We obtain projective spacesby adding the horizon at infinity to lines, planes etc.

Source: D. Cox, J. Little und D. O’Shea, Ideals, algorithms and varieties, S. 386

It is customary to denote by S the polynomial ring k [x0, . . . , xn] together with thestandard grading

⊕ν≥0 Sν , Sν = the k -span of monomials of total degree ν.

174. Definition. An algebraic set in Pnk is the common zero locus of a finite set of

homogeneous polynomials in S = k [x0, . . . , xn].

175. Algebraic sets and homogeneous ideals. For a homogeneous ideal I wedefine

Vp(I) = {[x0 : . . . : xn] ∈ Pnk | f (x0, . . . , xn) = 0 for all homogeneous f ∈ I}.

Conversely, for an algebraic set X ⊂ Pn we consider the homogeneous ideal

Ip(X ) = ideal generated by all homogeneous polynomials in Swhich vanish identically on X .

176. Remark. A homogeneous ideal I ⊂ S defining the projective algebraic set Xalso gives the affine algebraic set C(X ) := {x ∈ kn+1 | f (x) = 0 for all f ∈ I}, theso-called cone of X . Of course, this is just V(I) ⊂ An+1 if we consider I as an idealof k [x0, . . . , xn].

The cone C(X ) over the curve X = Vp(I) in P2k

Source: R. Hartshorne, Algebraic Geometry, Springer, 1977, p. 12

The following properties are also useful (cf. problem class or for instance SectionVII.2 in P. Samuel and O. Zariski, Commutative Algebra vol. 2, Springer, 1960):

(i) The sum, product, intersection and radical of homogeneous ideals is alsohomogeneous.

(ii) Let I be a homogeneous ideal of S = k [x0, . . . , xn]. Then I is prime if and onlyif for any pair of homogeneous elements f , g of S we have: f · g ∈ I ⇔ f ∈ I org ∈ I.

It follows that if {Iλ}λ∈Λ is a family of homogeneous ideals,⋂λ Vp(Iλ) = Vp(

⋃λ Iλ);

if I1,2 are two homogeneous ideals, Vp(I1) ∪ Vp(I2) = Vp(I1 ∩ I2). In particular,projective algebraic sets define a topology which is again Noetherian.

177. Definition. The Zariski topology on Pn is defined by algebraic sets Vp(I) ofPn, I ⊂ S homogeneous, as closed sets. An irreducible algebraic set in Pn is calleda projective variety.

178. Proposition [CLS, 8.3.6]. Every projective algebraic set X can bedecomposed into irreducible projective algebraic sets. Furthermore, a projectivealgebraic set X is a projective variety if and only if I(X ) is prime.

179. Examples.(i) If X is a closed projective set, then π−1(X ) = C(X ) ∩ An+1 \ {0} is closed in

An+1 \ {0}. It follows that π : An−1 \ {0} → Pn is continuous with respect tothe (induced) Zariski topologies. Moreover, it is easy to see that theidentification of Ui with kn is actually a homeomorphism Ui

∼= An between the(induced) Zariski topologies, that is, it is a bijective continuous map withcontinuous inverse (cf. Paragraph 171).

(ii) k -dimensional Linear varieties Λk of Pn are defined by linear homogeneousequations. Equivalently, they are of the form π(V \ {0}) for ak + 1-dimensional linear subspace V k+1 ⊂ kn+1. In particular, they given bythe equations x0 = . . . = xn−k = 0 for suitably chosen coordinates x0, . . . , xn.In particular, I(Λk ) is prime for k [x0, . . . , xn]/I(Λk ) ∼= k [xn−k+1, . . . , xn] is anintegral domain, and Λk ∼= Pk

n. For instance, Uci = V (xi) is closed so that the

standard covering is an open covering by the open subsets Ui ; the cellulardecomposition Pn = Ui ∪ V (xi) is a decomposition into an open subset Uihomeomorphic to the affine space An and a closed subset V (xi) which is the“horizon” at infinity of the affine space Ui

∼= An (cf. Paragraph 173).(ii) The (projective) twisted cubic is the projective algebraic set

