arXiv:1404.0352v1 [math.AG] 1 Apr 2014

CHERN CHARACTERS FOR TWISTED MATRIX
FACTORIZATIONS AND THE VANISHING OF THE HIGHER
HERBRAND DIFFERENCE
MARK E. WALKER
Abstract. We develop a theory of “ad hoc” Chern characters for twisted
matrix factorizations associated to a scheme X, a line bundle L, and a regular
global section W ∈ Γ(X, L).
As an application, we establish the vanishing, in certain cases, of hR
c (M, N ),
the higher Herbrand difference, and, ηcR (M, N ), the higher codimensional analogue of Hochster’s theta pairing, where R is a complete intersection of codimension c with isolated singularities and M and N are finitely generated Rmodules. Specifically, we prove such vanishing if R = Q/(f1 , . . . , fc ) has only
isolated singularities, Q is a smooth k-algebra, k is a field of characteristic 0,
the fi ’s form a regular sequence, and c ≥ 2.
Such vanishing was previously established in the general characteristic, but
graded, setting in [16].

Contents
1. Introduction
2. Chern classes for generalized matrix factorizations
2.1. The Jacobian complex
2.2. Matrix factorizations
2.3. Connections for matrix factorizations and the Atiyah class
2.4. Ad hoc Hochschild homology
2.5. Supports
2.6. Chern classes of matrix factorizations
2.7. Functorality of the Chern character and the top Chern class
3. Relationship with affine complete intersections
3.1. Euler characteristics and the Herbrand difference for hypersurfaces
3.2. Matrix factorizations for complete intersections
4. Some needed results on the geometry of complete intersections
5. The vanishing of Chern characters
5.1. The vanishing of the top Chern class
5.2. The Polishchuk-Vaintrob Riemann-Roch Theorem
5.3. Vanishing of the higher h and η Invariants
Appendix A. Relative Connections
A.1. The classical Atiyah class
A.2. The vanishing of the classical Atiyah class and connections
References
Date: April 2, 2014.
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MARK E. WALKER

1. Introduction
This paper concerns invariants of finitely generated modules over an affine complete intersection ring R. Such a ring is one that can be written as R = Q/(f1 , . . . , fc ),
where Q is a regular ring and f1 , . . . , fc ∈ Q form a regular sequence of elements.
For technical reasons, we will assume Q is a smooth algebra over a field k. More
precisely, this paper concerns invariants of the singularity category, Dsg (R), of R.
This is the triangulated category defined as the Verdier quotient of Db (R), the
bounded derived category of R-modules, by the sub-category of perfect complexes
of R-modules. More intuitively, the singularity category keeps track of just the
“infinite tail ends” of projective resolutions of finitely generated modules. Since R
is Gorenstein, Dsg (R) is equivalent to M CM (R), the stable category of maximal
Cohen-Macaulay modules over R.
Thanks to a Theorem of Orlov [19, 2.1], the singularity category of R is equivalent
to the singularity category of the hypersurface Y of Pc−1
= Proj Q[T1 , . . . , Tc ]
Q
P
c−1
is
cut out by the element W = i fi Ti ∈ Γ(PQ , O(1)). Since the scheme Pc−1
Q
regular Dsg (Y ) may in turn be given by the homotopy category of “twisted matrix
factorizations” associated to the triple (Pc−1
Q , O(1), W ) (see [21], [15], [18], [23],
[2]). In general, if X is a scheme (typically smooth over a base field k), L is a
line bundle on X, and W ∈ Γ(X, L) is regular global section of L, a twisted matrix
factorization for (X, L, W ) consists of a pair of locally free coherent sheaves E0 , E1
on X and morphisms d1 : E1 → E0 , d0 : E0 → E1 ⊗OX L such that each composite
d0 ◦ d1 and (d1 ⊗ idL ) ◦ d0 is multiplication by W . In the special case where
X = Spec(Q) for a local ring Q, so that L is necessarily the trivial line bundle
and W is an element of Q, a twisted matrix factorization is just a classical matrix
factorization as defined by Eisenbud [10].
Putting these facts together, we get an equivalence
X
fi Ti )
Dsg (R) ∼
,
O(1),
= hmf (Pc−1
Q
i

between the singularity category
Pof R and the homotopy category of twisted matrix
factorizations for (Pc−1
,
O(1),
i fi Ti ). The invariants we attach to objects of
Q
Dsg (R) for a complete intersection ring R are actually defined, more generally, in
terms of such twisted matrix factorizations. In detail, for a triple (X, L, W ) as
above, let us assume also that X is smooth over a base scheme S and L is pulledback from S. In the main case of interest, namely X = Pc−1
for a smooth k-algebra
Q
c−1
Q, the base scheme S is taken to be Pk . In this situation, we define a certain
hoc
ad hoc Hochschild homology group HHad
(X/S, L, W ) associated to (X, L, W ).
0
Moreover, to each twisted matrix factorization E for (X, L, W ) we attach a class
hoc
ch(E) ∈ HHad
(X/S, L, W ),
0

called the ad hoc Chern character of E. We use the term “ad hoc” since we offer no
justification that these invariants are the “correct” objects with these names given
by a more theoretical framework. But, if X is affine and L is the trivial bundle,
then our ad hoc Hochschild homology groups and Chern characters coincide with
the usual notions found in [9], [8], [22], [7], [25], [17], [26], [20]. In particular, in
this case, our definition of the Chern character of a matrix factorization is given by
the “Kapustin-Li formula” [14].

CHERN CHARACTERS FOR TWISTED MATRIX FACTORIZATIONS

3

When the relative dimension n of X → S is even, we obtain from ch(E) a “top
Chern class”
ctop (E) ∈ H 0 (X, JW ( n2 ))
where JW is the “Jacobian sheaf” associated to W . By definition, JW is the
dW ∧−
n−1
coherent sheaf on X given as the cokernel of the map ΩX/S
⊗ L−1 −−−−→ ΩnX/S
define by multiplication by the element dW of Γ(X, Ω1X/S ⊗OX L). The top Chern
class contains less information than the Chern character in general, but in the affine
case with L trivial and under certain other assumptions, the two invariants coincide.
The main technical result of this paper concerns the vanishing of the top Chern
class:
Theorem 1.1. Let k be a field of characteristic 0 and Q a smooth k-algebra of
even dimension. If f1 , . . . , fc ∈ Q is a regular sequence of elements with c ≥ 2 such
that the singular locus of R = Q/(f1 , . . . , fc ) is zero-dimensional, then
ctop (E) = 0 ∈ H 0 (X, JW ( n2 ))
P
for all twisted matrix factorizations E of (Pc−1
Q , O(1),
i fi Ti ).

As an application of this Theorem, we prove the vanishing of the invariants ηc
and hc for c ≥ 2 in certain cases. These invariants are defined for a pair of modules
M and N over a complete intersection R = Q/(f1 , . . . , fc ) having the property
that ExtiR (M, N ) and ToriR (M, N ) are of finite length for i ≫ 0. For example, if
the non-regular locus of R is zero-dimensional, then every pair of finitely generated
modules has this property. In general, for such a pair of modules, the even and odd
sequences of lengths of ExtiR (M, N ) and ToriR (M, N ) are governed, eventually, by
polynomials of degree at most c − 1: there exist polynomials
pev (M, N )(t) = ac−1 tc−1 + lower order terms
podd (M, N )(t) = bc−1 tc−1 + lower order terms
with integer coefficients such that
2i+1
lengthR Ext2i
(M, N ) = podd (M, N )(i)
R (M, N ) = pev (M, N )(i) and lengthR ExtR

for i ≫ 0, and similarly for the Tor modules. Following Celikbas and Dao [4,
3.3], we define the invariant hc (M, N ) in terms of the leading coefficients of these
polynomials:
ac−1 − bc−1
hc (M, N ) :=
c2c
The invariant ηc (M, N ) is defined analogously, using Tor modules instead of Ext
modules; see [5].
If c = 1 (i.e., R is a hypersurface) the invariants h1 and η1 coincide (up to a
factor of 12 ) with the Herbrand difference, defined originally by Buchweitz [1], and
Hochster’s θ invariant, defined originally by Hochster [12]. In this case, ExtiR (M, N )
and TorR
i (M, N ) are finite length modules for i ≫ 0 and the sequence of their
lengths are eventually two periodic, and h1 and η1 are determined by the formulas
R
R
R
2hR
1 (M, N ) = h (M, n) = length Ext2i (M, N ) − length Ext2i+1 (M, N ), i ≫ 0

and
R
2η1R (M, N ) = θR (M, n) = length TorR
2i (M, N ) − length Tor2i+1 (M, N ), i ≫ 0.

We can now state the main application of Theorem 1.1:

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MARK E. WALKER

Theorem 1.2. Let k be a field of characteristic 0 and Q a smooth k-algebra. If
f1 , . . . , fc ∈ Q is a regular sequence of elements with c ≥ 2 such that the singular locus of R = Q/(f1 , . . . , fc ) is zero-dimensional, then hR
c (M, N ) = 0 and
ηcR (M, N ) = 0 for all finitely generated R-modules M and N .
Acknowledgements. I am grateful to Jesse Burke, Olgur Celikbas, Hailong Dao,
Daniel Murfet, and Roger Wiegand for conversations about the topics of this paper.
2. Chern classes for generalized matrix factorizations
In this section, we develop ad hoc notions of Hochschild homology and Chern
classes for twisted matrix factorizations. The connection with the singularity category for complete intersection rings will be explained more carefully later.
Assumptions 2.1. Throughout this section, we assume
• S is a Noetherian scheme.
• p : X → S is a smooth morphism of relative dimension n.
• LS is a locally free coherent sheaf of rank one on S. Let LX := p∗ LS , the
pullback of LS along p.
• W ∈ Γ(X, LX ) is global section of LX .
Recall that the smoothness assumption means that p : X → S is flat and of
finite type and that Ω1X/S is a locally free coherent sheaf on X of rank n.
Assumptions 2.2. When discussing ad hoc Hochschild homology and Chern characters (see below), we will also be assuming:
• p is affine and
• n! is invertible in Γ(X, OX ).
It is useful to visualize Assumptions 2.1 by a diagram. Let q : LS → SL
denote the
geometric line bundle whose sheaf of sections is LS ; that is, LS = Spec( i≥0 L−i
S ).
W

Then we may interpret W as a morphism X −→ LS fitting into the commutative
triangle
X❈
❈❈
❈❈W
p
❈❈
❈!

S o q LS .

The data (p : X → S, L, W ) determines a closed subscheme Y of X, defined by
the vanishing of W . More formally, if z : S ֒→ LS denotes the zero section of the
line bundle LS , then Y is defined by the pull-back square
/X

Y

W


S

z



/ LS .

