CHERN CHARACTER FOR MATRIX FACTORIZATIONS VIA CHERN-WEIL

arXiv:1310.7337v1 [math.RA] 28 Oct 2013

XUAN YU

Abstract. Given a matrix factorization, we use the Atiyah class to give an algebraic Chern-Weil type
construction to its Chern character; this allows us to realize the Chern character in an explicit way. It also
generalizes the existing result to any smooth k-algebra Q (k a commutative ring containing Q) and any f ∈ Q,
which agrees with a recent result of Platt [8]. We also study some basic properties of the Chern character.

1. Introduction
Given a commutative ring Q and an element f ∈ Q, a matrix factorization of f is a finitely generated
Z/2-graded projective Q-module M = M0 ⊕ M1 together with an odd degree endomorphism dM such that
d2M = f ⋅ 1M . Matrix factorizations were introduced by Eisenbud [4] to study modules over the ring Q/f .
The theory of matrix factorizations is very active in recent years and one of the things that’s been heavily
studied is the Chern character. Polishchuk-Vaintrob [9] establishes a Chern character for matrix factorizations
when Q = k[[x1 , ⋯, xn ]] (k is a field of characteristic 0) and f is an isolated singularity. Dyckerhoff-Murfet [2]
produce the same Chern character by an explicit description of a local duality isomorphism (also later in the
frame work of pushing forward of matrix factorizations [3]). Carqueville-Murfet [1] studies the bicategory of
Landau-Ginzburg models and recover the Chern character in this setting. Recently, Platt [8] gives an explicit
formula for the boundary bulk map; and in the case when the matrix factorization admits a connection, an
explicit formula for the Chern character.
In this paper, we use the Atiyah class At of matrix factorizations to give an algebraic Chern-Weil type
construction to the Chern character. In order to do this, we need to require At to be a strict morphism
of matrix factorizations. Our observation is that At is not a strict morphism; however, id + At is. The
Chern-Weil type construction allows us to generalize the Chern character to any finitely generated smooth
k-algebra Q (k a commutative ring containing Q) and any element f of Q. Carqueville-Murfet [1] defines
and studies Atiyah classes in a more general setting, the one we use in this paper is the more elementary one
that has already been given out in an earlier paper of Dyckerhoff-Murfet [3]. We also show that the Chern
character is in fact a ring homomorphism from the Grothendieck ring of the homotopy category of matrix
factorizations to its Hochschild homology (Corollary 5.18). As a byproduct, this construction gives a very
concrete way of constructing a complex that can be used to calculate the Hochschild homology for matrix
factorizations [11]. We also prove the naive funtoriality (Proposition 5.21) at the end.
1

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Convention Throughout all rings are Noetherian, commutative and unital. All modules are finitely generated. All complexes are bounded.
2. Matrix factorizations
First we recall the theory of matrix factorizations.
Definition 2.1. Given a ring Q and an element f of Q, a matrix factorization M of f ∈ Q is a Z/2graded Q-module M = M0 ⊕ M1 , where M is a finitely generated projective Q-module, together with an odd
endomorphism

⎡
⎢0
⎢
d=⎢
⎢
⎢d0
⎣

such that d ○ d = f ⋅ 1M .

⎤
d1 ⎥
⎥
⎥
⎥
0⎥
⎦

Equivalently, a matrix factorization for (Q, f ) consists of a pair of finitely generated projective Q-modules
M0 and M1 and Q-linear maps d0 ∶ M0 → M1 and d1 ∶ M1 → M0 such that each composition is multiplication
by f :
d0 ○ d1 = f ⋅ 1M1 and d1 ○ d0 = f ⋅ 1M0 .
We write a matrix factorization as
 M0 ) or (M1 Ð→ M0 Ð→ M1 ).
(M1 ÐÔÐ
d1

d1

d0

d0

Note that the degree 1 part is on the left for the first version. For the second version, we have the degree 0
piece in the middle and degree 1 pieces elsewhere.
xi

