Generalized Syzygies for Commutative Koszul … Galvez...The algebra of syzygies (I) De nition: the algebra of syzygies Let A be a commutative C-algebra which is a module over S(V),

Generalized Syzygiesfor Commutative Koszul Algebras

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

Imma Galvez Carrillo

Universitat Politecnica de Catalunya, Departament de Matematica Aplicada III,EET, Terrassa

CSASC 2011 ”Categorical Algebra, Homotopy Theory, andApplications”

Donau Universitat KremsSeptember 27, 2011

Imma Galvez Carrillo UPC-Terrassa

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 .

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



Cohomology of L≥2

Theorem [4, 12, 9]

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.

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

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

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



Lie ideals and the algebra of syzygies

So 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.


















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-Terrassa


Berkovits complex (II)

Definition

The Berkovits complex of a commutative Koszul algebra A is

CB(A) := 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.



Generalized syzygies

Main Theorem [GGST 2011]

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

H∗(CB(A), dB) ∼= H∗(L≥3,C).

H∗(CB(A), dB) is called the algebra of generalized syzygies of A



Tools for the proof: A result on DG algebras

Key Lemma

Let (C , d) be a commutative DG algebra over C,nonnegatively graded and finitely generated in each degree.

Let B be a contractible DG subalgebra of C , withquasi-isomorphism ε : B → C.

Consider the DG ideal 〈B〉 of C generated by theaugmentation ideal B = ker(ε).

If 〈B〉 is freely generated as a B-module by a graded basis ofhomogeneous elements Z =

⋃i≥0 Zi , then C is

quasi-isomorphic to C/〈B〉.




Key Lemma










Key Lemma










Key Lemma









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.





(dC f )(x0, x1) = −f [x0, x1].




)∗and ψ ∈

(∧j L)∗

,

ϕ� ψ ∈(∧i+j L






(dC f )(x0, x1) = −f [x0, x1].




)∗and ψ ∈

(∧j L)∗

,

ϕ� ψ ∈(∧i+j L




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.

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

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

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-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.



Lifting the Berkovits differential

The setup

We have a basis {q1, . . . , qm} of L∗2 such that

qk =n∑

i ,j=1

Γkij{ai , aj},

and by construction,

dC (qk) =n∑

i ,j=1

Γkijaiaj

for k = 1, . . . ,m.



Lifting the Berkovits differential (II)

The Berkovits differential

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.



Lifting the Berkovits differential (III)

The correction term

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.



Lifting the Berkovits differential (IV)

Proposition

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



The Resolution of C inside Y (I)

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.



The Resolution of C inside Y (II)

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.

Step 3

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



Proof of Main Theorem (I)

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)).



Proof of Main Theorem (I)

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)).



Proof of Main Theorem (II)

Now, consider the filtration of Y

{0} ⊂ F0Y ⊂ F1Y ⊂ · · · ⊂ FnY ⊂ . . . ,

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.





{0} ⊂ F0Y ⊂ F1Y ⊂ · · · ⊂ FnY ⊂ . . . ,

given by

FpYq :=∑j≤p

∑i+j=p+q

Chi (L)⊗(∧

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


Since,∧






{0} ⊂ F0Y ⊂ F1Y ⊂ · · · ⊂ FnY ⊂ . . . ,

given by

FpYq :=∑j≤p

∑i+j=p+q

Chi (L)⊗(∧

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


Since,∧




Proof of main theorem (III)

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 dB .

Hence, the homology of this complex is also H∗(Cb(A), dB)the homology of the Berkovits complex.



Proof of main theorem (III)

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 dB .

Hence, the homology of this complex is also H∗(Cb(A), dB)the homology of the Berkovits complex.



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



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Gorodentsev, A.; Khoroshkin, A.; Rudakov A.; On syzygies ofhighest weight orbits, Amer. Math. Soc. Transl. 221 (2007).

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Generalized Syzygies for Commutative Koszul … Galvez...The algebra of syzygies (I) De nition: the algebra of syzygies Let A be a commutative C-algebra which is a module over S(V),