We often suppress the subscripts and write L to mean LS or LX . For brevity,
we write F (n) to mean F ⊗OX L⊗n
X if F is a coherent sheaf or a complex of such
on X. In our primary application, L will in fact by the standard very ample line
bundle O(1) on projective space, and so this notation is reasonable in this case.
The case of primary interest for us is described in the following example:

CHERN CHARACTERS FOR TWISTED MATRIX FACTORIZATIONS

5

Example 2.3. Suppose k is a field, Q is a smooth k-algebra of dimension n, and
f1 , . . . , fc is regular sequence of elements of Q. Let
X = Pc−1
= Proj Q[T1 , . . . , Tc ],
Q
S = Pc−1
= Proj k[T1 , . . . , Tc ]
k
and define
p:X →S
to be the map induced by k ֒→ Q. Finally, set LS := OS (1) and
X
fi Ti ∈ Γ(X, LX ).
W :=
i

Then the triple (p : X → S, L, W ) satisfies all the hypotheses in Assumptions
2.1, and moreover p is affine. The subscheme cut out by W is
Y = Proj Q[T1 , . . . , Tc ]/W
in this case.
2.1. The Jacobian complex. We construct a complex of locally free sheaves on X
that arises from the data (p : X → S, LS , W ). Regarding W as a map W : X → LS
as above, we obtain an induced map
W ∗ Ω1LS /S → Ω1X/S
on cotangent bundles. There is a canonical isomorphism Ω1LS /S ∼
= q ∗ L−1
S and hence
−1
∗ 1
∼
an isomorphism L−1
→
Ω1X/S , and
.
Using
this,
we
obtain
a
map
L
W
Ω
=
X
X
LS /S
upon tensoring with L we arrive at the map:
dW : OX → Ω1X/S ⊗OX LX =: Ω1X/S (1).
Note that

Λ·X/S (Ω1X/S ⊗OX L) ∼
=

M

ΩqX/S (q)

q

so that we may regard the right-hand side as a sheaf of graded-commutative gradedOX -algebras. Since dW is a global section of this sheaf lying in degree one, we may
form the Jacobian complex with differential given as repeated multiplication by dW :


dW
dW
dW
dW
(2.4)
Ω·dW := OX −−→ Ω1X/S (1) −−→ Ω2X/S (2) −−→ · · · −−→ ΩnX/S (n) .
We index this complex so that ΩjX/S (j) lies in cohomological degree j.

Example 2.5. Let us consider the case where X and S are affine and the line
bundle L is the trivial one. That is, suppose k is a Noetherian ring, A is a smooth
k-algebra, and f ∈ A, and set S = Spec(k), X = Spec(A), L = OS , and W = f ∈
Γ(X, OX ) = A. (The map p : X → S is given by the structural map k → A.)
Then the Jacobian complex is formed by repeated multiplication by the degree
one element df ∈ Ω1A/k :

 df ∧−
df ∧−
df ∧−
df ∧−
Ω·df = A −−−→ Ω1A/k −−−→ Ω2A/k −−−→ · · · −−−→ ΩnA/k .

If we further specialize to the case A = k[x1 , . . . , xn ], then we may identify Ω1A/k
with An by using the basis dx1 , . . . , dxn . Then Ω·df is the A-linear dual of the
∂f
∂f
Koszul complex on the sequence ∂x
, . . . , ∂x
of partial derivatives of f .
1
n

6

MARK E. WALKER

∂f
∂f
, . . . , ∂x
form an
If k is a field and f has only isolated singularities, then ∂x
1
n
A-regular sequence. In this case, the Jacobian complex has cohomology only in
degree n and
k[x1 , . . . , xn ]
H n (Ω·df ) ∼
dx1 ∧ · · · ∧ dxn .
= ∂f
∂f
i
h ∂x1 , . . . , ∂x
n

Example 2.6. With the notation of Example 2.3, the Jacobian complex is the
complex of coherent sheaves on Pc−1
associated to the following complex of graded
Q
modules over the graded ring Q[T ] := Q[T1 , . . . , Tc ]:
dW

dW

dW

0 → Q[T ] −−→ Ω1Q/k [T ](1) −−→ · · · −−→ ΩnQ/k [T ](n) → 0
Here, dW = df1 T1 +· · ·+dfc T1 , a degree one element in the graded module Ω1Q/k [T ].
P ∂f
Further specializing to Q = k[x1 , . . . , xn ], we have dfi = j ∂x
dxj and
j

 ∂f
 
∂fc
1
· · · ∂x
T1
∂x1
1
 ∂f1
∂f 

 ∂x2 · · · ∂xc2  
 T2 

·
.
dW = 

..   .. 
 ..
.
 .
. 
∂f1
∂xn

···

∂fc
∂xn

Tc

It is convenient to think of Ω·dW as representing a “family” of complexes of the
type occurring in Example 2.5 indexed by the Q points of Pc−1
Q . That is, given
a tuple a = (a1 , . . . , ac ) of elements of Q that generated the unit ideal, we have
an associated Q-point ia : Spec(Q) ֒→ Pc−1
of Pc−1
Q
Q . Pulling back along ia (and
∗
identifying ia O(1) with the trivial bundle in the canonical way) gives the Jacobian
P
complex of Example 2.5 for (Spec(Q) → Spec(k), O, i ai fi ).
We are especially interested in the cokernel (up to a twist) of the last map in
the Jacobian complex (2.4), and thus give it its own name:

Definition 2.7. The Jacobi sheaf associated to (p : X → S, L, W ), written J (X/S, L, W )
or just JW for short, is the coherent sheaf on X defined as


dW ∧−
n−1
JW = J (X/S, L, W ) := coker ΩX/S
(−1) −−−−→ ΩnX/S .
In other words, JW := Hn (Ω·dW )(−n), where Hn denote the cohomology of a
complex in the abelian category of coherent sheaves on X.
Example 2.8. If S = Spec(k), k is a field, X = Ank = Spec(k[x1 , . . . , xn ]), L = OX
and f a polynomial, then
k[x1 , . . . , xn ]
 dx1 ∧ · · · ∧ dxn .
Jf = 
∂f
∂f
,
.
.
.
,
∂x1
∂xn

Example 2.9. With the notation of 2.3, JW is the coherent sheaf associated to
the graded Q[T ]-module


dW
n−1
coker ΩQ/k
[T ](−1) −−→ ΩnQ/k [T ] .

If we further specialize to the case Q = k[x1 , . . . , xn ] = k[x], then using dx1 , . . . , dxn
as a basis of Ω1Q/k gives us that JW is isomorphic to the coherent sheaf associated

CHERN CHARACTERS FOR TWISTED MATRIX FACTORIZATIONS

to
⊕n

coker k[x, T ](−1)

(T1 ,...,Tc )



∂fi
∂xi

7

!

t

−−−−−−−−−−−→ k[x, T ] .

2.2. Matrix factorizations. We recall the theory of “twisted matrix factorizations” from [21] and [3].
Suppose X is any Noetherian scheme, L is a line bundle on X and W ∈ Γ(X, L)
any global section of it. (The base scheme S is not needed for this subsection, and
we write L for LX .) A twisted matrix factorization for (X, L, W ) consists of the
data E = (E0 , E1 , d0 , d1 ) where E0 and E1 are locally free coherent sheaves on X and
d0 : E0 → E1 (1) and d1 : E1 → E0 are morphisms such that d0 ◦ d1 and d1 (1) ◦ d0
are each given by multiplication by W . (Recall that E1 (1) denotes E1 ⊗OX L.) We
visualize E as a diagram of the form


d1
d0
E0 (1)
E1 −→
E = E0 −→
or as the “twisted periodic” sequence
d0 (−1)

d1 (−1)

d

d0 (1)

d

d1 (1)

1
0
E0 (1) −−−→ E1 (1) −−−→ · · · .
E1 −→
· · · −−−−→ E1 (−1) −−−−→ E0 −→

Note that the composition of any two adjacent arrows in this sequence is multiplication by W (and hence this is not a complex unless W = 0).
A strict morphism of matrix factorizations for (X, L, W ), say from E to E′ =
′
(E0 , E1′ , d′0 , d′1 ) consists of a pair of morphisms E0 → E0′ and E1 → E1′ such that the
evident pair of squares both commute. Matrix factorizations and strict morphisms
form an exact category, which we write as mf (X, L, W ), for which a sequence of
morphisms is declared exact if it is so in each degree.
If V ∈ Γ(X, L) is another global section, there is a tensor product pairing
− ⊗mf − : mf (X, L, W ) × mf (X, L, V ) → mf (X, L, W + V ).
In the special case L = OX and V = W = 0, this tensor product is the usual
tensor product of Z/2-graded complexes of OX -modules, and the general case is
the natural “twisted” generalization of this. We refer the reader to [2, §7] for a
more precise definition.
There is also a sort of internal Hom construction. Given W, V ∈ Γ(X, L) and
objects E ∈ mf (X, L, W ) and F ∈ mf (X, L, V ), there is an object Hommf (E, F) ∈
mf (X, L, V − W ). Again, if L = OX and W = V = 0, this is the usual internal
Hom-complex of a pair of Z/2-graded complexes of OX -modules, and we refer the
reader to [2, 2.3] for the general definition. If V = W , then Hommf (E, F) belongs
to mf (X, L, 0) and hence is actually a complex. In general, we refer to an object
of mf (X, L, 0) as a twisted two-periodic complex, since it is a complex of the form
d0 (−1)

d

d

d1 (1)

d0 (1)

0
1
E1 (1) −−−→ E0 (1) −−−→ · · · .
E0 −→
· · · −−−−→ E1 −→

In other words, such an object is equivalent to the data of a (homologically indexed)
complex E· of locally free coherent sheaves and a specified isomorphism E· (1) ∼
= E· [2].
(We use the convention that E· [2]i = Ei−2 .)
There is also a notion of duality. Given E ∈ mf (X, L, W ), we define E∗ ∈
mf (X, L, −W ) as Hommf (E, OX ) where here OX denotes the object of mf (X, L, 0)
given by the twisted two-periodic complex
· · · → 0 → OX → 0 → L → 0 → L⊗2 → · · · .

8

MARK E. WALKER

(In the notation introduced below, E∗ = Hommf (E, Fold OX ).) See [2, 7.5] for a
more explicit formula.
These constructions are related by the natural isomorphisms (cf. [2, §7])
Hommf (E, F) ∼
= E∗ ⊗mf F
and

Hommf (E ⊗mf G, F) ∼
= Hommf (E, Hommf (G, F)).
The construction of Hommf (E, F) is such that we have a natural identification
Homstrict (E, F) = Z0 Γ(X, Hommf (E, F)).

Here, Γ(X, Hommf (E, F)) is the complex of abelian groups obtained by regarding
Hommf (E, F) as an unbounded complex of coherent sheaves on X and applying
the global sections functor Γ(X, −) degree-wise, obtaining a complex of Γ(X, OX )modules, and Z0 denotes the cycles lying in degree 0 in this complex.
Let P(X) denote the category of complexes of locally free coherent sheaves on
X and let P b (X) denote the full subcategory of bounded complexes. There is a
“folding” functor,
Fold : P b (X) → mf (X, L, 0),
defined by taking direct sums of the even and odd terms in a complex. In detail,
given a complex (P, dP ) ∈ P b (X), let
M
M
P2i+1 (i).
P2i (i) and Fold(P )1 =
Fold(P )0 =
i

i

The required map Fold(P )1 → Fold(P )0 maps the summand P2i+1 (i) to the summand P2i (i) via dP (i) and Fold(P )0 → Fold(P )1 (1) maps P2i (i) to P2i−1 (i) also
via dP (i).
There is also an “unfolding” functor
Unfold : mf (X, L, 0) → P(X)
which forgets the two periodicity. In detail,
Unfold(E1 → E0 → E1 (1))2i = E0 (−i) and Unfold(E1 → E0 → E1 (1))2i+1 = E1 (−i)
with the evident maps.
These functors are related by the following adjointness properties: Given E ∈
mf (X, L, 0) and P ∈ P b (X), there are natural isomorphisms
∼ HomP(X) (P, Unfold E)
Hommf (X,L,0) (Fold P, E) =
and

HomP(X) (Unfold E, P) ∼
= Hommf (X,L,0) (E, Fold P).
(These functors do not technically form an adjoint pair, since Unfold takes values
in P(X), not P b (X).)

