Universal Algebraic Geometry Part 1omskconf2009.oscsbras.ru/daniyarova1.pdf · Universal Algebraic...

Universal Algebraic GeometryPart 1

Evelina Daniyarova1

based on joint results withAlexei Myasnikov2 and Vladimir Remeslennikov1

1 Sobolev Institute of Mathematics of the SB RAS, Omsk, Russia2 McGill University, Montreal, Canada

International Algebraic Workshop on “New Algebra-Logical Methods inSolution for Systems of Equations in Algebraic Structures”,

August 16 – 22, 2009, Omsk, Russia

Evelina Daniyarova Universal Algebraic Geometry

History

Algebraic geometry over algebraic structures ( = UniversalAlgebraic Geometry) is a new area of researching in algebratoday, but it has a deep genealogy. The equations playedimportant role in mathematics at all times.

Statement (Remeslennikov)No Equations =⇒ No Mathematics.

Algebraic geometry as mathematical discipline starts from 19thcentury. Classical algebraic geometry studies equations andtheir solutions over fields (the real number field, the complexnumber field, algebraic closed fields, finite fields, and so on).

History

Contemporary History

New needs and new problems in contemporary mathematicshave motivated to transfer classical algebraic geometry theoryto another algebraic structures (groups, rings, algebras over aring, semigroups, and so on). This generalization was started inwork:

G. Baumslag, A. Myasnikov, V. Remeslennikov, Algebraicgeometry over groups I: Algebraic sets and ideal theory,J. Algebra, 219, 1999, pp. 16–79.A. Myasnikov, V. Remeslennikov, Algebraic geometry overgroups II: Logical foundations, J. Algebra, 234, 2000,pp. 225–276.B. I. Plotkin, Some concepts of algebraic geometry inuneversal algebra, Algebra i Analiz, 9 (4), 1997,pp. 224–248.B. I. Plotkin, Algebras with the same (algebraic) geometry,Tr. Mat. Inst. Steklova, 242, 2003, pp. 176–207.

Contemporary History

New needs and new problems in contemporary mathematicshave motivated to transfer classical algebraic geometry theoryto another algebraic structures (groups, rings, algebras over aring, semigroups, and so on). This generalization was started inwork:

G. Baumslag, A. Myasnikov, V. Remeslennikov, Algebraicgeometry over groups I: Algebraic sets and ideal theory,J. Algebra, 219, 1999, pp. 16–79.A. Myasnikov, V. Remeslennikov, Algebraic geometry overgroups II: Logical foundations, J. Algebra, 234, 2000,pp. 225–276.B. I. Plotkin, Some concepts of algebraic geometry inuneversal algebra, Algebra i Analiz, 9 (4), 1997,pp. 224–248.B. I. Plotkin, Algebras with the same (algebraic) geometry,Tr. Mat. Inst. Steklova, 242, 2003, pp. 176–207.

Outstanding Results

At the present time the largest developments we can see inalgebraic geometry over groups where the most striking resultis the solution of the main problem of algebraic geometry onclassification of algebraic sets and coordinate groups in thecase of the free group. The classification of coordinate groupsis given in the language of free constructions. This wasobtained due to works of many specialists in the group theory.

O. Kharlampovich, A. Myasnikov, Irreducible affine varietiesover free group I: Irreducibility of quadratic equations andNullstellensatz, J. Algebra, 200 (2), 1998, pp. 472–516.O. Kharlampovich, A. Myasnikov, Irreducible affine varietiesover free group II: Systems in trangular quasi-quadraticform and description of residually free groups, J. Algebra,200(2), 1998, pp. 517–570.O. Kharlampovich, A. Myasnikov, Algebraic geometry overfree groups: Lifting solutions into generic points, Contemp.Math., 378, 2005, pp. 213–318.

Outstanding Results

At the present time the largest developments we can see inalgebraic geometry over groups where the most striking resultis the solution of the main problem of algebraic geometry onclassification of algebraic sets and coordinate groups in thecase of the free group. The classification of coordinate groupsis given in the language of free constructions. This wasobtained due to works of many specialists in the group theory.

O. Kharlampovich, A. Myasnikov, Irreducible affine varietiesover free group I: Irreducibility of quadratic equations andNullstellensatz, J. Algebra, 200 (2), 1998, pp. 472–516.O. Kharlampovich, A. Myasnikov, Irreducible affine varietiesover free group II: Systems in trangular quasi-quadraticform and description of residually free groups, J. Algebra,200(2), 1998, pp. 517–570.O. Kharlampovich, A. Myasnikov, Algebraic geometry overfree groups: Lifting solutions into generic points, Contemp.Math., 378, 2005, pp. 213–318.

What Is Universal Algebraic Geometry?

Universal algebraic geometry =

= transfer of general notions and ideas from concrete algebraicgeometries to the case of arbitrary algebraic structure +

+ formulation and proof of general results without help oftechnique and properties, specific for concrete algebraicstructures +

+ development of theory along with decision of new problemsarising in this area.

Language

A first-order language (or signature) is a set

L = {set of constant symbols c}∪{set of functional symbols F}.

Note: We consider only languages with no predicates.

Every functional symbol F is given with their arity nF .

Language

Algebras

An algebraic structure A in a language L is given by thefollowing data:

a non-empty set A called the universe of A;interpretation of symbols from L on the universe A:

a function F : AnF → A of arity nF for each function F ∈ L;an element c ∈ A for each constant c ∈ L.

We use notation A, B, C, . . . to refer to the universes of thestructures A,B, C, . . ..

Algebraic structures in a language with no predicates aretermed algebras.

So, for the current lecturesAlgebraic Structures = Algebras.

Algebras

Equations

Let X = {x1, . . . , xn} be a finite set of variables.

DefinitionEquation in the language L in variables X is an atomic formula

(t = s),

where t , s are terms.

DefinitionAny set S of atomic formulas forms a system of equations in L.

Equations

Let X = {x1, . . . , xn} be a finite set of variables.

DefinitionEquation in the language L in variables X is an atomic formula

(t = s),

where t , s are terms.

DefinitionAny set S of atomic formulas forms a system of equations in L.

Terms and Atomic Formulas

Terms in a language L in variables X are formal expressionsdefined recursively as follows:

variables x1, x2, . . . , xn and constants from L are terms;if t1, . . . , tn are terms and F (x1, . . . , xn) ∈ L is function thenF (t1, . . . , tn) is a term.

Atomic formulas are formulas of the form (t = s), where t , s areterms.

Terms and Atomic Formulas

Terms in a language L in variables X are formal expressionsdefined recursively as follows:

variables x1, x2, . . . , xn and constants from L are terms;if t1, . . . , tn are terms and F (x1, . . . , xn) ∈ L is function thenF (t1, . . . , tn) is a term.

Atomic formulas are formulas of the form (t = s), where t , s areterms.

Algebraic Sets

Let A be an algebra in a language L and S a system ofequations in L.

DefinitionThe solution of a system of equations S over A,

V(S) = { (a1, . . . , an) ∈ An |t(a1, . . . , an) = s(a1, . . . , an) ∀ (t = s) ∈ S },

is termed the algebraic set over A.

Radicals

DefinitionThe set of atomic formulas

Rad(S) = { (t = s) | t(a1, . . . , an) = s(a1, . . . , an)

∀ (a1, . . . , an) ∈ V(S) }

is termed the radical of the algebraic set V(S).

Rad(S) is the maximal system of equations which equivalent toS.

Radicals

DefinitionThe set of atomic formulas

Rad(S) = { (t = s) | t(a1, . . . , an) = s(a1, . . . , an)

∀ (a1, . . . , an) ∈ V(S) }

is termed the radical of the algebraic set V(S).

Rad(S) is the maximal system of equations which equivalent toS.

Coordinate Algebras

Let Y = V(S). Radical Rad(S) defines some congruence onthe absolutely free algebra (the algebra of terms) in thelanguage L in variables X :

t ∼ s ⇐⇒ (t = s) ∈ Rad(S), t , s are terms.

DefinitionWe say that factor-algebra defined by congruence ∼ above iscoordinate algebra of the algebraic set Y . We denote it Γ(Y ).

Coordinate Algebras

Let Y = V(S). Radical Rad(S) defines some congruence onthe absolutely free algebra (the algebra of terms) in thelanguage L in variables X :

t ∼ s ⇐⇒ (t = s) ∈ Rad(S), t , s are terms.

DefinitionWe say that factor-algebra defined by congruence ∼ above iscoordinate algebra of the algebraic set Y . We denote it Γ(Y ).

Algebra Signature

Typical form of equations

S Radical or algebraic set Coordinate algebra

Free group

11 2{ , ,1, , }a a−⋅

2 2 2 1x y z

, ,...,F a a a= ⟨ ⟩

= 1 2 3V( ) {( , , )|S f f f=

2 3 3 1] [ , ] [ , ]f f f f= =2 2 21 2 3 1}f f f

1 2[ , 1f f ==

3( )S F ∗Z Γ

1 2[ , ] [ , ]x y a a= V( )S is a very complex set

1 2[ , ] [ , ]( ) ,x y a aS F x y=Γ = ∗ ⟨ ⟩ is a free amalgamated product

Additive mo‐noid of natural

numbers N

{ , 0,1+3 4 7 7

x y z+ + = V( ) {(0,0,1),(1,1,0)}S = ( ) 1

x zS x z⊕⟨ ⟩⊕ ⟨ ⟩Γ = ⟨ + = ⟩

Free Lie algebra over a field k

,= ⟨ ⟩

{ ,[ , ], ,a b

[ , ] [ , ] 0x a y b = Remeslennikov, Stöhr:V( )S is an infinite dimen‐sional linear space over k

+,( ) id [ , ] [ , ]

L x yS ∗⟨ ⟩Γ = ⟨ x a x b+ ⟩

[[ , ],[ , ]]x a b a 0= Remeslennikov, Daniyarova: V( ) lin { , }kS a b=

1 2( ) ,S L t a t bΓ ⟨ +( )

⟩ , where x

t t k∈

∈∏ are algebraic

independent over k

Linear space over

ka field

{ , , ,k

α α+ ⋅ ∈, }v v V∈

x y vα β+ + + =, kα β ∈ , v V∈

1 ,..., rx x are free variables,

1 ,..., my y are dependent variables, V( ) {( , )| }S x y y Ax b= = +

1( ) lin ,...,k rS V x x

Γ ⊕ ⟨ ⟩

Model-Theoretic Foundation in Universal Algebraic Geometry

There are three segments in concrete algebraic geometries:Coefficient-free algebraic geometry;Diophantine algebraic geometry;Algebraic geometry with coefficients in some algebra Aand solutions in some extension A < B (usually, insaturated model).

Universal ApproachFrom the point of view of Universal Algebraic Geometry thesethree segments may be examined by means of one universaltechnique. It is just sufficient to make suitable choice ofsignature L.

Model-Theoretic Foundation in Universal Algebraic Geometry

There are three segments in concrete algebraic geometries:Coefficient-free algebraic geometry;Diophantine algebraic geometry;Algebraic geometry with coefficients in some algebra Aand solutions in some extension A < B (usually, insaturated model).

Universal ApproachFrom the point of view of Universal Algebraic Geometry thesethree segments may be examined by means of one universaltechnique. It is just sufficient to make suitable choice ofsignature L.

Example

Let us consider the language of groups Lg = {·,−1 , e}, a groupG and some extension G < H. Then:

Coefficient-free algebraic geometry over G is algebraicgeometry over G in the language Lg;Diophantine algebraic geometry over G is algebraicgeometry over G in the extended language

Lg,G = Lg ∪ {cg , g ∈ G};

Algebraic geometry over H with coefficients in G isalgebraic geometry over H in the extended language Lg,G.

