KOSZUL DUALITY BETWEEN BETTI AND COHOMOLOGY
NUMBERS IN CALABI-YAU CASE

arXiv:1711.06931v1 [math.AG] 18 Nov 2017

ALEXANDER PAVLOV

Abstract. Let X be a smooth projective Calabi-Yau variety and L a Koszul line bundle
on X. We show that for Betti numbers of a maximal Cohen-Macaulay module over
the homogeneous coordinate ring A of X there are formulas similar to the formulas for
cohomology number. This similarity is realized via the box-product resolution of the
diagonal ∆X ⊂ X × X.

1. Introduction
Let X be a smooth projective variety and L is a very ample line bundle and A =
L
0
m
m≥0 H (X, L ) is the homogeneous coordinate ring. The affine cone over X is an
isolated singularity. With this singularity one can associate the triangulated category of
singularity, in commutative algebra this category also known under the name of stable
category of maximal Cohen-Macaulay modules.
For Calabi-Yau varieties it was shown by Orlov [5] that the singularity category of A is
equivalent as to the bounded derived category D b (X) = D b (Coh(X)) of coherent sheaves
on X.
With a coherent sheaf F on X or more generally an object of D b (X) one can associate the
following numerical invariants
hij (F ) = dim Hi (X, F (j)),
we call this the cohomology numbers of F . With a finitely generated module M over A
one can associate numerical invariants known as Betti numbers
βij (M ) = dim ToriA (M, k(j)).
The goal of this paper is to show that if the homogeneous coordinate ring A is additionally
Koszul then there are formulas for Betti numbers related to the definition of cohomology
numbers in the same way as Koszul dual algebra B = A! is related to A.
Note that if L is any ample line bundle then a sufficiently large power Ld is a Koszul line
bundle meaning that corresponding homogeneous coordinate ring is Koszul.
A geometric manifestation of Koszulity of the homogeneous coordinate ring A is the resolution of the structure sheaf of the diagonal ∆X ⊂ X × X that has the box-product
structure in each term
· · · → O(−m) ⊠ Rm → O(−m + 1) ⊠ Rm−1 → . . .
→ O(−1) ⊠ R1 → OX ⊠ OX → O∆X → 0,
2010 Mathematics Subject Classification. Primary: 13D02 Secondary: 14F05, 14J32.
Key words and phrases. Betti numbers, Gorenstein ring, Koszul Duality, Maximal Cohen-Macaulay
modules.
1

2

ALEXANDER PAVLOV

where Rm are sheaves on X and the box-product is by definition F ⊠ G = π1∗ (F ) ⊗X×X
π2∗ (G), where π1 and π2 are projections of X × X to X.
If we rewrite formula for cohomology numbers as follows
hij (F ) = dim Hom(O(−j), F [i]),
where Hom stands for Hom-space in the derived category D b (X). Then (up to a shift)
Betti numbers are given by the same formula, but O(−j) need to be replaced by its partner
R−j in box-product resolution above.
More precisely we have the following
Theorem 1.1. Let X be a smooth projective Calabi-Yau variety, and let L be a Koszul
line bundle s.t. H i (X, O(j)) = 0 for i, j > 0. If F ∈ D b (X) and M is the corresponding
MCM module then
βij (M ) = dim Hom(R−j , F [d − 1 + j − i]),
for i, j ∈ Z.
The papers is organized as follows. We give brief preliminaries on maximal CohenMacaulay modules and singularity category, details can be found in [2] and [5]. Then
we summarize application of Orlov’s equivalence to Betti numbers as it was developed in
[6]. For the resolution of the diagonal we closely follow [3].
2. Maximal Cohen-Macaulay modules and Singularity Category
L
Let A = m≥0 Am be a graded commutative algebra over an algebraically closed field k
of characteristic zero. In this paper A is connected that is A0 = k and Gorenstein
RHomA (k, A) ∼
= k(a)[−n],
where the parameter a ∈ Z is called the Gorenstein parameter of A.
If X is a smooth Calabi-Yau projective variety and L is a very ample line bundle on X
then the homogeneous coordinate ring
M
M
A=
Am =
H 0 (X, Lm )
m≥0

m≥0

is generated in degree on, connected, Gorenstein with parameter a = 0.
The abelian category of coherent sheaves Coh(X) is equivalent, by Serre’s theorem, to the
quotient category qgr(R)
Coh(X) ∼
= qgr(A),
here the abelian category qgr(A) is defined as a quotient of the abelian category of finitely
generated graded A-modules by the Serre subcategory of torsion modules
qgr(A) = modgr(A)/ torsgr(A),
where torsgr(A) is the category of graded modules that are finite dimensional over k.
The bounded derived category of the abelian category of finitely generated graded Amodules D b (gr−A) = D b (modgr(A)) has a full triangulated subcategory consisting of objects that are isomorphic to bounded complexes of projectives. The latter subcategory
can also be described as the derived category of the exact category of graded projective
modules, we denote it D b (grproj−R). The triangulated singularity category of R is defined
as the Verdier quotient
gr
DSg
(A) = D b (gr−A)/D b (grproj−A).

KOSZUL DUALITY BETWEEN BETTI AND COHOMOLOGY NUMBERS IN CALABI-YAU CASE

3

In commutative algebra the singularity category is known under the name stable category
of maximal Cohen-Macaulay modules. It was introduced and studied by R.-O. Buchweitz
[2].
If M is a finitely generated module over A, then depth of M is bounded by the dimension
of M
depth(M ) ≤ dim(M ) ≤ dim(A).
If depth(M ) = dim(M ), then the module M is called Cohen-Macaulay, and if depth(M ) =
dim(A) the module M is called a maximal Cohen-Macaulay (MCM). In the stable category
of MCM modules objects are MCM modules and stable homomorphism groups are defined
by
Hom(M, N ) = Hom(M, N )/{morphisms that factor through a projective}.
In particular, all projective modules (and projective summands) in the stable category
are identified with the zero module. We denote this category MCMgr(A). Note that the
operation of taking syzygies is functorial on the stable category, we denote this functor syz .
The functor syz is an autoequivalence of MCMgr(A), the inverse of this functor we denote
cosyz . Moreover, if A is an isolated singularity the stable category of MCM modules is
Hom-finite and admits a structure of triangulated category with autoequivalence cosyz.
One of the main results of [2] is the following
Theorem 2.1. There is an equivalence of triangulated categories
gr
MCM (A) ∼
= D (A) .
gr

Sg

The triangulated singularity category of A is related to the bounded derived category of
X, as was shown in [5]. We briefly summarize the result in one special case used in this
paper.
The truncated derived global sections functor RΓ≥l : D b (X) → D b (gr−A≥l ) is given by
the following direct sum:
M
RΓ≥l (F ) =
RHom(OX , F (i)),
i≥l

where D b (gr−A≥l ) denotes the derived category of the category of finitely generated modules concentrated in degrees l and higher.
gr
The functor st : D b (gr−A≥i ) → DSg
(A) is the stabilization functor constructed by Orlov.

The special case of Orlov’s theorem (see Theorem 2.5 in [5]) that we need in this paper
can be formulated as follows.
Theorem 2.2. Let X be a smooth projective Calabi-Yau variety, A its homogeneous coordinate ring. Then for any parameter l ∈ Z the composition of functors
gr
Φl = st ◦RΓ≥l : D b (X) → DSg
(A)

is an equivalence of triangulated categories.
Set Φ = Φ1 .
Abusing notation, we denote the composition of Orlov’s and Buchweitz’s equivalences by
the same Φ.
Φ : D b (X) → MCMgr(A).
This equivalence of categories is our main tool for computing Betti numbers of MCM
modules.