Example 2.10. If E is a locally free coherent sheaf on X, we write E[0] for the
object in P(X) with E in degree 0 and 0’s elsewhere. Then
Fold(E[0]) = (0 → E → 0)
= (· · · → E(−1) → 0 → E → 0 → E(1) → · · · )
indexed so that E lies in degree 0. Letting E[j] = E[0][j], we have
Fold(E[2i]) = (0 → E(i) → 0)
= (· · · → E(i − 1) → 0 → E(i) → 0 → E(i + 1) → · · · )

CHERN CHARACTERS FOR TWISTED MATRIX FACTORIZATIONS

9

with E(i) in degree 0, and
Fold(E[2i + 1]) = (E(i) → 0 → E(i + 1))
= (· · · → E(i − 1) → 0 → E(i) → 0 → E(i + 1) → · · · )
with E(i) in homological degree 1.
If P ∈ P b (X) is a bounded complex and E ∈ mf (X, L, W ) is a twisted matrix
factorization, we define their tensor product to be
P ⊗ E := Fold(P ) ⊗mf E ∈ mf (X, L, W ).
Example 2.11. We will use this construction, in particular, for the complex P =
Ω1X/S (1)[−1] consisting of Ω1X/S (1) concentrated in homological degree −1. Since
Ω1 lies in odd degree, the sign conventions give us


− id ⊗d1 (1)
− id ⊗d0
Ω1X/S ⊗ E1 (1) −−−−−−−→ Ω1X/S ⊗ E0 (1) .
Ω1X/S (1)[−1] ⊗ E = Ω1X/S ⊗ E0 −−−−−→



dW ∧−
We will also apply this construction to the complex P := OX −−−−→ Ω1X/S (1)
indexed so that OX lies in degree 0 to form the matrix factorization
dW ∧−

(OX −−−−→ Ω1X/S (1)) ⊗OX E
which is given explicitly as
E1

⊕

Ω1X/S ⊗E0





d1
0

dW − id ⊗d0
−−−−−−−−−−−−−→


E0

⊕

Ω1X/S ⊗E1 (1)





d0
0

dW − id ⊗d1
−−−−−−−−−−−−−→


E1 (1)

⊕

Ω1X/S ⊗E0 (1)

.

Building on the identity
Homstrict (E, F) = Z0 Γ(X, Hommf (E, F))
we obtain a natural notion of homotopy: Two strict morphisms are homotopic if
their difference lies in the image of the boundary map in the complex Γ(X, Hommf (E, F)).
That is, the group of equivalence classes of strict morphisms up to homotopy from
E to F is H0 Γ(X, Hommf (F, E)). We define [mf (X, L, W )]naive , the “naive homotopy category”, to be the category with the same objects as mf (X, L, W ) and with
morphisms given by strict maps modulo homotopy; that is,
Hom[mf ]naive (E, F) := H0 Γ(X, Hommf (F, E)).
Just as the homotopy category of chain complexes is a triangulated category, so
too is the category [mf (X, L, W )]naive ; see [21, 1.3] or [3, 2.5] for details.
The reason for the pejorative “naive” in this definition is that this notion of
homotopy equivalence does not globalize well: two morphisms can be locally homotopic without being globally so. Equivalently, an objects can be locally contractible
with being globally so. To rectify this, we define the homotopy category of matrix
factorizations for (X, L, W ), written hmf (X, L, W ), to be the Verdier quotient of
[mf (X, L, W )]naive by the thick subcategory consisting of objects that are locally
contractible. (The notion appears to be originally due to Orlov, but see also [21,
3.13] where this category is referred to as the “derived category of matrix factorizations”.)
We list the properties of hmf (X, L, W ) needed in the rest of this paper:

10

MARK E. WALKER

(1) There is a functor mf (X, L, W ) → hmf (X, L, W ), and it sends objects
that are locally contractible to the trivial object.
(2) If α : E → F is strict morphism and it is locally null-homotopic, then α is
sent to the 0 map in hmf (X, L, W ).
(3) When W = 0, hmf (X, L, 0) is the Verdier quotient of [mf (X, L, 0]naive obtained by inverting quasi-isomorphisms of twisted two-periodic complexes.
(4) At least in certain cases, the hom sets of hmf (X, L, W ) admit a more
explicit description. In general, for objects E, F ∈ mf (X, L, W ) there is a
natural map
Homhmf (E, F) → H0 (X, Unfold Hommf (E, F))
where H0 (X, −) denotes sheaf hyper-cohomology [3, 3.5]. This map is an
isomorphism when
(a) W = 0 or
(b) X is projective over an affine base scheme and L = OX (1) is the
standard line bundle; see [3, 4.2].
2.3. Connections for matrix factorizations and the Atiyah class. See Appendix A for recollections on the notion of a connection for a locally free coherent
sheaf on a scheme. Much of the material in this subsection represents a generalization to the non-affine case of constructions found in [26], which were in turn
inspired by constructions in [9].
Definition 2.12. With (p : X → S, L, W ) as in Assumptions 2.1, given E ∈
mf (X, L, W ), a connection on E relative to p, written as ∇E or just ∇, is a pair of
connections relative to p
∇1 : E1 → Ω1X/S ⊗OX E1
and
∇0 : E0 → Ω1X/S ⊗OX E0 .
There is no condition relating ∇E and d1 , d0 .
Since each ∇i is p∗ OS -linear and L is pulled back from S, we have induced
connections
∇1 (j) : E1 (j) → Ω1X/S ⊗OX E1 (j)
and
∇0 (j) : E0 (j) → Ω1X/S ⊗OX E0 (j)
for all j.
Example 2.13. Suppose S = Spec k and X = Spec Q are affine where Q is a
smooth k-algebra of dimension n, L = OX so that W ∈ Q, and thecomponents of

A
E are free Q-modules. Upon choosing bases, we may represent E as Qr −===
− Qr
B

where A, B are r × r matrices with entries in Q such that AB = BA = W Ir . The
choice of basis leads to an associated “trivial” connection
⊕r

∇ = d : Qr → Ω1Q/k ⊗Q Qr = Ω1Q/k
given by applying exterior differentiation to the components of a vector.

As noted in Appendix A, if p is affine, then every vector bundle on X admits a
connection relative to p and so every matrix factorization admits a connection in
this case.

CHERN CHARACTERS FOR TWISTED MATRIX FACTORIZATIONS

11

Definition 2.14. Given a connection ∇ relative to p for a twisted matrix factorization E ∈ mf (X, L, W ), the associated Atiyah class is defined to be the map
AtE,∇ : E → Ω1X/S (1)[−1] ⊗ E
given as the “commutator” [∇, δE ]. In detail, it is given by the pair of maps
∇1 (1) ◦ δ − (id ⊗δ) ◦ ∇0 : E0 → Ω1 (1) ⊗ E1
and
∇0 ◦ δ − (id ⊗δ) ◦ ∇1 : E1 → Ω1 ⊗ E0 .
AtE,∇ is not a morphism of matrix factorizations in general, but by Lemma A.6
we have that AtE,∇ is OX -linear.
Example 2.15. Keep the notations and assumptions of Example 2.13, so that


A
r
E = Qr −===
−Q .
B

Then

1

Ω (1)[−1] ⊗ E =
1



Ω1Q/k

⊕r

⊕r 

−B
1
−==

==− ΩQ/k
−A

and the map At : E → Ω (1)[−1] ⊗ E obtained by choosing the trivial connections
is represented by the following diagram:
A

Qr
dA


⊕r

1
ΩQ/k

/ Qr

B

dB

−B


⊕r

1
/ Ω
Q/k

/ Qr
dA

−A


⊕r

1
/ Ω
Q/k

Here dA and dB denote the r×r matrices with entires in Ω1Q/k obtained by applying
d entry-wise to A and B.
This is not a map of matrix factorizations since the squares do not commute.
Indeed, the differences of the compositions around these squares are (dB)A+B(dA)
and (dA)B + A(dB). Since AB = BA = W , these expression both equal dW , a
fact which is relevant for the next construction.
Definition 2.16. Given a connection ∇ for a matrix factorization E , define


dW ∧−
ΨE,∇ : E → OX −−−−→ Ω1X/S (1) ⊗OX E
dW ∧−

as idE + At∇ . (Here, OX −−−−→ Ω1X/S (1) is the complex with OX in degree 0.) In
other words, ΨE,∇ is the morphism whose composition with the canonical projection


dW ∧−
OX −−−−→ Ω1X/S (1) ⊗OX E −−։ E
is the identity and whose composition with the canonical projection


dW ∧−
OX −−−−→ Ω1X/S (1) ⊗OX E −−։ Ω1 (1)[−1] ⊗OX E
is AtE,∇ .

Lemma 2.17. The map ΨE,∇ is a (strict) morphism of matrix factorizations, and
it is independent in the naive homotopy category [mf (X, L, W )]naive (and hence the
homotopy category too) of the choice of connection ∇.

12

MARK E. WALKER

Proof. The proofs found in [26], which deal with the case where X is affine and L
is trivial, apply nearly verbatim. The homotopy relating ΨE,∇ and ΨE,∇′ for two
different connections ∇ and ∇′ on E is given by the map
∇ − ∇′ : E → Ω1X/S ⊗ E,
which is OX -linear by Lemma A.6.



Example 2.18. Continuing with Examples 2.13 and 2.15, the map Ψ in this case
is represented by the diagram
A

Qr




1
dA





Qr

 ⊕ r
Ω1Q/k



A
dW

B

/ Qr

0
−B



/



1
dB

/ Qr





Qr

 ⊕ r
Ω1Q/k



B
dW



0
−A

/





1
dA

Qr

 ⊕ r
Ω1Q/k .

This diagram commutes, confirming that Ψ is indeed a morphism of matrix factorizations.
2.4. Ad hoc Hochschild homology. For an integer j ≥ 0, we define the twisted
two-periodic complex
(j)

ΩdW :=


(j−2)dW
(j−1)dW
jdW
2dW
dW
−→ ΩjX/S (j)) .
Fold OX −−→ Ω1X/S (1) −−−−−→ Ω2X/S (2) −−−−−→ · · · −−→ Ωj−1
X/S (j − 1) −
(j)

Explicitly, ΩdW is the twisted periodic complex
 1

 1



ΩX/S
ΩX/S (1)
OX
 ⊕ 
 ⊕ 
 ⊕ 
 3

 3

 2

ΩX/S (1)
ΩX/S (2)
ΩX/S (1)












· · · −→  ⊕  −→  4 ⊕  −→  ⊕  −→ · · ·
Ω5 (2)

Ω5 (3)
Ω
(2)
 X/S 
 X/S 
 X/S 
 ⊕ 
 ⊕ 
 ⊕ 






.
..
..
.
.
.
.

with

Li=⌊ 2j ⌋
i=0

Ω2i
X/S (i) in degree 0 and

Li=⌊ j−1
2 ⌋
i=0

Ω2i+1
X/S (i) in degree 1.

(j)

The coefficients appearing in the maps of ΩdW are necessarily to make the following statement hold true. (We omit its straight-forward proof.)
Lemma 2.19. The pairings
ΩiX/S (i) ⊗OX ΩlX/S (l) → Ωi+l
X/S (i + l)
defined by exterior product induce a morphism
(j)

(l)

(j+l)

ΩdW ⊗mf ΩdW → ΩdW .
in mf (X, L, 0).

CHERN CHARACTERS FOR TWISTED MATRIX FACTORIZATIONS

13

If j! is invertible in Γ(X, OX ) (for example, if X is a scheme over a field k with
char(k) = 0 or char(k) > j), then there is an isomorphism


dW
dW
dW
(j)
j
ΩdW ∼
= Fold OX −−→ Ω1X/S (1) −−→ · · · −−→ ΩX/S (j))
1
n(n−1)···(n−i)

given by collection of isomorphisms ΩiX/S (i) −−−−−−−−−→ ΩiX/S (i), 0 ≤ i ≤ j. In
particular, if n! is invertible in Γ(X, OX ), then there is an isomorphism
(n)
ΩdW ∼
= Fold Ω·dW .