 Q), for all i.
Example 2.2. Let Q = C[[x]] and f = xn , then we have matrix factorizations (Q ÐÔÔÐ
xn−i

1
 Q2 ), with
Example 2.3. Given Q = C[[x, y, z]], f = xy + yz + zx, then it’s easy to check that (Q2 ÐÔÐ

d

⎡
⎤
⎡
⎢z
⎢x + y
y ⎥
⎢
⎥
⎢
⎥ and d0 = ⎢
d1 = ⎢
⎢
⎥
⎢
⎢x −x − y ⎥
⎢ x
⎣
⎦
⎣

⎤
y⎥
⎥
⎥ is a matrix factorization of f .
⎥
−z ⎥
⎦

d0

Definition 2.4. A strict morphism of matrix factorizations from M to N is a Z/2-graded Q-linear map of
degree zero α ∶ M → N such that dN ○ α = α ○ dM
Equivalently, a strict morphism is a pair of Q-linear maps α0 ∶ M0 → N0 and α1 ∶ M1 → N1 such that the
two evident squares commute.
We write M F (Q, f ) for the category of all matrix factorizations of (Q, f ) with morphisms given by the
set of strict morphisms.

CHERN CHARACTER FOR MATRIX FACTORIZATIONS VIA CHERN-WEIL

3

Definition 2.5. Two strict morphisms α, β ∶ M → N are homotopic if there exists morphisms h1 ∈
Hom(M1 , N0 ) and h0 ∈ Hom(M0 , N1 ) such that
dN ○ h + h ○ dM = α − β.
We visual a homotopy as the following:
dM

M1
α−β

// M0

dM

// M1

α−β
h0

 zz
N1

dN

 zz
// N0

α−β
h1
dN


// N1

This is an equivalent relation and is preserved by composition of strict morphism. The homotopy category
[M F (Q, f )] is obtained from M F (Q, f ) by modding out the hom sets by this equivalence relation.

A strict morphism α ∶ M → N that becomes an isomorphism in [M F (Q, f )] is called a homotopy equiv-

alence, i.e., α is a homotopy equivalence if and only if there exists a strict morphism β ∶ N → M such that
α ○ β and β ○ α are each homotopic to the appropriate identity map.
Definition 2.6. For M ∈ M F (Q, f ), the shift M[1] ∈ M F (Q, f ) is defined to be:
1
0
 M0 ) [1] = (M0 ÐÔÔ
 M1 ) .
(M1 ÐÔÐ
Ð

d

−d

d0

−d1

Definition 2.7. The cone of a strict morphism α ∶ M → N is the following matrix factorization of (Q, f ):
⎤
⎡
⎢ N
⎛
⎞
⎢d
α0 ⎥⎥⎥
⎢ 1
⎜
⎟
⎥
⎢
⎥
⎢
⎜
⎟
⎢
⎜
⎟
M⎥
⎥
⎢ 0
−d0 ⎥⎦
⎢
⎜
⎟
⎣
⎜
 N0 ⊕ M 1 ⎟
⎟.
cone(α) = ⎜N1 ⊕ M0 ÐÔÔÔÔÔÔÔÐ
⎜
⎟
⎤
⎡
⎜
⎟
⎥
⎢ N
⎥
⎢
⎜
⎟
d0
α1 ⎥
⎢
⎜
⎟
⎥
⎢
⎥
⎢
⎜
⎟
⎥
⎢
M⎥
⎢ 0
−d
⎥
⎢
⎝
⎠
1
⎦
⎣

There are canonical maps
N → cone(α) and cone(α) → M[1],
as the classic situation of chain complexes. These will give the “distinguished triangles” in the triangulated
structure discussed in the next proposition.
Proposition 2.8. For any ring Q and element f ∈ Q, the category [M F (Q, f )] is a triangulated category.