4

ALEXANDER PAVLOV

3. The tautological formula for Betti numbers
Let M be a finitely generated graded A-module, and let P • → M be a minimal free
resolution of M over R
· · · → P 1 → P 0 → M → 0,
where each term P i of the resolution is a direct sum of free modules with generators in
various degrees
M
A(−j)βi,j .
Pi ∼
=
j

The exponents βi,j are positive integers, that are called the (graded) Betti numbers. The
Betti numbers contain information about the shape of a minimal free resolution.
We start with the simple observation that the graded Betti numbers are given by the
formula
βi,j (M ) = dim Exti (M, k(−j)) .
The extension groups Exti (M, k(−j)) are isomorphic to the cohomology groups of the
complex Hom(P • , k(−j)), but the resolution P • is minimal, therefore, differentials in the
complex Hom(P • , k(−j)) vanish.
By the depth lemma k-syzygy of a finitely generated module are MCM modules for k ≥
k0 > 0, so up to finitely many terms problem of computing Betti numbers is equivalent to
a problem of computing Betti numbers of MCM modules. In this paper we deal with the
MCM case.
In Orlov’s equivalence the stable category of MCM modules is used, so our next step is to
express the Betti numbers in terms of dimensions of stable extension groups.
Lemma 3.1. Let M be a finitely generated MCM module. Then the graded Betti numbers
are equal to the dimensions of the following stable extension groups
βi,j (M ) = dim Exti (M, kst (−j)),
for i ≥ 0 .
Proof. See [6] for the proof.



Remarks. The formula above also can be used to compute graded Betti numbers of an
MCM module M for i < 0. This means that we replace the projective resolution of M by
a complete minimal resolution
CR(M )• = . . . ← P−2 ← P−1 ← P0 ← P1 ← P2 ← . . . ,
and we use such a complete resolution to extend the definition of Exti (M, k(−j)) for i < 0.
See [2] for details.
Let us denote σ the endofunctor σ : D b (X) → D b (X) corresponding to the shift in internal
degree functor on the MCM side
σ = Φ−1 ◦ (1) ◦ Φ.
We have the following tautological formula for Betti numbers.
Theorem 3.2. The graded Betti numbers of an MCM module M are given by
βi,j (M ) = dim HomDb (X) (Φ−1 (M ), σ −j (Φ−1 (kst ))[i]) .

KOSZUL DUALITY BETWEEN BETTI AND COHOMOLOGY NUMBERS IN CALABI-YAU CASE

5

The advantage of the formula is that answers are presented in the derived category, which
is easier to understand than stable category of MCM modules.
We can give more detailed description of σ and Φ−1 (kst ). The following lemma was proved
in [6] for elliptic curves but proof for Calabi-Yau varieties goes verbatim.
Lemma 3.3. Let X be a smooth projective Calabi-Yau variety then Φ−1 (kst ) ∼
= O[1].
Let E ∈ D b (X) be an object then we can define the spherical twist functor TE on D b (X).
It is defined as cone of the derived evaluation map
TE (F ) = cone(RHom(E, F ) ⊗k E → F )
On a Calabi-Yau variety object E ∈ D b (X) is called spherical if Ext• (E, E) ∼
= H • (S dim X , k).
For example, the structure sheaf O is a spherical object. Spherical objects and functors
were introduced in [7] and the following important theorem were proved
Theorem 3.4. Let X be a smooth projective Calabi-Yau variety. If E is a spherical object,
then the spherical twist functor is an auto equivalence.
Let L : D b (X) → D b (X) be an autoequivalence defined by L(F ) = F ⊗ O(1). Finally for
functor σ we have the following
Lemma 3.5. There is an isomorphism of functors
σ∼
= TO ◦ L.
Proof. This is essentially reformulation of Lemma 5.2.1 in [4] for i = 1.