(Recall that n is the relative dimension of X of S, and hence is the rank of Ω1X/S .
It follows that Ωm
X/S = 0 for m > n.)
Definition 2.20. Let p : X → S, L, W be as in Assumptions 2.1 and assume
also that n! is invertible in Γ(X, OX ). We define the (degree 0) ad hoc Hochschild
homology of mf (X, L, W ) relative to S to be
hoc
(X/S, L, W ) := H0 (X, Unfold Fold Ω·dW ).
HHad
0
hoc
As we will see in the next section, HHad
(X/S, L, W ) is the target of what we
0
term the “ad hoc Chern character” of a twisted matrix factorization belonging to
mf (X, L, W ).
We use the adjective “ad hoc” for two reasons. The first is that we offer here no
justification that this definition deserves to be called “Hochschild homology”. See,
however, [22, 24] for the affine case and [20] for the non-affine case with L = OX .
The second reason is that whereas the support of every object of mf (X, L, W )
is contained in the singular locus of the subscheme Y cut out by W (as will be
justified in the next subsection), the twisted two-periodic complex Fold Ω·dW is not
so supported in general. A more natural definition of Hochshild homology would
thus be given by H0Z (X, Unfold Fold Ω·dW ), where Z is the singular locus of Y and
H0Z refers to hypercomology with supports (i.e., local cohomology). Since this more
sensible definition of Hochschild homology is not necessary for our purposes and
hoc
only adds complications to what we do, we will stick with using HHad
.
0
Recall that the Jacobi sheaf is defined as


dW
n−1
JW = J (X/S, L, W ) = coker ΩX/S
(−1) −−→ ΩnX/S

so that there is a canonical map

Ω·dW → Hn (Ω·dW )[−n] = JW (n)[−n].
From it we obtain the map
Fold(Ω·dW ) → Fold(JW (n)[−n]) ∼
= Fold(JW ( n2 )).
Applying Unfold and using the canonical map Unfold Fold(JW ( n2 )) → JW ( n2 ) results in a map Unfold Fold Ω·dW → JW ( n2 ). Finally, applying H0 (X, −) yields the
map
(2.21)

hoc
HHad
(X/S, L, W ) → H 0 (X, JW ( n2 )).
0

This map will play an important role in the rest of this paper. Let us observe that
in certain situations, it is an isomorphism:

14

MARK E. WALKER

Proposition 2.22. Assume S = Spec k for a field k, X = Spec(Q) for a smooth
k-algebra Q, and hence LX = OX and W ∈ Q. If the morphism of smooth varieties
W : X → A1 has only isolated critical points and n is even, then the canonical map
HHad
0

hoc

(X/S, W ) → H 0 (X, JW ( n2 )) =

ΩnQ/k
n−1
dW ∧ ΩQ/k

is an isomorphism.
Proof. These conditions ensure that the complex of Q-modules Ω·dW is exact except
on the far right where it has homology J (n). The result follows since X is affine. 
Example 2.23. Assume S = Spec(k) for a field k, X = Ank = Spec k[x1 , . . . , xn ],
and the morphism f : Ank → A11 associated to a given polynomial f ∈ k[x1 , . . . , xn ]
has only isolated critical points. Then
hoc
HHad
(X/S, f ) ∼
= H 0 (X, Jf ( n2 )) =
0

ΩnQ/k
df ∧

n−1
ΩQ/k

=

k[x1 , . . . , xn ]
dx1 ∧ · · · ∧ dxn .
∂f
∂f
( ∂x
, . . . , ∂x
)
1
n

2.5. Supports. We fix some notation. For a Noetherian scheme Y , the non-regular
locus of Y is
Nonreg(Y ) := {y ∈ Y | the local ring OY,y is not a regular local ring}.
Under mild additional hypotheses (e.g., Y is excellent), Nonreg(Y ) is a closed subset
of Y .
Assume g : Y → S is a morphism of finite type. Recall that g is smooth near
y ∈ Y if and only if it is flat of relative dimension n near y and the stalk of Ω1Y /S
at y is a free OY,y -module of rank n. We define singular locus of g : Y → S to be
the subset
Sing(g) = Sing(Y /S) = {y ∈ Y | g is not smooth near y}.
At least for a flat morphism g : Y → S of finite type, the singular locus of g is a
closed subset of Y by the Jacobi criterion. When S = Spec(k) for a field k and
there is no danger of confusion, we write Sing(Y ) instead of Sing(Y / Spec(k)). If
Y is finite type over a field k, then Nonreg(Y ) ⊆ Sing(Y ) = Sing(Y / Spec(k)), and
equality holds if k is perfect.
Proposition 2.24. Assume X is a Noetherian scheme, L is a line bundle on X,
and W is a regular global section of L. Let Y ⊆ X be the closed subscheme cut
out by W . Then for every E, F ∈ mf (X, L, W ), the twisted two-periodic complex
Hommf (E, F) is supported Nonreg(Y ).
Proof. For x ∈ X, we have
Hommf (E, F)x ∼
= Hommf (Ex , Fx )
where the hom complex on the right is for the category mf (Spec(OX,x ), Lx , Wx ).
By choosing a trivialization OX ∼
= Lx , we may identify this category with mf (Q, f )
for a regular local ring Q and non-zero-divisor f . The assertion thus becomes that
if f is non-zero-divisor in regular local ring Q and Q/f is also regular, then the
complex Hommf (E, F) is acyclic for all objects E, F ∈ mf (Q, f ). This holds since the
∗
d Q/f (coker(E), coker(F));
cohomology modules of the complex Hommf (E, F) are Ext
see Theorem 3.2 below.


CHERN CHARACTERS FOR TWISTED MATRIX FACTORIZATIONS

15

Example 2.25. Suppose k is a field, Q is a smooth k-algebra, m is a maximal
ideal of Q, and f1 , . . . , fc ∈ m form a regular sequence such that the non-regular
locus of the ring R := Q/(f1 , . . . , fc ) is {m}. Let X = Pc−1
= Proj Q[T1 , . . . , Tc ],
Q
P
L = OX (1), and W = i fi Ti . Then
c−1
Nonreg(Y ) = Pc−1
Q/m ⊆ PQ .

2.6. Chern classes of matrix factorizations. If E ∈ mf (X, L, W ) is a matrix
factorization that admits a connection (for example, if p is affine), we define strict
morphisms
(j)
(j)
ΨE,∇ : E → ΩdW ⊗ E, for j ≥ 0,
recursively, by letting
(1)

Ψ(0) = idE , ΨE,∇ = ΨE,∇ ,
(j)

and, for j ≥ 2, defining ΨE,∇ as the composition of
(j−1)

idΩ(1) ⊗ΨE,∇

(1)

(j−1)

E −−−−−−−−−→ ΩdW ⊗ ΩdW

∧⊗id

(j)

E
⊗ E −−−−→
ΩdW ⊗ E.

(j)

By Lemma 2.17, ΨE,∇ is independent up to homotopy of the choice of ∇.
Example 2.26. Continuing with Examples 2.13, 2.15 and 2.18, the map Ψ(∧j)
arising from the trivial connection is in degree zero given by
j factors

}|
{
z
(1 + dA) · · · (1 + dA)(1 + dB)

for j even and by

j factors

}|
{
z
(1 + dB) · · · (1 + dA)(1 + dB)
for j odd, where the product is taken in the ring Matr×r (Ω·Q/k ).
By the duality enjoyed by the category mf (X, L, Q), we obtain a morphism of
objects of mf (X, L, 0) of the form
(j)

(j)

Ψ̃E : Endmf (E) = E∗ ⊗mf E → ΩdW .
Moreover, as a morphism in the category [mf (X, L, 0)]naive , it is independent of
the choice of connection. Take j = n and assume n! is invertible, so that we have
an isomorphism
(n)
ΩdW ∼
= Fold Ω·dW .
We obtain the morphism
Endmf (E) → Fold Ω·dW
in mf (X, L, 0) and hence a map of unbounded complexes
(2.27)

chE : Unfold Endmf (E) → Unfold Fold Ω·dW .
id

E
The identity morphism on E determines a morphism OX −−→
Unfold Endmf (E),
0
that is, an element of H (X, Unfold Endmf (E)).

Definition 2.28. Under Assumptions 2.1 and 2.2, define the ad hoc Chern character of E ∈ mf (X, L, W ) relative to p : X → S to be
hoc
ch(E) = chE (idE ) ∈ HHad
(X/S, L, W ).
0

16

MARK E. WALKER

hoc
Recall that when n is even, we have the map HHad
(X/S, L, W ) → H 0 (X, JW ( n2 ))
0
defined in (2.21), and hence we obtain an invariant of E in H 0 (X, JW ( n2 )). Since
this invariant will be of crucial importance in the rest of this paper, we give it a
name:

Definition 2.29. Under Assumptions 2.1 and 2.2 and with n even, define the ad
hoc top Chern class of E ∈ mf (X, L, W ) relative to p : X → S to be the element
ctop (E) ∈ H 0 (X, JW ( n2 )) given as the image of ch(E) under (2.21).
Example 2.30. With the notation and assumptions of Example 2.13, we have


dW
−−→ Ω2i+1
M ker Ω2i
Q/k
Q/k
hoc
hoc

.
HHad
(X/S, L, W ) = HHad
(Spec(Q)/ Spec(k), O, W ) =
0
0
2i−1 dW
2i
i im ΩQ/k −−→ ΩQ/k

and, when n is even, H 0 (X, JW ( n2 )) is the last summand:
H 0 (X, JW ( n2 )) =

ΩnQ/k
n−1
dW ∧ ΩQ/k

.

We have
n factors

}|
{
z
2
tr(dAdB · · · dB).
n!
Here dA and dB are r × r matrices with entries in Ω1 − Q/k and the product is
occurring in the ring Matr×r (Ω·Q/k ), and the map tr : Matr×r (ΩnQ/k ) → ΩnQ/k is
the usual trace map.
Our formula for ctop (E) coincides with the “Kapustin-Li” formula [14] found in
many other places, at least up to a sign: In the work of Polishchuck-Vaintrob [22,
n
Cor 3.2.4], for example, there is an additional a factor of (−1)( 2 ) in their formula
for the Chern character.
ctop (E) =

Since the top Chern class of E ∈ mf (X, L, W ) is defined as the image of idE
under the composition
ch

E
Unfold Endmf (E) −−→
Unfold Fold Ω·dW → JW ( n2 )

of morphisms of complexes of coherent sheaves on X, the following result is an
immediate consequence of Proposition 2.24. Recall that for a quasi-coherent sheaf
F on X and a closed subset Z of X, HZ0 (X, F ) denotes the kernel of H 0 (X, F ) →
H 0 (X \ Z, F ).
Proposition 2.31. For any E ∈ mf (X, L, W ), its top Chern class is supported on
Nonreg(Y ):
0
n
ctop (E) ∈ HNonreg(Y
) (X, JW ( 2 )).
Remark 2.32. It is also follows from what we have established so far that the Chern
character of any E is supported on Nonreg(Y ) in the sense that it lifts canonically
to an element of
H0Nonreg(Y ) (X, Unfold Fold Ω·dW ),
where in general H0Z denote local hyper-cohomology of a complex of coherent
sheaves.