The shift functor is M ↦ M [1] and the distinguished triangles are those isomorphic (in [M F (Q, f )]) to
triangles of the form
α

MÐ
→N Ð
→ cone(α) Ð
→ M[1]

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for any strict morphism α.
Definition 2.9. Given f, f ′ ∈ Q and matrix factorizations M ∈ M F (Q, f ) and N ∈ M F (Q, f ′), the tensor
product of M and N is
 (M0 ⊗Q N0 ) ⊕ (M1 ⊗Q N1 )),
M ⊗mf N ∶= ((M1 ⊗Q N0 ) ⊕ (M0 ⊗Q N1 ) ÐÔÔÔÐ
dM ⊗N

dM ⊗N

where dM⊗N (m ⊗ n) = dM (m) ⊗ n + (−1)∣m∣ m ⊗ dN (n) for simple, homogeneous tensor m ⊗ n. The tensor
product is a matrix factorization of f + f ′ . For further details, see [12].
The tensor product − ⊗mf − of matrix factorizations is well-defined on the homotopy category of matrix
factorizations.
Proposition 2.10. (Lemma 2.2 of [12])
Given any three matrix factorizations M, N and L, we have
(M ⊕ N ) ⊗mf L ≅ (M ⊗mf L) ⊕ (N ⊗mf L)
For a complex of Q-modules, we have the following definition.
⋅
Definition 2.11. Given any complex C ⋅ of Q-modules we denote by CZ/2
the Z/2-folding, which has ⊕ C i in
i∈2Z

degree zero and

⊕ C i in degree one, together with the obvious differentials. This is a matrix factorization

i∈2Z+1

of 0.
Remark 2.12. We can talk about tensor product (in the sense of definition 2.9) between complexes of projective Q-modules and matrix factorizations. If one of the factors in the tensor product is simply a projective
Q-module, or more generally a complex of projective Q-modules, we first view it as a matrix factorization of
zero using the Z/2-folding (for the case of a single module, we follow the usual convention by placing it in
⋅
the degree 0 piece of a complex), then tensor everything as matrix factorizations, i.e., P ⋅ ⊗ M ∶= PZ/2
⊗mf M

for P ⋅ a complex. We have the following proposition addressing the problem of compatibility.
Proposition 2.13. Given two complexes X ⋅ and Y ⋅ of projective Q-modules, we have (X ⋅ ⊗cx Y ⋅ )Z/2 =
⋅
⋅
XZ/2
⊗mf YZ/2
, where ⊗cx stands for the usual tensor product of complexes.
⋅
⋅
Proof. For note that the underlying modules for (X ⋅ ⊗cx Y ⋅ )Z/2 and XZ/2
⊗mf YZ/2
are identical.

Indeed, we have
((X ⋅ ⊗cx Y ⋅ )Z/2 )1 =

i
j
⊕ ( ⊕ (X ⊗ Y ))

k is odd

i+j=k

CHERN CHARACTER FOR MATRIX FACTORIZATIONS VIA CHERN-WEIL

5

and
(XZ/2 ⊗mf YZ/2 )1 = [( ⊕ X i ) ⊗ ( ⊕

j is even

i is odd

Y j )] ⊕[( ⊕

i is even

X i ) ⊗ ( ⊕ Y j )] =
j is odd

i
j
⊕ ( ⊕ (X ⊗ Y )).

k is odd

i+j=k

⋅
⋅
Similarly, ((X ⋅ ⊗cx Y ⋅ )Z/2 )0 = (XZ/2
⊗mf YZ/2
)0 .

The fact that the differentials are the same can be seen by carefully keeping track of where elements go.

3. Algebraic Chern-Weil Theory
We review the basic Chern-Weil theory from the algebraic point of view in this section, which will be
used later in our construction. From now on, Q is a finitely generated commutative k-algebra, where k is a
commutative ring. All modules are finitely generated. Also, let Ω1Q/k be the Q-module of differential 1-forms
and ΩnQ/k ∶= ∧nQ Ω1Q/k , the Q-module of differential n-forms. For details, see [7].
Definition 3.1. Let Q be a commutative k-algebra and E a Q-module. A connection on the Q-module E
is a k-linear map ∇ ