Koszul Duality and Cohomology: Generalized Syzygies for ... · commutative algebras. There, the...

Koszul Duality and Cohomology: GeneralizedSyzygies for Commutative Koszul Algebras

Joint Work in Progress with Vassily Gorbounov (Aberdeen)Zain Shaikh (Cologne) and Andrew Tonks (Londonmet)

Imma Galvez Carrillo

UPC-EET, Terrassa

UAB Algebra and Combinatorics Seminar 2011CRM

May 26, 2011

Imma Galvez Carrillo UPC-EET, Terrassa

Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras

Towards generalized syzygies

Definition: Koszul homology

Let {a1, . . . , an} be a sequence of elements in a commutativeC-algebra A. Let W be an n-dimensional complex vector spacewith basis {θ1, . . . , θn}. The Koszul homology of A with respectto the sequence {a1, . . . , an} is the homology of the complex

A⊗∧

W ,

where A has homological degree zero, each θi has homologicaldegree one, and the Koszul differential is given by the formula

dK =n∑

i=1

ai∂

∂θi.



Koszul Algebras (I)

Definition: Quadratic algebra

Let A be a positively graded connected algebra, locallyfinite-dimensional. A is called quadratic if it is determined by avector space of generators V = A1 and subspace of quadraticrelations I ⊂ A1 ⊗ A1

Definition: Koszul quadratic dual

The Koszul dual algebra A! associated with a quadratic algebra Ais

A! = T (V ∗)/I⊥,

where I⊥ ⊂ V ∗ ⊗ V ∗ is the annihilator of I . Clearly A!! = A.



Koszul algebras (II)

Definition: Koszul algebra

A quadratic as above is a Koszul algebra iff

A! ∼= Ext∗A(k,k)



Lie superalgebras (I)

Definition: Lie superalgebra

A Lie superalgebra over C is a Z/2-graded vector space (over C)L = L(0) ⊕ L(1) with a map [·, ·] : L⊗ L→ L of Z/2-graded spaces,satisfying:

1 (anti-symmetry) [x , y ] = −(−1)|x ||y |[y , x ] for all homogeneousx , y ∈ L,

2 (Jacobi identity)(−1)|x ||z|[x , [y , z ]]+(−1)|y ||x |[y , [z , x ]]+(−1)|z||y |[z , [x , y ]] = 0for all homogeneous x , y , z ∈ L.



Lie superalgebras (II)

Definition: Lie superalgebra (continued)

Here |x | is the parity of x : |x | = i when x ∈ L(i) for i = 0, 1. Anelement x in L(0) or L(1) is termed even or odd respectively. Werecover the familiar definition of a Lie algebra (over C) in the caseL = L(0).

Definition: graded Lie superalgebra

A graded Lie superalgebra is a Lie superalgebra L together witha grading compatible with the bracket and supergrading. That is,L =

⊕m≥1 Lm such that [Li , Lj ] ⊂ Li+j and L(i) =

⊕m≥1 L2m−i

for i = 0, 1.



Koszul dual as universal envelope

Koszul dual as universal envelope

Assume that A = T (V )/I is commutative, hence∧2 V ⊂ I .

Therefore I⊥ is contained in S2(V ∗), and so is generated by certainlinear combinations of anti-commutators [a∗i , a

∗j ] = a∗i a∗j + a∗j a∗i .

As a consequence, the Koszul quadratic dual of A can be describedas the universal envelope of a graded Lie superalgebra,

A! = U(L), L =⊕m≥1

Lm = L(V ∗)/J, (1)

where L is the free Lie superalgebra functor, the space of (odd)generators V ∗ is concentrated in degree 1, and J is the Lie idealwith the same generators as I⊥ but viewed as linear combinationsof supercommutators.



Some Lie ideals

The following Lie ideals are main characters of our work.

Definition

For k ≥ 2, we define the graded Lie superalgebras

L≥k =⊕m≥k

Lm.