CHERN CHARACTERS FOR TWISTED MATRIX FACTORIZATIONS

17

2.7. Functorality of the Chern character and the top Chern class. The
goal of this subsection is to prove Chern character is functorial in S. Throughout,
we suppose Assumptions 2.1 and 2.2 hold.
Let i : S ′ → S be a morphism of Noetherian schemes and write L′ , X ′ , W ′ for
the evident pull-backs: X ′ = X ×S S ′ , L′S = i∗ (LS ), and W ′ = i∗ (W ). The typical
application will occur when i is the inclusion of a closed point of S.
There is a functor
(2.33)

i∗ : mf (X, L, W ) → mf (X ′ , L′ , W ′ )

induced by pullback along the map X ′ → X induced from i. There is a canonical
isomorphism i∗ Ω·dW ∼
= Ω·dW ′ and hence an hence an induced map
hoc
hoc
i∗ : HHad
(X/S, L, W ) → HHad
(X ′ /S ′ , L′ , W ′ ).
0
0

Proposition 2.34. For E ∈ mf (X, L, W ), we have
hoc
i∗ (ch(E)) = ch(i∗ (E)) ∈ HHad
(X ′ /S ′ , L′ , W ′ ).
0

Proof. Recall that the Chern character of E is determined by a map of matrix
factorizations
E → E ⊗ Ω·dW
which is itself determined by a choice of connection Ej → Ej ⊗OX Ω1X/S for j = 0, 1.
Since our connections are OS -linear, we have the pull-back connection
i∗ Ej → i∗ (Ej ) ⊗OX ′ Ω1X ′ /S ′ , j = 0, 1.
which we use to define the map
i∗ E → i∗ E ⊗ Ω·dW ′ .
Using also that i∗ End(E) = End(i∗ E), it follows that the square
End(E)

/ Fold Ω·
dW


Ri∗ End(i∗ E)


/ Ri∗ Fold Ω· ′
dW

commutes. Applying H0 (X, Unfold(−)) gives the commutative square
H0 (X, Unfold End(E))

hoc
/ HHad
(X/S, L, W )
0

i∗


H0 (X ′ , Unfold End(i∗ E))

i∗



hoc
/ HHad
(X ′ /S ′ , L′ , W ′ ).
0

The result follows from the fact that i∗ : H0 (X, Unfold End(E)) → H0 (X ′ , Unfold End(i∗ E))
sends idE to idi∗ E .

Recall from (2.21) that, when n is even, there is a natural map
hoc
HHad
(X/S, L, W ) → Γ(X, JX/S,L,W ( n2 ))
0

Since i∗ JX/S,L,W ∼
= JX ′ /S ′ ,L′ ,W ′ , we obtain a map
i∗ : Γ(X, JX/S,L,W ( n2 )) → Γ(X ′ , JX ′ /S ′ ,L′ ,W ′ ( n2 )).

18

MARK E. WALKER

Corollary 2.35. When n is even, for any E ∈ mf (X, L, W ), we have
i∗ (ctop (E)) = ctop (i∗ (E)) ∈ Γ(X, JX ′ /S ′ ,L′ ,W ′ ( n2 )).
Proof. This follows from Proposition 2.34 and the fact that
hoc
HHad
(X/S, L, W )
0

/ Γ(X, JX/S,L,W ( n2 ))

i∗

i∗




/ Γ(X ′ , JX ′ /S ′ ,L′ ,W ′ ( n2 ))

hoc
HHad
(X ′ /S ′ , L′ , W ′ )
0

commutes.


3. Relationship with affine complete intersections

We present some previously known results that relate twisted matrix factorizations with the singularity category of affine complete intersections. We start with
some basic background.
For any Noetherian scheme X, we write Db (X) for the bounded derived category
of coherent sheaves on X and Perf(X) for the full subcategory of perfect complexes
— i.e., those complexes of coherent sheaves on X that are locally quasi-isomorphic
to bounded complexes of finitely generated free modules. We define Dsg (X) to be
the Verdier quotient Db (X)/ Perf(X). For a Noetherian ring R, we write Dsg (R)
for Dsg (Spec(R)).
The category Dsg (X) is triangulated. For a pair of finitely generated R-modules
M and N , we define their stable Ext-modules as
i

d (M, N ) := HomD (R) (M, N [i]),
Ext
R
sg

where on the right we are interpreting M and N [i] as determining complexes (and
hence objects of Dsg (R)) consisting of one non-zero component, lying in degrees 0
and −i, respectively. Note that since Db (R) → Dsg (R) is a triangulated functor by
construction and the usual ExtR -modules are given by HomDb (R) (M, N [i]), there
is a canonical map
i
d R (M, N )
ExtiR (M, N ) → Ext

that is natural in both variables.

Proposition 3.1. [1, 1.3] For all finitely generated R-modules M and N , the natural map
i
d (M, N )
Exti (M, N ) → Ext
R

R

is an isomorphism for i ≫ 0.

3.1. Euler characteristics and the Herbrand difference for hypersurfaces.
Throughout this subsection, assume Q is a regular ring and f ∈ Q is a non-zeroα
divisor, and let R = Q/f . Given an object E = (E1 −==
− E0 ) ∈ mf (Q, f ), define
β

α

→ E0 ). The Q-module coker(E) is annihilated by f and
coker(E) to be coker(E1 −
we will regard it as an R-module.
Theorem 3.2 (Buchweitz). [1] If R = Q/f where Q is a regular ring and f is a
non-zero-divisor, there is an equivalence of triangulated categories
∼
=

hmf (Q, f ) −
→ Dsg (R)

CHERN CHARACTERS FOR TWISTED MATRIX FACTORIZATIONS

19

induced by sending a matrix factorization E to coker(E), regarded as an object of
Dsg (R).
In particular, we have an isomorphisms
i

and

d R (coker(E), coker(F)), for all i,
Extihmf (E, F) := Homhmf (E, F[i]) ∼
= Ext
Extihmf (E, F) ∼
= ExtiR (coker(E), coker(F)), for i ≫ 0.

Since the category hmf (Qp , f ) is trivial for all f ∈ p such that Rp is regular,
Homihmf (E, F) is an R-module supported on Nonreg(R), for all i. If we assume
Nonreg(R) is a finite set of maximal ideals, then Homihmf (E, F) has finite length as
an R-module, allowing us to make the following definition.
Definition 3.3. Assume Q is a regular ring and f ∈ Q is a non-zero-divisor such
that Nonreg(R) is a finite set of maximal ideals, where R := Q/f . For E, F ∈
mf (Q, f ), define the Euler characteristic of (E, F) to be
χ(E, E) = length Hom0hmf (E, F) − length Hom1hmf (E, F).
In light of Theorem 3.2, if M and N are the cokernels of E and F, then
χ(E, F) = hR (M, N ),

(3.4)

where the right-hand side is the Herbrand difference of the R-modules M and N ,
defined as
0

1

d R (M, N ) − length Ext
d R (M, N )
hR (M, N ) = length Ext

2i+1
(M, N ), i ≫ 0.
= length Ext2i
R (M, N ) − length ExtR

Lemma 3.5. Suppose φ : Q → Q′ is a flat ring homomorphism between regular
rings, f is a non-zero-divisor in Q, and f ′ := φ(f ) is a non-zero-divisor of Q′ .
Assume Nonreg(Q/f ) = {m} and Nonreg(Q′ /f ′ ) = {m′ } for maximal ideals m, m′
satisfying the condition that mQ′ is m′ -primary.
Then, for all E, F ∈ mf (Q, f ),
χmf (Q,f ) (E, F) = λ · χmf (′ Q,f ′ ) (E ⊗Q Q′ , F ⊗Q Q′ ),
where λ := lengthQ′ (Q′ /mQ′ ).
Proof. Let R = Q/f and R′ = Q′ /f ′ . The induced map R → R′ is flat and hence
for any pair of R-modules M and N , we have an isomorphism
ExtiR (M, N ) ⊗R R′ ∼
= ExtiR′ (M ⊗R R′ , N ⊗R R′ )
of R′ -modules. For i ≫ 0, ExtiR (M, N ) is supported on {m} and ExtiR′ (M ⊗R
R′ , N ⊗R R′ ) is supported on {m′ }. It follows that
lengthR′ ExtiR′ (M ⊗R R′ , N ⊗R R′ ) = lengthR′ (ExtiR (M, N )⊗R R′ ) = λ lengthR ExtiR (M, N ).
′

Hence hR (M ′ , N ′ ) = λhR (M, N ) and the result follows from (3.4).



20

MARK E. WALKER

3.2. Matrix factorizations for complete intersections. Using a Theorem of
Orlov [19, 2.1], one may generalize Theorem 3.2 to complete intersection rings. The
precise statement is the next Theorem. Versions of it are found in [21], [15], [18],
[23]; the one given here is from [2, 2.11].
Theorem 3.6. Assume Q is a regular ring of finite Krull dimension and f1 , . . . , fc
is a regular sequence of elements of Q. Let R = Q/(f1 , . . . , fc ), define X = Pc−1
=
Q
P
Proj Q[T1 , . . . , Tc ] and set W = i fi Ti ∈ Γ(X, O(1)). There is an equivalence of
triangulated categories
∼
=

Dsg (R) −
→ hmf (X, O(1), W ).
The isomorphism of Theorem 3.6 has a certain naturality property that we need.
Suppose a1 , . . . , ac is a sequence of element of Q that generate the unit ideal, and
let i : Spec(Q) ֒→ X be the associated closed immersion. Then we have a functor
X
ai fi ).
i∗ : hmf (X, O(1), W ) → hmf (Q,
i

∗
∗
Technically, the right hand side should
P be hmf (Spec(Q), i O(1), i (W )), but it is
canonically isomorphic to hmf (Q, i aP
i fi ) because there is a canonical isomorphism i∗ O(1) ∼
i∗ (W ) to i ai fi .
= Q that sends P
Also, the quotient map Q/( i ai fi ) −−։ R has finite projective dimension, since,
locally on Q, it is given by modding out by a regular sequence of length c − 1. We
thus an induced functor
X
ai f i )
res : Dsg (R) → Dsg (Q/
i

on singularity categories, given by restriction of scalars.
Proposition 3.7. With the notation above, the square
∼
=

Dsg (R)

/ hmf (Pc−1 , O(1), W )
Q

res

Dsg (Q/


P

i

i∗
∼
=

ai f i )

/ hmf (Q,

commutes up to natural isomorphism.

P

i

ai f i )

Proof. To prove this, we need to describe the equivalence of Theorem 3.6 explicitly.
Let Y ⊆ Pc−1
denote the closed subscheme cut out by W . Then there are a pair of
Q
equivalences
(3.8)

∼
=

∼
=

Dsg (R) −
→ Dsg (Y ) ←
− hmf (X, O(1), W ),
∼
=

which give the equivalence Dsg (R) −
→ hmf (X, O(1), W ) of the Theorem.
The left-hand equivalence of (3.8) was established by Orlov [19, 2.1] (see also
[2, A.4]), and it is induced by the functor sending a bounded complex of finitely
generated R-modules M · to β∗ π ∗ (M · ), where π : Pc−1
→ Spec(R) is the evident
R
projection and β : Pc−1
֒→
Y
is
the
evident
closed
immersion.
R
The right-hand equivalence of (3.8) is given by [3, 6.3], and it sends a twisted
α

β

→ E1 (1)) to coker(α) (which may be regarded
→ E0 −
matrix factorization E = (E1 −
as a coherent sheaf on Y .)

CHERN CHARACTERS FOR TWISTED MATRIX FACTORIZATIONS

For brevity, set g =
the diagram
(3.9)

P

i

ai fi . The Proposition will follow once we establish that

β∗ ◦π ∗
∼
=

/ Dsg (Y ) o
s
ss
ss
k∗
s
ss ∗
 yss Lj
Dsg (Q/g) o
∼
Dsg (R)

21

∼
=

hmf (Pc−1
Q , O(1), W )
i∗


hmf (Q, g)

=

commutes up to natural isomorphism, where k : Spec(R) ֒→ Spec(Q/g) is the
canonical closed immersion and j : Spec(Q/g) → Y is the restriction of i. (The
closed immersion j is locally a complete intersection and thus of finite flat dimension. It follows that it induces a morphism on singularity categories, which we write
Lj ∗ .)
The right-hand square of (3.9) commutes since for a matrix factorization E =
(E1 → E0 → E1 (1)) we have
j ∗ coker(E1 → E0 ) ∼
= coker(i∗ E1 → i∗ E0 )
and
j ∗ coker(E1 → E0 ) ∼
= Lj ∗ coker(E1 → E0 ).
The first isomorphism is evident. The second holds since j has finite flat dimension
and coker(E1 → E0 ) is an “infinite syzygy”; that is, coker(E1 → E0 ) has a right
resolution by locally free sheaves on Y , namely
γ ∗ E1 (1) → γ ∗ E0 (1) → γ ∗ E1 (2) → · · · ,
where γ : Y ֒→ X is the canonical closed immersion.
To show the left-hand triangle commutes, we consider the Cartesian square of
closed immersions
c−1 /
PR
O

β

j

iR

O
Spec(R) /

/Y
O

k

O
/ Spec(Q/g),

where iR denote the restriction of i to Spec(R). This square is Tor-independent
Y
(i.e., TorO
i (β∗ OPc−1 , j∗ OSpec(Q/g) ) = 0 for i > 0), from which it follows that
R

Lj ∗ ◦ β∗ = k∗ ◦ Li∗R .
Since Li∗R ◦ π ∗ is the identity map, the left-hand triangle of (3.9) commutes.