Major ProblemOne of the major problems of algebraic geometry over given analgebra A in a language L consists in classifying algebraic setsover A with accuracy up to isomorphism.

One can classify algebraic sets by means of three languages,which are equivalent to each other:

1 in geometric language, by describing algebraic setsdirectly;

2 in the language of radical ideals;3 and in algebraic language, by classifying coordinate

algebras of algebraic sets.

FactEvery algebraic set may be restored in unique manner from itsradical and it may be restored from its coordinate structure justup to isomorphism.

First Universal Result

Let A be an algebra in a language L.

TheoremThe category of algebraic sets over algebra A and the categoryof coordinate algebras of algebraic sets over A are duallyequivalent.

E. Daniyarova, A. Miasnikov, V. Remeslennikov, Unificationtheorems in algebraic geometry, Algebra and DiscreteMathematics, 1 (2008), and on arxiv.org.

First Universal Result

Let A be an algebra in a language L.

TheoremThe category of algebraic sets over algebra A and the categoryof coordinate algebras of algebraic sets over A are duallyequivalent.

E. Daniyarova, A. Miasnikov, V. Remeslennikov, Unificationtheorems in algebraic geometry, Algebra and DiscreteMathematics, 1 (2008), and on arxiv.org.

Unification Theorem A

TheoremLet A be an equationally Noetherian algebra in a language L(with no predicates). Then for a finitely generated algebra C ofL the following conditions are equivalent:

1 Th∀(A) ⊆ Th∀(C), i.e., C ∈ Ucl(A);2 Th∃(A) ⊇ Th∃(C);3 C embeds into an ultrapower of A;4 C is discriminated by A;5 C is a limit algebra over A;6 C is an algebra defined by a complete atomic type in the

theory Th∀(A) in L;7 C is the coordinate algebra of a non-empty irreducible

algebraic set over A defined by a system of equations inthe language L.

Unification Theorem B

TheoremLet A be an equationally Noetherian algebra in a language L(with no predicates). Then for a finitely generated algebra C ofL the following conditions are equivalent:

1 C ∈ Qvar(A), i.e., Thqi(A) ⊆ Thqi(C);2 C ∈ Pvar(A);3 C embeds into a direct power of A;4 C is separated by A;5 C is a subdirect product of finitely many limit algebras overA;

6 C is an algebra defined by a complete atomic type in thetheory Thqi(A) in L;

7 C is the coordinate algebra of a non-empty algebraic setover A defined by a system of equations in the language L.

Equationally Noetherian Algebras

DefinitionAn algebra A is equationally Noetherian, if for any positiveinteger n and any system of equations S(x1, . . . , xn) thereexists a finite subsystem S0 ⊆ S such that V(S) = V(S0).

RemarkIn above definition the information on some language L ishidden. When saying that some group G is equationallyNoetherian it is necessary to point out the implicit language.

Equationally Noetherian Algebras

DefinitionAn algebra A is equationally Noetherian, if for any positiveinteger n and any system of equations S(x1, . . . , xn) thereexists a finite subsystem S0 ⊆ S such that V(S) = V(S0).

RemarkIn above definition the information on some language L ishidden. When saying that some group G is equationallyNoetherian it is necessary to point out the implicit language.

Equationally Noetherian: Yes or No?

1 Noetherian commutative ring Yes

2 Linear group over Noetherian ring (free group, polycyclic group, finitely generated metabelian group)

3 Torsion‐free hyperbolic group Yes

4 Free solvable group Yes

5 Finitely generated metabelian (or nilpotent) Lie algebra Yes

6 Infinitely generated nilpotent group No

7 Wreath product A ζ B of a non‐abelian group A and an infinite group B No

8 The Grigorchuk group Γ No

9 The min‐max structures ⟨ ;max,min, , , ,0,1+ − ⋅ ⟩R ;max,min, ,0,1⟨ + ⟩N and No

10 Free Lie algebra ?

11 Free anti‐commutative algebra ?

12 Free associative algebra ?

13 Free products of equationally Noetherian groups ?

Equationally Noetherian algebras

LemmaLet A be an equationally Noetherian algebra. Then thefollowing algebras are equationally Noetherian too:

1 every subalgebra of A;2 every filterpower (ultrapower, direct power) of A;3 coordinate algebra Γ(Y ) of an algebraic set Y over A;

Corollaryevery algebra from Qvar(A);every algebra from Ucl(A).