The algebra of syzygies (I)

Definition: the algebra of syzygies

Let A be a commutative C-algebra which is a module over S(V ),and suppose we have a minimal free resolution of A,

· · · → F2 → F1 → F0 → A→ 0.

This is an exact sequence of graded free S(V )-modules,

Fp =⊕q≥mp

Rpq ⊗ S(V ),

where Rpq is the finite dimensional vector spaces of p-th ordersyzygies of degree q for A, and mp is the minimum degreeamong the p-th order syzygies.



The algebra of syzygies (II)

Syzygies and the Koszul complex

Since the chosen resolution is minimal, the differentialvanishes on tensoring this complex with the trivialS(V )-module C.

HenceTor

S(V )p (A,C) = Rp :=

⊕q≥mp

Rpq.



The algebra of syzygies (III)

The functor TorS(V ) may also be calculated by resolving theother argument C.

The Koszul complex K (S(V )) of the symmetric algebraS(V ) = C[a1, . . . , an], is given by (S(V )⊗

∧W , dK ), that is,

(C[a1, . . . , an]⊗∧

(θ1, . . . , θn), dK ). (2)

with the Koszul differential dK of the sequence {a1, . . . , an}.



The algebra of syzygies (IV)

Since this complex is a resolution of C we can calculate thesyzygies Rp of the quadratic algebra A = T (V )/I by the homologyof the complex

A⊗S(V ) K (S(V )) = A⊗S(V ) S(V )⊗C∧

W = A⊗∧

W . (3)

This homology inherits a multiplication from K (S(V )) andbecomes an associative algebra.



Lie ideals and the algebra of syzygies

H∗(L≥2,C) gives indeed the algebra of syzygies of A.

The interpretation of the algebras L≥k for k > 2 was outlinedby Berkovits in [4].

We concentrate on the case k = 3.



Introduction: Berkovits’ example

In his work on string quantization and string/gauge theoryduality [4], Berkovits uses crucially Koszul duality theory forcommutative algebras.

There, the coordinate algebra of the orthogonal GrassmannianA = OGr(5, 10), related to the spinor representation of thegroup SO(10,C) is considered.

This algebra A is quadratic, and hence we know that itsKoszul dual A! is U(L) the universal enveloping algebra of acertain (graded) Lie superalgebra L =

⊕i≥1 Li .

[4] Berkovits, N.; Cohomology in the pure spinor formalism for thesuperstring, J. High Energy Phys. 9 (2000).



Berkovits’ first observation, made in the context of string andgauge theories, was that the algebra of syzygies of A isisomorphic to the cohomology H∗(L≥2,C) of the Liesuperalgebra L≥2 =

⊕i≥2 Li .

He proposed an extension of the Koszul complex of thecoordinate algebra of OGr(5, 10), relevant for his quantizationprocedure, and considered the question of calculating itshomology.



This extension is worthy of study for more general algebras.

Berkovits’ construction is an iteration of the Koszul homologyof a sequence of elements of an algebra, first applied to thealgebra A and then applied to its syzygies.

As we recalled there is a notion of Koszul homology withrespect to a sequence of elements {a1, .., an} of acommutative algebra A.

And we recalled as well that if A is finitely generated andpresented as C[a1, ..., an]/I , the Koszul homology with respectto the sequence {ai} gives the algebra of syzygies of A.



The generators Γj of the ideal I represent syzygies in thelowest degree.

The Koszul homology of the algebra of syzygies with respectto the sequence Γj is the homology studied by Berkovits. Thisis what we call the algebra of generalized syzygies.

It turns out that if A is Koszul, the generalized syzygies canalso be described explicitly in terms of the Lie superalgebra L.This was shown by Movshev and Schwarz in [12] for the caseof the coordinate algebra of the OGr(5, 10).

There was proved that in that case the homology of theBerkovits complex is isomorphic to the cohomologyH∗(L≥3,C) of the Lie superalgebra L≥3 =

⊕i≥3 Li .