4. Some needed results on the geometry of complete intersections
Let k be a field and Q a smooth k-algebra of dimension n. Set V = Spec(Q), a
smooth affine k-variety. Given f1 , . . . , fc ∈ Q, we have the associated morphism
f := (f1 , . . . , fc ) : V → Ack
of smooth k-varieties, induced by the k-algebra map k[t1 , . . . , tc ] → Q sending ti to
fi . Define the Jacobian of f , written Jf , to be the map
Jf : Qc → Ω1Q/k

22

MARK E. WALKER

given by (df1 , . . . , dfc ). (More formally, Jf is the map f ∗ Ω1Ac /k → Ω1V /k induced by
k
f , but we identify f ∗ Ω1Ac /k with Qc by using the basis dt1 , . . . , dtc of Ω1Ac /k . Also,
k
k
Jf is really the dual of what is often called the Jacobian of f .)
For example, if Q = k[x1 , . . . , xn ], then Jf is given by the n× c matrix (∂fi /∂xj ),
using the basis dx1 , . . . , dxn of Ω1Q/k .
For a point x ∈ V , let κ(x) denote its residue field and define
Jf (x) = Jf ⊗Q κ(x) : κ(x)c → Ω1Q/k ⊗Q κ(x) ∼
= κ(x)n
to be the map on finite dimensional κ(x)-vector spaces induced by Jf . Define
(4.1)

Vj := {x ∈ V | rank Jf (κ(x)) ≤ j} ⊆ V,

so that we have a filtration
∅ = V−1 ⊆ V0 ⊆ · · · ⊆ Vc = V.
Note that the set Vc−1 is the singular locus of the map f .
For example, if Q = k[x1 , . . . , xn ], then Vj is defined by the vanishing of the
(j + 1) × (j + 1) minors of the matrix (∂fi /∂xj ).
Example 4.2. Let k be a field of characteristic not equal to 2, let Q = k[x1 , . . . , xn ]
so that V = Ank , let c = 2, and define f1 = 21 (x21 + · · · + x2n ) and f2 = 12 (a1 x21 +
· · · + an x2n ), where a1 , . . . , an ∈ k are such that ai 6= aj for all i 6= j. Thus f is
a morphism between two affine spaces: f : Ank → A2k . For the usual basis, the
Jacobian is the matrix


x1 a1 x1
 x2 a2 x2 


Jf =  .
..  .
 ..
. 
xn

an xn

Then V0 is just the origin in V = Ank and V1 is the union of the coordinate axes.

In the previous example, we have dim(Vj ) ≤ j for all j < c. This will turn out
to be a necessary assumption for many of our result.
Let us summarize the notations and assumptions we will need in much of the
rest of this paper:
Assumptions 4.3. Unless otherwise indicated, from now on we assume:
(1) k is field.
(2) V = Spec(Q) is smooth affine k-variety of dimension n.
(3) f1 , . . . , fc ∈ Q is a regular sequence of elements. We write
f : V → Ack
for the flat morphism given by (f1 , . . . , fc ). (Note that c ≤ n.)
(4) The singular locus of f −1 (0) = Spec(Q/(f1 , . . . , fc )), is zero-dimensional
— say Sing(f −1 (0)) = {v1 , . . . , vm } where the vi ’s are closed points of V .
(5) dim(Vj ) ≤ j for all j < c, where Vj is defined in (4.1).
Observe that the singular locus of f −1 (0) may be identified with Vc−1 ∩ f −1 (0).
For the results in this paper, we will be allowed to shrink about the singular
locus of f −1 (0). In this situation, if char(k) = 0, the last assumption in above list
is unnecessary, as we now prove:

CHERN CHARACTERS FOR TWISTED MATRIX FACTORIZATIONS

23

Lemma 4.4. If all of Assumptions 4.3 hold except possibly (5) and char(k) = 0,
then for a suitably small affine open neighborhood V ′ of the set {v1 , . . . , vm }, we
have dim(Vj′ ) ≤ i for all j ≤ c − 1.
Proof. Since the induced map Sing(f ) → Ac has finite type, the set
B := {y ∈ Sing(f ) | Sing(f ) ∩ dim(f −1 (f (y)) > 0}
is a closed subset of Sing(f ) by the upper-semi-continuity of fiber dimensions
[EGAIV, 13.1.3]. Since Sing(f −1 (0)) = f −1 (0) ∩ Sing(f ) is a finite set of closed
points, it follows that {v1 , . . . , vm } ∩ B = ∅. Thus V ′ := V \ B is an open neigh′
is quasi-finite over Ac .
borhood of the vi ’s and Sing(f |V ′ ) = Sing(f ) ∩ V ′ = Vc−1
′
Shrinking V further, we may assume it is affine.
′
′ ) is
→ f (Vc−1
By [11, III.10.6], we have dim(f (Vj′ )) ≤ j for all j. Since Vc−1
quasi-finite, so is Vj′ → f (Vj′ ) for all j ≤ c − 1, and hence dim(Vj′ ) ≤ i for all
j ≤ c − 1.

Example 4.5. Suppose char(k) = p > 0, Q = k[x1 , . . . , xc ], and let f1 (x) =
xp1 , · · · , fc (x) = xpc . Then Jf : Qc → Ω1Q/k ∼
= Qc is the zero map, and yet
p
p
k[x1 , . . . , xc ]/(x1 , . . . , xc ) has an isolated singularity at (x1 , . . . , xc ). This shows
that the characteristic 0 hypothesis is necessary in Lemma 4.4.
Given the set-up in Assumptions 4.3, we let
S = Pc−1
= Proj k[T1 , . . . , Tc ]
k
and

X = Pc−1
×k V = Proj Q[T1 , . . . , Tc ],
k

and we define
W = f1 T1 + · · · + fc Tc ∈ Γ(X, O(1)).
As before, we have a map
dW : OX → Ω1X/S (1),
or, equivalently, a global section dW ∈ Γ(X, Ω1X/S (1)). Recall that Ω1S/X (1) is the
coherent sheaf associated to the graded module Ω1Q/k [T1 , . . . , Tc ](1), and dW may
be given explicitly as the degree one element
dW = df1 T1 + · · · + dfc Tc ∈ Ω1Q/k [T1 , . . . , Tc ],
or, in other words,

(4.6)




T1
 T2 
 
dW = Jf ·  .  .
 .. 
Tc

Proposition 4.7. With Assumptions 4.3, dW is a regular section of Ω1X/S (1).
Proof. Since X = Pc−1
×k V is smooth of dimension n + c − 1 and Ω1X/S is locally
k
free of rank n, it suffices to prove the subscheme Z cut out by dW has dimension
at most c − 1. Using (4.6), we see that the fiber of Z → V over a point x ∈ V is a
linear subscheme of Pc−1
κ(x) of dimension c − 1 − rank(Jf (x)). That is, if x ∈ Vj \ Vj−1 ,
then the fiber over Z → Y over y has dimension c − 1 − j. (This includes the case
j = c in the sense that the fibers over Vc \ Vc−1 are empty.) Since dim(Vj ) ≤ j if
j < c by assumption, it follows that dim(Z) ≤ c − 1.


24

MARK E. WALKER

Corollary 4.8. If Assumptions 4.3 hold, then the Jacobian complex Ω·dW is exact
everywhere except in the right-most position, and hence determines a resolution of
JW (n) by locally free coherent sheaves.
We will need the following result in the next section.
Proposition 4.9. Under Assumptions 4.3, there is an open dense subset U of
Pc−1
such that for every k-rational point [a1 : · · · : ac ] ∈ U , the singular locus of
k
the morphism
X
ai fi : V → A1k
i

has dimension 0.

Proof. Let Z be the closed subvariety of Pc−1
×k V defined by the vanishing of
k
dW (see (4.6)). The fiber of the projection map π : Z → Pc−1
over a k-rational
k
point
[a
:
·
·
·
:
a
]
may
be
identified
with
the
singular
locus
of
the morphism
1
c
P
1
a
f
:
V
→
A
.
The
claim
is
thus
that
there
is
a
open
dense
subset
U of Pc−1
over
i
i
k
i
k
which the fibers of π have dimension zero. Such a U exists because dim(Z) ≤ c − 1,
as we established in the proof of Proposition 4.7. (U may be taken to be the
complement of π(B) where B is the closed subset {z ∈ Z | dim π −1 π(z) ≥ 1} of
Z.)

5. The vanishing of Chern characters
In this section, we prove that the top Chern class of a twisted matrix factorization
vanishes in certain situations. We combine this with a Theorem of PolishchukVaintrob to establish the vanishing of hc and ηc under suitable hypotheses.
5.1. The vanishing of the top Chern class. The following theorem forms the
key technical result of this paper.
Theorem 5.1. If Assumptions 4.3 hold and in addition
Pn is even, char(p) ∤ n!,
c−1
and c ≥ 2, then ctop (E) = 0 for any E ∈ mf (PQ
, O(1), i fi Ti ).

Proof. The key point is that, under Assumptions 4.3 with c ≥ 2, we have
(5.2)

HP0c−1 ×{v1 ,...,vm } (Pc−1
×k V, JW (i)) = 0
k

for all i < n.
To prove this, since
HP0c−1 ×{v1 ,...,vm } (Pc−1 ×k V, JW (i)) =

m
M

HP0c−1 ×{vi } (Pc−1 ×k V, JW (i))

i=1

and since we may replace V = Spec(Q) by any open neighborhood of vi without
affecting the Q-module HP0c−1 ×{vi } (Pc−1 ×k V, JW (i)), we may assume m = 1 and
that there exists a regular sequence of elements x1 , . . . , xn ∈ m that generate the
maximal ideal m of Q corresponding to v = v1 .
Write C := C(x1 , . . . , xn ) for the “augmented Cech complex”


M  1 
M 1
1
Q
→
→ ··· → Q
,
Q
Q→
xi
xi xj
x1 · · · xn
i,j
i

CHERN CHARACTERS FOR TWISTED MATRIX FACTORIZATIONS

25

with Q in cohomological
degree 0. Equivalently C = C(x1 ) ⊗Q · · · ⊗Q C(xn ), where
h i
c−1
1
, we have
C(xi ) = (Q ֒→ Q xi ). For any coherent sheaf F on PQ
HP0c−1 ×{v} (Pc−1 ×k V, F ) = Γ(Pc−1 ×k V, H0 (F ⊗Q C)).