Equationally Noetherian algebras

LemmaLet A be an equationally Noetherian algebra. Then thefollowing algebras are equationally Noetherian too:

1 every subalgebra of A;2 every filterpower (ultrapower, direct power) of A;3 coordinate algebra Γ(Y ) of an algebraic set Y over A;

Corollaryevery algebra from Qvar(A);every algebra from Ucl(A).

Zariski Topology

There are three perspectives for investigation in algebraicgeometry over algebra A: algebraic, geometrical and logic.Geometrical approach is connected with examination of affinespace An as topological space.

We define Zariski topology on An, where algebraic sets over Aform a subbase of closed sets, i.e., closed sets in this topologyare obtained from the algebraic sets by finite unions and(arbitrary) intersections.

Zariski Topology

There are three perspectives for investigation in algebraicgeometry over algebra A: algebraic, geometrical and logic.Geometrical approach is connected with examination of affinespace An as topological space.

We define Zariski topology on An, where algebraic sets over Aform a subbase of closed sets, i.e., closed sets in this topologyare obtained from the algebraic sets by finite unions and(arbitrary) intersections.

Zariski Topology for Equationally Noetherian Algebra

LemmaFor an algebra A the following conditions are equivalent:

1 A is equationally Noetherian;2 for any natural number n the Zariski topology on An is

Noetherian (satisfies the descending chain condition onclosed subsets);

3 every chain

Γ(Y1) → Γ(Y2) → Γ(Y3) → . . .

of proper epimorphisms of coordinate algebras of algebraicsets Y1, Y2, Y3, . . . over A is finite.

Irreducible Algebraic Sets

TheoremEvery algebraic set Y over equationally Noetherian algebra Acan be expressed as a finite union of irreducible algebraic sets(irreducible components):

Y = Y1 ∪ Y2 ∪ . . . ∪ Ym.

Furthermore, this decomposition is unique up to permutation ofirreducible components and omission of superfluous ones.

DefinitionAn algebraic set Y is irreducible if it is not a finite union ofproper algebraic subsets.

Irreducible Algebraic Sets

TheoremEvery algebraic set Y over equationally Noetherian algebra Acan be expressed as a finite union of irreducible algebraic sets(irreducible components):

Y = Y1 ∪ Y2 ∪ . . . ∪ Ym.

Furthermore, this decomposition is unique up to permutation ofirreducible components and omission of superfluous ones.

DefinitionAn algebraic set Y is irreducible if it is not a finite union ofproper algebraic subsets.

Irreducible Coordinate Algebras

TheoremLet A be an equationally Noetherian algebra in a language L.A finitely generated algebra C in L is the coordinate algebra ofan algebraic set over A if and only if it is a subdirect product offinitely many coordinate algebras of irreducible algebraic setsover A.

CorollaryClassification of irreducible algebraic sets (and/or theircoordinate algebras) is the essential problem of algebraicgeometry over equationally Noetherian algebra.

Irreducible Coordinate Algebras

TheoremLet A be an equationally Noetherian algebra in a language L.A finitely generated algebra C in L is the coordinate algebra ofan algebraic set over A if and only if it is a subdirect product offinitely many coordinate algebras of irreducible algebraic setsover A.

CorollaryClassification of irreducible algebraic sets (and/or theircoordinate algebras) is the essential problem of algebraicgeometry over equationally Noetherian algebra.

Universal Algebraic Geometry Part 1omskconf2009.oscsbras.ru/daniyarova1.pdf · Universal Algebraic...

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