[12] Movshev, M.; Schwarz, A.; On maximally supersymmetric Yang-Millstheories, Nuclear Physics B 681 (2004)



Our aim is to extend these statements about H∗(L≥3,C) fromthe coordinate algebra of the orthogonal Grassmannian to anarbitrary finitely generated commutative Koszul algebra.

Further examples are supplied by the orbits of the highestweight vector in an irreducible representation of a semisimplecomplex Lie group, as studied by Gorodentsev, Khoroshkinand Rudakov [9].

We prove in our Main Theorem that the cohomology of theLie algebra L≥3 is isomorphic to the Berkovits homology of A.

[9] Gorodentsev, A.; Khoroshkin, A.; Rudakov A.; On syzygies of highestweight orbits, Amer. Math. Soc. Transl. 221 (2007).



Since A is finitely generated, Hilbert’s syzygy theorem saysthat the algebra of syzygies of A is finite dimensional.

We expect the Lie superalgebra L≥k ⊂ L≥3 to be a free Liesuperalgebra for some k .

Movshev and Schwarz, in [12], proved that L≥3 is a free Liesuperalgebra on an infinite set of generators in the case of thecoordinate algebra of pure spinors.

From the point of view of the gauge theory, freeness of L≥k isimportant because it implies that the solution to theEuler-Lagrange equations for an abelian gauge group can bedeformed to the solutions for a non-abelian gauge group.



A commutative Koszul algebra A has a nice resolution in thecategory of algebras, given by the Chevalley complex of theLie superalgebra L.

Our main idea is to lift the Koszul differential which calculatesthe generalized syzygies to this resolution and subsequentlycalculate its homology.

This requires a correction term to the naively lifted differentialto guarantee that the lift squares to zero.

This makes heavy use of the homological perturbation lemmaa la Barnes and Lambe [3].

[3] Barnes, D.; Lambe, L.; A fixed point approach to homologicalperturbation theory, Proc. Amer. Math. Soc. 112 (1991), no. 3, 881–892.



Berkovits homology or generalized syzygies

Main Theorem

The algebra of generalized syzygies of a commutative Koszulalgebra A is isomorphic to H∗(L≥3,C).



A result on DG algebras

Key Lemma

Let (C , d) be a commutative DG algebra over C, nonnegativelygraded and finitely generated in each degree. Let B be acontractible DG subalgebra of C , with quasi-isomorphismε : B → C, and consider the DG ideal 〈B〉 of C generated by theaugmentation ideal B = ker(ε). If 〈B〉 is freely generated as aB-module by a graded basis of homogeneous elementsZ =

⋃i≥0 Zi , then C is quasi-isomorphic to C/〈B〉.



The Chevalley complex

Assume that A is a commutative Koszul algebra with A! = U(L) asabove. We construct a resolution of A in the category of DGalgebras from the Chevalley complex of L.

Definition: the Chevalley complex

The Chevalley complex of L is the cochain complex with

Chi (L) =

(∧iL

)∗and the differential dC : Chk(L)→ Chk+1(L)

(dCϕ)(x0, . . . , xk) =∑i<j

(−1)j+ε(i ,j)ϕ(x0, . . . , xi−1, [xi , xj ], xi+1, . . . , xj , . . . , xk)

if each xr is homogeneous, where xj means omit xj andε(i , j) = |xj |(|xi+1|+ · · ·+ |xj−1|).



Some properties of the Chevalley complex

dC : Ch1(L)→ Ch2(L) is the map that is dual to the bracket,

(dC f )(x0, x1) = −f [x0, x1].

The Chevalley complex is a cochain complex that calculatesthe cohomology of L with trivial coefficients, H∗(L,C).

This complex admits an algebra structure that descends toone on the cohomology of L so that the differential dC is aderivation with respect to its product.

That is, given ϕ ∈(∧i L

)∗and ψ ∈

(∧j L)∗

,

ϕ� ψ ∈(∧i+j L

)∗and dC (ϕ� ψ) = dCϕ� ψ ± ϕ� dCψ.