(5.3)

The complex C is exact in all degrees except in degree n, and we set E = H n (C).
Explicitly,
i
h
1
Q x1 ···x
n
h
i.
E=
1
Σi Q x1 ···xˆi ···xn

(The localization of E at m is an injective hull of the residue field, but we do not
∼
need this fact.) Thus we have a quasi-isomorphism C −
→ E[−n].
By Corollary 4.8, there is a quasi-isomorphism
∼

Ω·dW (−n)[n] −
→ JW .
Since C is a complex of flat modules, the map
∼

→ JW ⊗Q C
Ω·dW (−n)[n] ⊗Q C −
∼

is also a quasi-isomorphism. Combining this with the quasi-isomorphism C −
→
E[−n] gives an isomorphism in the derived category
Ω· (−n) ⊗Q E ∼
= JW ⊗ C.
dW

For any i we obtain an isomorphism
H0 (JW (i) ⊗ C) ∼
= H0 (Ω·dW (i − n) ⊗Q E)
of quasi-coherent sheaves.
We thus obtain from (5.3) the isomorphism

Pc−1 ×k V, JW (i) ∼
(5.4)
Γ c−1
= Γ Pc−1 ×k V, H0 (Ω·
Pk

×{v}

k

dW (i

k

But H0 (Ω·dW (i − n) ⊗Q E) is the kernel of


− n) ⊗Q E)

dW ∧−

OPc−1 (i − n) ⊗Q E −−−−→ Ω1Pc−1 /Pc−1 (i − n + 1) ⊗Q E.
Q

Q

k

Since c − 1 ≥ 1, the coherent sheaf OPc−1 (i − n) has no global section when i < n,
Q
and hence

Γ Pc−1
×k V, H0 (Ω·dW (i − n) ⊗Q E) = 0,
k
which establishes (5.2).
P
Let Y be the closed subscheme of X = Pc−1
× V cut out by W =
i fi Ti .
k
c−1
We claim Sing(Y
/k)
⊆
P
×
{v
,
.
.
.
,
v
}.
Indeed,
Sing(Y
/k)
is
defined
by
the
1
m
k
P
1
d(f
)T
=
0
and
f
=
0,
i
=
1,
.
.
.
c,
where
d(f
)
∈
Ω
.
The
equations
i i
i
i
i
Q/k
first equation cuts out a subvariety contained in Pc−1
×
V
,
where,
recall,
Vc−1
k
c−1
k
denotes the singular locus of f : V → Ack . The equations fi = 0, i = 1, . . . c, cut out
Pc−1
×k f −1 (0). But Vc−1 ∩ f −1 (0) = Sing(f −1 (0)) = {v1 , . . . , vm }.
k
Since Nonreg(Y ) ⊆ Sing(Y /k), we deduce from (5.2) that
c−1
0
×k V, JW (i)) = 0.
HNonreg(Y
) (Pk

The vanishing of ctop (E) now follows from Proposition 2.31.
Corollary 5.5. If
• k is a field of characteristic 0,



26

MARK E. WALKER

• Q is a smooth k-algebra of even dimension,
• f1 , . . . , fc ∈ Q forms a regular sequence of elements with c ≥ 2, and
• the singular locus of Q/(f1 , . . . , fc ), is zero-dimensional,
P
then ctop (E) = 0 for all E ∈ mf (Pc−1
Q , O(1),
i fi Ti ).

Proof. Let {v1 , . . . , vm } be the singular locus of Q/(f1 , . . . , fc ) and let V ′ = Spec(Q′ )
be an affine open neighborhood of it such that dim(Vi′ ) ≤ i for all i ≤ c − 1. The
existence of such a V ′ is given by Lemma 4.4. The result follows from the (proof
of the) Theorem, since the map
X
X
fi Ti )
fi Ti ) → HP0c−1 ×{v ,...,v } (Pc−1
HP0c−1 ×{v ,...,v } (Pc−1
Q′ , O,
Q , O,
k

1

m

i

k

1

m

i

is an isomorphism.



5.2. The Polishchuk-Vaintrob Riemann-Roch Theorem. We recall a Theorem of Polishchuk-Vaintrob that relates the Euler characteristic of (affine) hypersurfaces with isolated singularities to Chern characters. This beautiful theorem
should be regarded as a form of a “Riemann-Roch” theorem. Since PolishchukVaintrob work in a somewhat different setting that we do, we begin by recalling
their notion of a Chern character.
For a field k and integer n, define Q̂ := k[[x1 , . . . , xn ]], a power series ring in
n variables, and let f ∈ Q̂ be such that the only non-regular point of Q̂/f is its
maximal ideal. Define Ω̂·Q̂/k to be the exterior algebra on the free Q̂-module
Ω̂1Q̂/k = Q̂dx1 ⊕ · · · ⊕ Q̂dxn .
In other words,
Ω̂·Q̂/k = Ω·k[x1 ,...,xn ]/k ⊗k[x1 ,...,xn ] Q̂.
Finally, define
Note that



n−1 df ∧−
Jˆ(Q̂/k, f ) = coker Ω̂Q̂/k
−−−→ Ω̂nQ̂/k .

k[[x1 , . . . , xn ]]
dx1 ∧ . . . ∧ dxn .
Jˆ(Q̂/k, f ) ∼
= ∂f
∂f
h ∂x1 , . . . , ∂x
i
n

Definition 5.6. With the notation above, given a matrix factorization E ∈ mf (Q̂, f ),
A
upon choosing bases, it may be written as E = (Q̂r −===
− Q̂r ) for r × r matrices
B

A, B with entries in Q̂. When n is even, we define the Polishchuk-Vaintrob Chern
character of E to be


n factors
}|
{
z
n
chPV (E) = (−1)( 2 ) tr dA ∧ dB ∧ · · · ∧ dB  ∈ Jˆ(Q̂/k, f ).

Theorem 5.7 (Polishchuk-Vaintrob). Assume k is a field of characteristic 0 and
f ∈ Q̂ = k[[x1 , . . . , xn ]] is a power series such that R := Q̂/f has in an isolated
singularity (i.e., Rp is regular for all p 6= m).
If n is odd, then χ(E, F) = 0 for all E, F ∈ mf (Q̂, f ).
If n is even, there is a pairing h−, −i on Jˆ(Q̂, f ), namely the residue pairing,
such that for all E, F ∈ mf (Q̂, f ), we have
χ(E, F) = hchPV (E), chPV (F)i.

CHERN CHARACTERS FOR TWISTED MATRIX FACTORIZATIONS

27

In particular, χ(E, F) = 0 if either chPV (E) = 0 or chPV (F) = 0.
It is easy to deduce from their Theorem the following slight generalization of it,
which we will need to prove Theorem 5.11 below.
Corollary 5.8. Assume k is a field of characteristic 0, Q is a smooth k-algebra of
dimension n, f ∈ Q is a non-zero-divisor such that the singular locus of R := Q/f
is zero-dimensional. Given E ∈ mf (Q, f ), if either n is odd or n is even and
n−1
ctop (E) = 0 in J (Q/k, f ) = ΩnQ/k /df ∧ ΩQ/k
, then
χ(E, F) = 0
for all F ∈ mf (Q, f ).
Proof. We start be reducing to the case where R has just one singular point m and
it is k-rational. We have
X
χmf (Q,f ) (E, F) =
χmf (Qm ,f ) (Em , Fm )
m

where the sum ranges over the singular points of R. If k is an algebraic closure of
k, then for each such m we have
∼ Hom∗
Hom∗hmf (Qm ,f ) (Em , Fm ) ⊗k k =
(Em ⊗k k, Fm ⊗k k)
hmf (Qm ⊗k k,f ⊗1)
and hence
χmf (Qm ⊗k k,f ⊗1) (Em ⊗k k, Fm ⊗k k) = [Q/m : k] · χmf (Qm ,f ) (Em , Fm ).
It thus suffices to prove the Corollary for a suitably small affine open neighborhood
of each maximal ideal m of Q ⊗k k at which Q ⊗k k/f is singular.
In this case, a choice of a regular sequence of generators x1 , . . . , xn of the
maximal ideal of Qm allows us to identify the completion of Q along m with
Q̂ = k[[x1 , . . . , xn ]]. By Lemma 3.5,
χmf (Q,f ) (E, F) = χmf (Q̂,f ) (Ê, F̂)
where Ê = E⊗Q Q̂ and F̂ = F⊗Q Q̂. Moreover, under the evident map J (Qm /k, f ) →
Jˆ(Q̂/k, f ), ctop (E) is sent to a non-zero multiple of chPV (Ê) by Example 2.30. The
result now follows from the Polishchuk-Vaintrob Theorem.

5.3. Vanishing of the higher h and η Invariants. We apply our vanishing
result for ctop to prove that the invariants hc and ηc vanish for codimension c ≥ 2
isolated singularities in characteristic 0, under some mild additional hypotheses.
First, we use the following lemma to prove the two invariants are closely related.
I thank Hailong Dao for providing me with its proof.
Lemma 5.9 (Dao). Let R = Q/(f1 , . . . , fc ) for a regular ring Q and a regular
sequence of elements f1 , . . . , fc ∈ Q with c ≥ 1. Assume M and N are finitely geni
erated R-modules and that M is MCM. If TorR
i (M, N ) and ExtR (HomR (M, R), N )
have finite length for i ≫ 0, then
ηcR (M, N ) = (−1)c hR
c (HomR (M, R), N ).
In particular, if the non-regular locus of R is zero dimensional and hc vanishes
for all pairs of finitely generated R-modules, then so does ηc .
Remark 5.10. The conditions that TorR
i (M, N ) has finite length for i ≫ 0 and that
ExtiR (HomR (M, R), N ) has finite length for i ≫ 0 are equivalent; see [13, 4.2].

28

MARK E. WALKER

Proof. We proceed by induction on c. The case c = 1 follows from the isomorphism
[6, 4.2]
i
R
d R (HomR (M, R), N ) ∼
d −i−1 (M, N ).
Ext
= Tor

Let S = Q/(f1 , . . . , fc−1 ) so that R = S/(fc ) with fc a non-zero-divisor of S.
By [5, 4.3] and [4, 3.4], TorSi (M, N ) and ExtiS (HomR (M, R), N ) have finite length
for i ≫ 0 and
1 S
ηcR (M, N ) = ηc−1
(M, N )
2c
1 S
hR
h (HomR (M, R), N ).
c (HomR (M, R), N ) =
2c c−1
Let S n −−։ M be a surjection of S-modules with kernel M ′ so that we have the
short exact sequence
0 → M ′ → S n → M → 0.

Observe that M ′ is an MCM S-module. Since fc ·M = 0 and fc is a non-zero-divisor
∼
=
of S, we have HomS (M, S n ) = 0. Using also the isomorphism HomR (M, R) −
→
1
ExtS (M, S) of S-modules coming from the long exact sequence of Ext modules
fc

associated to the sequence 0 → S −→ S → R → 0, we obtain the short exact
sequence
0 → S n → HomS (M ′ , S) → HomR (M, R) → 0.
In particular, it follows from these short exact sequences that TorSi (M ′ , N ) and
ExtiS (HomS (M ′ , S), N ) have finite length for i ≫ 0. Moreover, since hSc−1 vanishes
on free S-modules and is additive for short exact sequences, we have
hSc−1 (HomS (M ′ , S), N ) = hSc−1 (HomR (M, R), N ).
Using the induction hypotheses, we get
1
1 S
S
h (HomS (M ′ , S), N ) = (−1)c−1 ηc−1
(M ′ , N ).
hR
c (HomR (M, R), N ) =
2c c−1
2c
S
S
Finally, ηc−1
(M ′ , N ) = −ηc−1
(M, N ), since M ′ is a first syzygy of M . Combining
these equations gives
ηcR (M, N ) = (−1)c hR
c−1 (HomR (M, R), N ).
The final assertion holds since ηc is completely determined by its values on MCM
modules.

Theorem 5.11. Assume
• k is field of characteristic 0,
• Q is a smooth k-algebra,
• f1 , . . . , fc ∈ Q form a regular sequence of elements,
• the singular locus of R := Q/(f1 , . . . , fc ) is zero-dimensional, and
• c ≥ 2.
R
Then hR
c (M, N ) = 0 and ηc (M, N ) = 0 for all finitely generated R-modules M and
N.
Remark 5.12. Since k is a perfect field, the hypotheses that Q is a smooth k
algebra is equivalent to Q being a finitely generated and regular k-algebra. Likewise,
Sing(R/k) = Nonreg(R).