X = Vp(ty − x2, t2z − x3, xz − y2) ⊂ P3. If Ut = {[t : x : y : z] | t 6= 0} ⊂ P3,then Ut ∩ X is mapped to (x ′, y ′, z ′) with x ′ = x/t , y ′ = y/t and z ′ = z/t withy ′ − x ′2 = 0, z ′ − x ′3 = 0 and x ′z ′ − y ′2 = 0. The last equation is redundant sothat we find the affine twisted cubic discussed in Example 156. If we intersectwith the “hyperplane at infinity” given by H = Vp(t), then H ∩ X =Vp(t , ty − x2, t2z − x3, xz − y2) = {[0 : 0 : 0 : 1]} is the point “added atinfinity”. X is irreducible as the closure of the irreducible affine twistedcubic [CLS, 8.4.7].

180. Theorem (projective weak Nullstellensatz) [CLS, 8.3.9 and 8.3.10]. IfI ⊂ S is a homogeneous ideal with Vp(I) 6= ∅, then

Ip(Vp(I)) =√

I.

In particular, Ip and Vp set up inclusion-reversing bijections between nonemptyprojective varieties and radical homogeneous ideals properly contained in(x0, . . . , xn).

181. The algebra-geometry dictionary. (A = k [x1, . . . , xn], S = k [x0, x1, . . . , xn])

algebraic sets in An ←→ radical ideals of Aaffine varieties in An ←→ prime ideals of Apoints in An ←→ maximal ideals of Amorphisms of affine varieties X → Y ←→ k -algebra morphisms

A(Y )→ A(X )algebraic sets in Pn ←→ radical ideals of S

contained in (x0, . . . , xn)ascending chain condition ←→ descending chain condition√

I =⋂

I⊂P P minimal primes ←→ X =⋃

Xi irreducibles

182. Remark. Note that unlike the (affine) coordinate ring A(X ), the gradedhomogeneous coordinate ring S(X ) := S/I(X ) is not a ring of functions. Thiswill be remedied by introducing the sheaf of regular functions, see the courseAlgebraic Geometry 1.

Let I ⊂ S be a homogeneous ideal so that the Hilbert function and thus the Hilbertpolynomial of the graded ring S/I are defined.

183. Definition. The dimension of a nonempty projective algebraic set X is thedegree of the Hilbert polynomial of S(X ).

184. Example. According to Example 136, dimPn = n which matches ourgeometric intuition.

185. The dimension of an algebraic set. We will now discuss the geometricmeaning of this definition. Here, the idea is that dimension should be invariantunder certain deformations or degenerations to a geometrically obvious case witha “natural” dimension. For instance, consider I = (x2y , x3) ⊂ k [x , y ]. Then

V(I) = V(x2y) ∩ V(x3) = (V(x) ∪ V(y)) ∩ V(x) = V(x).

Regarded as an affine algebraic set in A2, V(x) ∼= A1, so its dimension should be1. However, regarded as a projective algebraic set in P1, V(x) = {[0 : 1]}, so itsdimension should be 0.

To generalise the last example we call a projective algebraic set monomial if it isof the form Vp(I) for some monomial ideal I ⊂ S = k [x0, . . . , xn].

186. Proposition [CLS, 9.1.1]. If I ⊂ S is a monomial ideal, then Vp(I) is a finiteunion of linear varieties of Pn.

187. Definition. Let I ⊂ S be a monomial ideal. Then the dimension of Vp(I)denoted by dimVp(I) is the maximal dimension of its linear varieties. Note thatdimVp(I) = Vp(

√I).

Assume we are given a monomial ideal I generated by monomialsm1, . . . ,mt ∈ k [x0, . . . , xn]. To compute the dimension we need to determine thecomponent of

⋂V (mi) of largest dimension. If xi0 , . . . , xir are variables such that at

least one appears in each monomial mj , then xi0 = . . . = xir = 0 is contained inV(I). We therefore let

Mj = {` ∈ {0, . . . ,n} | x` divides the monomial mj}

andM = {J ⊂ {0, . . . ,n} | J ∩Mj 6= ∅ for all 0 ≤ j ≤ t}.

188. Proposition [CLS, 9.1.3]. We have

dimVp(I) = n −min(|J| | J ∈M),

where |J| is the number of elements of the set J.

189. Example. If I = (x2y , x3) is as above, then M1 = {0,1} and M2 = {0}. Itfollows thatM = {{0}, {0, 1}}, whence dimVp(I) = 0.