The Chevalley complex as a chain complex

As well as being a cochain complex whose cohomology is thatof L, the Chevalley complex may also be considered a chaincomplex which defines a resolution of A.

The chain complex is given by defining L∗p to havehomological grading p − 1, so that the homological andcohomological gradings together give the total degree in

∧L∗.



The Chevalley complex as a chain complex (II)

We illustrate the Chevalley complex as a chain complex withhomological grading as follows:

Ch3(L)dC // Ch2(L)

dC // Ch1(L)dC // Ch0(L) // 0

0 // L∗1// 0

0 // L∗2//∧2 L∗1

// 0

0 // L∗3// L∗2 ∧ L∗1

//∧3 L∗1

// 0

The original cohomological grading is seen on the diagonals.Imma Galvez Carrillo UPC-EET, Terrassa


The Chevalley complex as resolution

In our situation of a graded Lie superalgebra L with A! = U(L) weobserve that

Ch0(L) =∧

L∗1 = S(V ).

Proposition

For a commutative Koszul algebra A with A! = U(L), the chaincomplex given by the Chevalley complex of L with homologicalgrading is a resolution of A.



The Chevalley complex as resolution (II)

Proof of proposition

The Chevalley complex of L with cohomological degree given bythe number of exterior powers calculates

H i (L,C)j ∼= ExtijU(L)(C,C).

Since A is a commutative Koszul algebra, we have

A =⊕i≥0

ExtiiA!(C,C)

and ExtijA!(C,C) = 0 for i 6= j



Chevalley complex as resolution (III)

Proof (continued)

But A! = U(L) and

Hi (Ch(L))j = H j−i (L,C)j ,

where on the left hand side we are using our homological gradingof Ch(L). Therefore,

Hi (Ch(L)) =

{A if i = 00 if i 6= 0

and the Chevalley complex of L is a resolution of A.



Cohomology of L≥2

Theorem

Let A = T (V )/I be a commutative Koszul algebra, R = ⊕pRp isits algebra of syzygies and A! := T (V ∗)/I⊥ = U(L) is its Koszuldual. Then Rpq

∼= Hq−p(L≥2,C)q as algebras.



Idea of proof

Denote by X the tensor product Ch(L)⊗∧

W withgenerators of W in degree one.

Consider the differential dC + dK on X , which one can checkmakes this into a complex.

We put an algebra structure on the complex X by thefollowing rule: if α, β ∈ Ch(L) and η, ξ ∈

∧W , then

(α⊗ η) · (β ⊗ ξ) := (−1)|η||β|α ∧ β ⊗ η ∧ ξ.



Idea of proof continued

Since Ch0(L) = S(V ), this implies thatK (S(V )) = Ch0(L)⊗

∧W is a subalgebra in X .

The restriction of the differential to this subalgebra is dK , andhence is a resolution of C.

This satisfies the conditions of Key Lemma above and X isquasi-isomorphic to X/〈K (S(V ))〉, where 〈K (S(V ))〉 is theDG ideal in X generated by the augmentation ideal ofK (S(V )). Hence,Hi (X ) = Hi (X/〈K (S(V ))〉) = Hi (Ch(L≥2)), asCh0(L) = Ch(L1).



Idea of proof continued

Now, consider the following filtration of X

{0} ⊂ F0X ⊂ F1X ⊂ · · · ⊂ FnX ⊂ . . . ,given by

FpXq :=∑j≤p

∑i+j=p+q

Chi (L)⊗j∧

W .

The differential on the E0-term of the spectral sequenceassociated to this filtration is dC . Since

∧W is a vector space

over C, it is flat as a C-module. Further, Ch(L) is a resolutionfor A.We can conclude that the E1-term of the spectral sequence iscontained in one line given by A⊗

∧W with differential dK .

Hence, Ri∼= Hi (X ).



Idea of proof ended

It is clear that both interpretations of the homology respectthe multiplicative structure: X/〈K (S(V ))〉 inherits itsmultiplicative structure from X and the spectral sequence willalso respect it.