4. Resolution of the Diagonal
In this section we follow Kawamata’s presentation [3] of the resolution of the diagonal.
Let X be a projective variety, an ample line bundle L is called Koszul if the homogeneous
coordinate ring
M
A=
H 0 (X, Lm )
m≥0

is Koszul.
We define vector spaces Bm (m ≥ 0) by B0 = A0 , B1 = A1 and
Bm = Ker(Bm−1 ⊗ A1 → Bm−2 ⊗ A2 )
for m ≥ 2, where the homomorphism is induced from the multiplication in A. We set
Am = Bm = 0 for m < 0.
One of the ways to say that A is Koszul algebra is to require that the sequence of natural
homomorphisms
· · · → Bm ⊗ A(−m) → Bm−1 ⊗ A(−m + 1) → . . .
(1)
→ B1 ⊗ A(−1) → A → k → 0
is exact, where the shifted module A(j) is defined by A(j)k = Aj+k .
L
By [1] Theorem 2, the subring A(d) = m≥0 Adm of A is a Koszul algebra for a sufficiently
large integer d, i.e. Ld is a Koszul line bundle.
We define sheaves Rm (m ≥ 0) on X by R0 = OX and
Rm = Ker(Bm ⊗ OX → Bm−1 ⊗ L)

6

ALEXANDER PAVLOV

for m ≥ 1, where the homomorphism is induced from the natural homomorphisms Bm →
Bm−1 ⊗ A1 and A1 ⊗ OX → L.
If we take the sequence of associated sheaves to the tensor product of (1) with A(m), we
obtain an exact sequence
(2)

0 → Rm → Bm ⊗ OX → Bm−1 ⊗ L → · · · → B1 ⊗ Lm−1 → Lm → 0.

Lemma 4.1. There is an exact sequence
0 → A0 ⊗ Rm → A1 ⊗ Rm−1 →
· · · → Am−1 ⊗ R1 → Am ⊗ R0 → Lm → 0.
On the product X × X by Lemma 4.1, we obtain a homomorphism L−m ⊠ Rm → L−m+1 ⊠
Rm−1 as a composition of the following homomorphisms
(3)

L−m ⊠ Rm → A1 ⊗ (L−m ⊠ Rm−1 ) → L−m+1 ⊠ Rm−1 .

Theorem 4.2. Let X be a projective variety, L be a Koszul line bundle, and ∆X ⊂ X × X
the diagonal subvariety of the direct product. Then the complex of sheaves on X × X
· · · → L−m ⊠ Rm → L−m+1 ⊠ Rm−1 → . . .
→ L−1 ⊠ R1 → OX ⊠ OX → O∆X → 0
is exact.
Proofs of the lemma and the theorem can be found in [3].
5. The Duality formula
We assume that X is a smooth projective variety and L is an ample line bundle on X,
then for sufficiently large integer d line bundle O(1) = Ld and all its powers are Koszul
and have trivial high cohomology H i (X, O(j)) = 0.
The following short exact sequence immediately follows from the definition of the sheaves
Rm
0 → Rm → Bm ⊗ O → Rm−1 (1) → 0.
Lemma 5.1. For m ≥ 0 there is an isomorphism Bm ∼
= H 0 (X, Rm−1 (1)).
Proof.
H 0 (X, Rm−1 (1)) = ker(Bm−1 ⊗ H 0 (X, O(1)) → Bm−2 ⊗ H 0 (X, O(2)) =
ker(Bm−1 ⊗ A1 → Bm−2 ⊗ A2 ) = Bm .

Lemma 5.2. For m ≥ 0, i > 0 and j > 0 the cohomology groups H i (X, Rm (j)) vanish
H i (X, Rm (j)) = 0.
Proof. We prove vanishing by induction on m. If m = 0 then Rm = O and vanishing
follows from assumptions on O(1). If m > 0 then tensoring short exact sequence
0 → Rm → Bm ⊗ O → Rm−1 (1) → 0.
with O(j) and considering long exact sequence of cohomology groups
. . . →H i−1 (X, Rm−1 (j + 1)) →
→ H i (X, Rm (j)) → H i (X, O(j)) ⊗ Bm →H i (X, Rm−1 (j + 1)) → . . .

KOSZUL DUALITY BETWEEN BETTI AND COHOMOLOGY NUMBERS IN CALABI-YAU CASE

7

we get vanishing for i > 1 by assumption of induction and assumptions of O(1). For i = 1
vanishing follows from assumption that A is Koszul.