190. Coordinate subspaces. Consider a monomial ideal I ⊂ k [x0, . . . , xn]. Weconsider again the complement of monomials C(I) := {α ∈ Nn | xα 6∈ I}. Further,setting ei = (0, . . . ,1, . . . ,0) as usual, we define the coordinate subspace of Nn

determined by ei1 , . . . ,eir as follows:

[ei1 , . . . ,eir ] = {r∑

j=1

ajeij | aj ∈ N}

Its dimension is r by definition. Its translate by α ∈ Nn is the set

α + [ei1 , . . . ,eir ] = {α + β | β ∈ [ei1 , . . . ,eir ]}.

191. Example.

Let I = (y6, x5y3, x7y2, x12). The exponents in C(I) are given by open circles.

Source: D. Cox, J. Little and D. O’Shea, Ideals, algorithms and varieties, pp. 474

192. Example.

Let I = (y6, x5y3, x7y2). The exponents in C(I) are the open circles.


193. Example.

Let I = (x3y6, x5y3, x7y2). The exponents in C(I) are the open circles.


194. Example. Consider a proper monomial ideal I ⊂ k [x , y ]. Then Vp(I) is either(i) empty, and xa and yb ∈ I for some positive integers a and b. Further, C(I)

consists of a finite number of points, cf. Example 191.(ii) or the point [x : y ] = [1 : 0], and xa 6∈ I for all a > 0, but yb ∈ I for some b.

Further, C(I) consists of a finite number of translates of [e1] and possibly of afinite number points not contained in these translates, cf. Example 192.

(iii) or the point [x : y ] = [0 : 1], and yb 6∈ I for all b > 0, but xa ∈ I for some a.Further, C(I) consists of a finite number of translates of [e2] and possibly of afinite number points not contained in these translates.

(iv) or the union of the points [1 : 0] and [0 : 1], and xy must divide everymonomial in I. Further, C(I) consists of a finite number of translates of [e1], of[e2], and possibly of a finite number points not contained in these translates,cf. Example 193.

195. Proposition [CLS, 9.2.1]. Let I ⊂ S be a proper monomial ideal.(i) The linear variety Vp(xi | i 6∈ {i0, . . . , ir}) is contained in Vp(I) if and only if

[ei0 , . . . ,eir ] ⊂ C(I).(ii) dimVp(I) + 1 = the maximal dimension of the coordinate subspaces in C(I).

196. Theorem [CLS, 9.2.3]. If I ⊂ S is a proper monomial ideal, then the setC(I) ⊂ Nn+1 of exponents of monomials not lying in I can be written as a finiteunion of translates of (possibly 0-dimensional) coordinate subspaces of Nn+1.

197. Lemma [CLS, 9.2.5]. The number of points in a coordinate translateα + [ei0 , . . . ,eim ] of total degree s is equal to(

m + s − |α|s − |α|

)−(

m + s − 1− |α|s − 1− |α|

),

provided that s > |α|+ 1.

198. Proposition [CLS, 9.3.3]. Let I be a proper monomial ideal inS = k [x0, . . . , xn]. For ν ≥ 0, HS/I(ν) is the number of monomials not in I and oftotal degree s. Furthermore, deg PS/I = dimVp(I).

199. Corollary (Dimension theorem) [CLS, 9.3.8]. Let I ⊂ S = k [x0, . . . , xn] be ahomogeneous ideal. Then

dimVp(I) = dimVp(LM(I)).

200. Remark. In a similar way we can consider monomial algebraic subsets in An

and “affine” Hilbert functions to define the dimension of affine varieties, see [CLS,Section 9.3].

References

[CLS] D. COX, J. LITTLE AND D. O’SHEA

Ideals, algorithms and varietiesSpringer, 2007

[DP] W. DECKER AND G. PFISTER

A First Course in Computational Algebraic GeometryLecture notes

[EH] V. ENE AND J. HERZOG

Gröbner Bases in Commutative AlgebraAMS, 2012

[GP] G.-M. GREUEL AND G. PFISTER

A SINGULAR introduction to commutative algebraSpringer, 2002.

Groups, Algorithms, Geometries & Applications A · 2019. 11. 19. · Groups, Algorithms, Geometries...

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