The Chevalley complex Ch(L) calculates the cohomology ofLie superalgebra L, where the cohomological grading of Ch(L)is given by the number of exterior powers.

Therefore, Rpq = Hp(X )q = Hp(Ch(L≥2))q = Hq−p(L≥2,C)q.



Berkovits Complex

Let A = C[a1, . . . , an]/I be commutative Koszul, with minimal setof generators {Γ1, . . . , Γm} of I representing lowest degree syzygies.

Lemma

If the quadratic relations for A are defined by the formulas

Γk =n∑

i ,j=1

Γkijaiaj ,

for k = 1, . . . ,m, then the representative for the homology class inthe algebra of syzygies defined by the sequence {Γ1, . . . , Γm} is

Γk =n∑

i ,j=1

Γkijaiθj , k = 1, . . . ,m

.Imma Galvez Carrillo UPC-EET, Terrassa


Berkovits complex (II)

Definition

The Berkovits complex of a commutative Koszul algebra A is

A⊗∧

(θ1, . . . , θn)⊗ C[y1, . . . , ym]

equipped with the Berkovits differential

dB = dK + dBer =n∑

i=1

ai∂

∂θi+

m∑k=1

n∑i ,j=1

Γkijaiθj

∂

∂yk,

where the yk have homological degree two.(One can prove that d2

B = 0).



Lifting the differential

The setup

As before L =⊕

Li where U(L) = A!. The vector space L2 hasthe same dimension as S2(V ∗)/I⊥ ∼= I ∗ and the generators of I arelinearly independent, so we have a basis {q1, . . . , qm} of L∗2 suchthat

qk =n∑

i ,j=1

Γkij{ai , aj},

and by construction,

dC (qk) =n∑

i ,j=1

Γkijaiaj

for k = 1, . . . ,m.



Lifting the differential (II)

The idea

Define Y as:

Ch(L)⊗∧

(θ1, . . . , θn)⊗ C[y1, . . . , ym],

with an obvious graded algebra structure.We can try to lift the Berkovits differential to Y as follows:dC + dK + dBer where

dBer =n∑

i ,j=1

m∑k=1

Γkijaiθj

∂

∂yk.



However, one checks that

(dC + dK + dBer )2 =n∑

i ,j=1

m∑k=1

Γkijaiaj

∂

∂yk6= 0.

In order for the differential to square to zero, we define a correctionto the differential as

dS = −m∑

k=1

qk∂

∂yk.



Proposition

(dC + dK + dBer + dS)2 = 0.



dKdS + dSdK = dBerdS + dSdBer = d2S = 0,

and

dCdS+dSdC = −n∑

i ,j=1

m∑k=1

Γkijaiaj

∂

∂yk+

m∑k=1

qk∂

∂ykdC−

m∑k=1

qk∂

∂ykdC .

Therefore,

(dC +dK +dBer +dS)2 = (dC +dK +dBer )2−n∑

i ,j=1

m∑k=1

Γkijaiaj

∂

∂yk= 0.



The Resolution of C inside Y

Proposition

The subalgebra T of Y given by:

Ch(L1, L2)⊗∧

(θ1, . . . , θn)⊗ C[y1, . . . , ym]

is a resolution of C.

This subcomplex is equipped with the differential

n∑i=1

ai∂

∂θi+

m∑k=1

n∑i ,j=1

Γkijaiθj − qk

∂

∂yk+

m∑k=1

n∑i ,j=1

Γkijaiaj

∂

∂qk.



Idea of the proof

Step 1

The Koszul complex (P, dK ) of the sequence θ1, . . . , θn in Ch(L1)and the Koszul complex (Q, dS) of y1, . . . , ym in Ch(L2) arecontractible, as is the product (P ⊗ Q, dK + dS).

Step 2

We perturb the differential on P ⊗ Q to the differential on T byusing the homological perturbation lemma of [3].

Step 3

The perturbed homotopy is well-defined, so T has a strongdeformation retraction to (C, 0) concentrated in degree zero.