Combining these two lemmas we get
Theorem 5.3. Sheaves Rm satisfy recursive relation in the derived category D b (X)
Rm [1] = (TO ◦ L)(Rm−1 ).
Proof. By definition of spherical twist TO we have distinguished triangle
⊕i H i (X, Rm−1 (1)) ⊗ O → Rm−1 (1) → (TO ◦ L)(Rm−1 ) → . . .
By two previous lemmas ⊕i H i (X, Rm−1 (1)) ∼
= Bm . Short exact sequence
0 → Rm → Bm ⊗ O → Rm−1 (1) → 0.
gives distinguished triangle
Rm → Bm ⊗ O → Rm−1 (1) → Rm [1]
Noticing that these two distinguished triangles have the same objects and maps we conclude
that (TO ◦ L)(Rm−1 ) ∼

= Rm [1].
We can reformulate the result as follows
Corollary 5.4. Sheaves Rm form the orbit of the functor γ = TO ◦ L[−1] for m ≥ 0
Rm = γ m (O).
Proof. Iterating identity Rm [1] = (TO ◦ L)(Rm−1 ) we get Rm [m] = (TO ◦ L)m (O).



The key observation is that two functors σ used in the formula for Betti numbers and γ
are related by σ = γ[1].
Let us set by definition Rm = γ m (O) for m ∈ Z. For m < 0 we get objects of the derived
category Rm ∈ D b (X), for example R−1 = O(−1)[d], where d = dim X.
If F is any object in the derived category D b (X) we define cohomology numbers of F as
the following dimensions of the hypercohomology groups
hij (F ) = dim Hi (X, F (j)),
where i, j ∈ Z.
Combining all our observations together we get the following version of Koszul duality between cohomology numbers of F and Betti numbers of the MCM module M corresponding
to F under Orlov’s equivalence. This duality is controlled by two families of objects O(i)
and R(j) appearing in the box-product resolution of the diagonal
· · · → O(−m) ⊠ Rm → O(−m + 1) ⊠ Rm−1 → . . .
→ O(−1) ⊠ R1 → OX ⊠ OX → O∆X → 0.
Theorem 5.5. Let X be a smooth projective Calabi-Yau variety, and let L be a Koszul
line bundle s.t. H i (X, O(j)) = 0 for i, j > 0. If F ∈ D b (X) and M is the corresponding
MCM module then
βij (M ) = dim Hom(R−j , F [d − 1 + j − i]),
hij (F ) = dim Hom(O(−j), F [i]).

8

ALEXANDER PAVLOV

for i, j ∈ Z.
Proof. We know that βij (M ) = dim Hom(F, σ −j (O)[i + 1]) = dim Hom(F, R−j [i − j + 1]).
Applying Serre’s duality we get βij (M ) = dim Hom(R−j , F [d − 1 + j − i]).
The second formula is clear.


References

[1] J. Backelin. On the rate of growth of the homologies of Veronese subrings. Lecture Notes in Math.
1183(1986), 79–100.
[2] R.-O. Buchweitz, Maximal Cohen-Macaulay Modules and Tate Cohomology over Gorenstein Rings,
preprint, https://tspace.library.utoronto.ca/handle/1807/16682.
[3] Y. Kawamata, Equivalences of derived catgories of sheaves on smooth stacks, American Journal of
Mathematics, Vol. 126, No. 5 (Oct., 2004), pp. 1057–1083
[4] B. Keller, D. Murfet, M. Van den Bergh On Two Examples by Iyama and Yoshino, Compos. Math.
147 (2011), no. 2, 591-612.
[5] D. Orlov, Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities, in
”Algebra, Arithmetic and Geometry: In Honor of Yu. I. Manin”, Vol 2, Birkhöuser (2009), 503-531.
[6] A. Pavlov, Betti Tables of Maximal Cohen-Macaulay modules over the cone a plane cubic,
http://arxiv.org/abs/1511.05089
[7] P. Seidel, R. Thomas, Braid Group Actions on Derived Categories of Coherent Sheaves, Duke Math.
J. 108(2001), 37-108.
University of Wisconsin, 480 Lincoln Dr., Madison, WI, USA
E-mail address: pavlov@math.wisc.edu