CHERN CHARACTERS FOR TWISTED MATRIX FACTORIZATIONS

29

Proof. By Lemma 5.9, it suffices to prove hR
c (M, N ) = 0. The value of this invariant
is unchanged if we semi-localize at Nonreg(R). So, upon replacing Spec(Q) with
a suitably small affine open neighborhood of Nonreg(R), Lemma 4.4 allows us to
assume all of Assumptions 4.3 hold.
Let EM , EN ∈ mf (Pc−1
Q , O(1), W ) be twisted matrix factorizations corresponding to the classes of M, N in Dsg (R) under the equivalence of Theorem 3.6. Now
choose a1 , . . . , ac ∈ k as in Proposition
P 4.9 so that the singular locus of Q/g is zero
dimensional, where we define g := i ai fi .
A key fact we use is that hR
c is related to the classical Herbrand difference of the
hypersurface Q/g: By [4, 3.4], for all finitely generated R-modules M and N , we
have
1
Q/g
hR
(M, N ) = hQ/g (M, N ),
c (M, N ) = h1
2
and thus it suffices to prove hQ/g (M, N ) = 0. By Proposition 3.7 the affine matrix
factorizations i∗ EM , i∗ EN ∈ hmf (Q, g) represent the classes of M, N in Dsg (Q/g),
where i : Spec(Q) ֒→ Pc−1
is the closed immersion associated to the k-rational
Q
point [a1 : · · · , ac ] of Pc−1
.
It
thus follows from (3.4) that
k
hQ/g (M, N ) = χ(i∗ EM , i∗ EN ).
By Corollary 5.8, it suffices to prove ctop (i∗ EM ) = 0 in J (Qm /k) (when n is even).
But ctop (i∗ EM ) = i∗ ctop (EM ) by Corollary 2.35 and ctop (EM ) = 0 by Corollary
5.5.

Appendix A. Relative Connections
We record here some well known facts concerning connections for locally free
coherent sheaves. Throughout, S is a Noetherian, separated scheme and p : X → S
is a smooth morphism; i.e., p is separated, flat and of finite type and Ω1X/S is locally
free.
Definition A.1. For a vector bundle (i.e., locally free coherent sheaf) E on X, a
connection on E relative to p is a map of sheaves of abelian groups
∇ : E → Ω1X/S ⊗OX E
on X satisfying the Leibnitz rule on sections: given an open subset U ⊆ X and
elements f ∈ Γ(U, OX ) and e ∈ Γ(U, E), we have
∇(f · e) = df ⊗ e + f ∇(e) in Γ(U, Ω1X/S ⊗OX E),
where d : OX → Ω1X/S denotes exterior differentiation relative to p.
Note that
imply that ∇ is OS -linear — more precisely, p∗ (∇) :

 the hypotheses
p∗ E → p∗ Ω1X/S ⊗OX E is a morphism of quasi-coherent sheaves on S.

A.1. The classical Atiyah class. Let ∆ : X → X ×S X be the diagonal map,
which, since X → S is separated, is a closed immersion, and let I denote the
sheaf of ideals cutting out ∆(X). Since p is smooth, I is locally generated by
a regular sequence. Recall that I/I 2 ∼
= ∆∗ Ω1X/S . Consider the coherent sheaf
P̃X/S := OX×S X /I 2 on X ×S X. Observe that P̃X/S is supported on ∆(X), so
that (πi )∗ P̃X/S is a coherent sheaf on X, for i = 1, 2, where πi : X ×S X → X
denotes projection onto the i-th factor.

30

MARK E. WALKER

The two push-forwards (πi )∗ P̃X/S , i = 1, 2 are canonically isomorphic as sheaves
of abelian groups, but have different structures as OX -modules. We write PX/S = P
for the sheaf of abelian groups (π1 )∗ P̃ = (π2 )∗ P̃ regarded as a OX − OX -bimodule
where the left OX -module structure is given by identifying it with (π1 )∗ P̃X/S the
right OX -module structure is given by identifying it with (π2 )∗ P̃X/S .
Locally on an affine open subset U = Spec(Q) of X lying over an affine open
−·−
subset V = Spec(A) of S, we have PU/V = (Q⊗A Q)/I 2 , where I = ker(Q⊗A Q −−−→
Q) and the left and right Q-module structures are given in the obvious way.
There is an isomorphism of coherent sheaves on X × X
∼
∆∗ Ω1
= I/I 2
X/S

given locally on generators by dg 7→ g ⊗ 1 − 1 ⊗ g. From this we obtain the short
exact sequence
0 → Ω1X/S → PX/S → OX → 0.

(A.2)

This may be thought of as a sequence of OX − OX -bimodules, but for Ω1X/S and
OX the two structures coincide.
Locally on open subsets U and V as above, we have Ω1Q/A ∼
= I/I 2 , and (A.2)
takes the form
0 → I/I 2 → (Q ⊗A Q)/I 2 → Q → 0.
Viewing (A.2) as either a sequence of left or right modules, it is a split exact
sequence of locally free coherent sheaves on X. For example, a splitting of PX/S →
OX as right modules may be given as follows: Recall that as a right module,
PX/S = (π2 )∗ P and so a map of right modules OX → PX/S is given by a map
π2∗ OX → PX/S . Now, π2∗ OX = OX×S X , and the map we use is the canonical
surjection. We refer to this splitting as the canonical right splitting of (A.2).
Locally on subsets U and V as above, the canonical right splitting of (Q ⊗A
Q)/I 2 −−։ Q is given by q 7→ 1 ⊗ q.
Given a locally free coherent sheaf E on X, we tensor (A.2) on the right by E to
obtain the short exact sequence
π

i

→E →0
→ PX/S ⊗OX E −
0 → Ω1X/S ⊗OX E −

(A.3)

of OX − OX -bimodules. Taking section on affine open subsets U and V as before,
letting E = Γ(U, E), this sequence has the form
0 → Ω1Q/A ⊗Q E → (Q ⊗A E)/I 2 · E → E → 0.
Since (A.2) is split exact as a sequence of right modules and tensor product preserves
split exact sequences, (A.3) is split exact as a sequence of right OX -modules, and
the canonical right splitting of (A.2) determines a canonical right splitting of (A.3),
which we write as
can : E → PX/S ⊗OX E.
The map can is given locally on sections by e 7→ 1 ⊗ e.
In general, (A.3) need not split as a sequence of left modules. Viewed as a
sequence of left modules, (A.3) determines an element of
Ext1 (E, Ω1 ⊗O E) ∼
= H 1 (X, Ω1 ⊗O EndO (E)),
OX

X/S

X

X/S

X

X

sometimes called the “Atiyah class” of E relative to p. To distinguish this class
from what we have called the Atiyah class of a matrix factorization in the body of

CHERN CHARACTERS FOR TWISTED MATRIX FACTORIZATIONS

31

this paper, we will call this class the classical Atiyah class of the vector bundle E,
and we write it as
(E) ∈ Ext1OX (E, Ω1X/S ).
Atclassical
X/S
The sequence (A.3) splits as a sequence of left modules if and only if Atclassical
= 0.
X/S
Lemma A.4. If p : X → S is affine, then the classical Atiyah class of any vector
bundle E on X vanishes, and hence (A.3) splits as a sequence of left modules.
Proof. Since p is affine, p∗ is exact. Applying p∗ to (A.3) results in a sequence
of OS − OS bimodules (which are quasi-coherent for both actions). But since
p ◦ π1 = p ◦ π2 these two actions coincide. Moreover, since (A.3) splits as right
modules, so does its push-forward along p∗ .
It thus suffices to prove the following general fact: If
F := (0 → F ′ → F → F ′′ → 0)
is a short exact sequence of vector bundles on X such that p∗ (F ) splits as a sequence of quasi-coherent sheaves on S, then F splits. To prove this, observe that F
determines a class in H 1 (X, HomOX (F ′′ , F ′ )) and it is split if and only if this class
vanishes. We may identify H 1 (X, HomOX (F ′′ , F ′ )) with H 1 (S, p∗ HomOX (F ′′ , F ′ ))
since p is affine. Moreover, the class of F ∈ H 1 (S, p∗ HomOX (F ′′ , F ′ )) is the image
of the class of p∗ (F ) ∈ H 1 (S, HomOS (p∗ F ′′ , p∗ F ′ )) under the map induced by the
canonical map
HomOS (p∗ F ′′ , p∗ F ′ ) → p∗ HomOX (F ′′ , F ′ ).
But by our assumption the class of p∗ (F ) vanishes since p∗ F splits.



A.2. The vanishing of the classical Atiyah class and connections. Suppose
σ : E → PX/S ⊗OX E is a splitting of the map π in (A.3) as left modules and recall
can : E → PX/S ⊗OX E is the splitting of π as a morphism of right modules given
locally by e 7→ 1 ⊗ e. Since σ and can are splittings of the same map regarded as
a map of sheaves of abelian groups, the difference σ − can factors as i ◦ ∇σ for a
unique map of sheaves of abelian groups
∇σ : E → Ω1X/S ⊗OX P.
Lemma A.5. The map ∇σ is a connection on E relative to p.
Proof. The property of being a connection may be verified locally, in which case
the result is well known.
In more detail, restricting to an affine open U = Spec(Q) of X lying over an
affine open V = Spec(A) of S, we assume E is a projective Q-module and that we
are given a splitting σ of the map of left Q-modules (Q ⊗A E)/I 2 · E −−։ E. The
map ∇σ = (σ − can) lands in I/I 2 ⊗Q Q = Ω1Q/A ⊗Q E, and for a ∈ A, e ∈ E we
have
∇σ (ae) = σ(ae) − 1 ⊗ ae
= aσ(e) − 1 ⊗ ae
= aσ(e) − a ⊗ e + a ⊗ e − 1 ⊗ ae
= a(σ(e) − 1 ⊗ e) + (a ⊗ 1 − 1 ⊗ a) ⊗ e
= a∇σ (e) + da ⊗ e,
since da is identified with a ⊗ 1 − 1 ⊗ a under Ω1Q/A ∼
= I/I 2 .



32

MARK E. WALKER

Lemma A.6. Suppose E, E ′ are locally free coherent sheaves on X and ∇, ∇′ are
connections for each relative to p. If g : E → E ′ is a morphisms of coherent sheaves,
then the map
∇′ ◦ g − (id ⊗g) ◦ ∇ : E → Ω1X/S ⊗OX E ′
is a morphism of coherent sheaves.
Proof. Given an open set U and elements f ∈ Γ(U, OX ), e ∈ Γ(U, E), the displayed
map sends f · e ∈ Γ(U, E) to
∇′ (g(f e)) − (id ⊗g)(df ⊗ e − f ∇(e) = ∇′ (f g(e)) − df ⊗ g(e) − f (id ⊗g)(∇(e))
= df ⊗ g(e) + f ∇′ (g(e)) − df ⊗ g(e) − f (id ⊗g)(∇(e))
= f ∇′ (g(e)) − f (id ⊗g)(∇(e))
= f ((∇′ ◦ g − (id ⊗g) ◦ ∇) (e)) .

Proposition A.7. For a vector bundle E on X, the function σ 7→ ∇σ determines a
bijection between the set of splittings of the map π in (A.3) as a map of left modules
and the set of connections on E relative to p. In particular, E admits a connection
relative to p if and only if Atclassical
(E) = 0.
X/S
Proof. From Lemma A.6 with g being the identity map, the difference of two connections on E is OX -linear. By choosing any one splitting σ0 of (A.3) and its
associated connection ∇0 = ∇σ0 , the inverse of
σ 7→ ∇σ
is given by
∇ 7→ ∇ − ∇σ0 + σ0 .

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