Step 1

Lemma

The complexes (P, dK ) and (Q, dS) are contractible.In fact, there are strong deformation retractions

C[a1, . . . , an]⊗∧

(θ1, . . . , θn)ϕK

++ ε // Cι

oo

∧(q1, . . . , qm)⊗ C[y1, . . . , ym]

ϕS

++ ε // Cι

oo

where the right hand side is concentrated in degree zero, withtrivial differential.The same is true for the complex (P ⊗ Q, dK + dS).



Step 2

However, ∂ = dK + dS is not the differential we are interested in;we want to consider instead the complex with a perturbeddifferential,

(P ⊗ Q, ∂ + ∂′).

Here ∂′ is given by dC + dBer ,

m∑k=1

n∑i ,j=1

Γkijaiθj

∂

∂yk+ Γk

ijaiaj∂

∂qk.

Then the complex is precisely T .



Step 2 continued

In order to show the complex with the new differential is alsocontractible we apply the homological perturbation lemma (see forexample [3]). This says there is a unique perturbation of thedeformation retraction

(P ⊗ Q, ∂)ϕ++ ε // C

ιoo

to a deformation retraction

(P ⊗ Q, ∂ + ∂′)ϕ′

++ ε // Cι

oo

if and only if the original contracting homotopy ϕ and theperturbation of the differential ∂′ satisfy a local nilpotencecondition.



Step 2 ended

That is, for each element γ the expression

(∂′ϕ)N(γ)

is zero for sufficiently large N = N(γ). In this case

ϕ

( ∞∑n=0

(∂′ϕ)n)

is well-defined and will be the required contracting homotopy ϕ′.



Step 3

We proceed to prove the local nilpotency condition required.

Suppose α⊗β ∈ P ⊗Q with β ∈∧≤s(q1, . . . , qm)⊗C[y1, . . . , ym].

If s = 0 then ∂′ϕ(α⊗ β) = 0, and if s > 0 then

∂′ϕ(α⊗ β) ∈ P ⊗∧≤(s−1)(q1, . . . , qm)⊗ C[y1, . . . , ym]

Hence (∂′ϕ)m+1 = 0.



Generalized syzygies

Main Theorem

Let A be a commutative Koszul algebra and A! = U(L). Then,

H∗Ber (A) ∼= H∗(L≥3,C).



Idea of proof

We have that T is a subalgebra in Y .

We showed in a Proposition above that T is a resolution of C.This satisfies the conditions of the Key Lemma and Y isquasi-isomorphic to Y /〈T 〉, where 〈T 〉 is the DG ideal in Ygenerated by the augmentation ideal of T . Hence,Hi (Y ) = Hi (Y /〈T 〉) = Hi (Ch(L≥3)).



Idea of proof, continued

Now, consider the filtration of X

{0} ⊂ F0X ⊂ F1X ⊂ · · · ⊂ FnX ⊂ . . . ,

given by

FpYq :=∑j≤p

∑i+j=p+q

Chi (L)⊗(∧

(θ1, . . . , θn)⊗ C[y1, . . . , ym])j.

The differential on the E0-term of the spectral sequenceassociated to this filtration is dC .

Since,∧

(θ1, . . . , θn)⊗ C[y1, . . . , ym] is a vector space over C,it is flat as a C-module.



Idea of proof, ended

As Ch(L) is a resolution for A, we can conclude that theE1-term of the spectral sequence is contained in one line andis given by,

A⊗∧

(θ1, . . . , θn)⊗ C[y1, . . . , ym]

with precisely the Berkovits differential.

Hence, the homology of this complex is also H∗Ber (A).

The result follows by converting the grading to thecohomological grading of the Chevalley complex.



Aisaka, Y.; Kazama, Y.; A New First Class Algebra,Homological Perturbation and Extension of Pure SpinorFormalism for Superstring, UT-Komaba 02-15hep-th/0212316 Dec, 2002.

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Berkovits, N.; Cohomology in the pure spinor formalism for thesuperstring, J. High Energy Phys. 9 (2000).



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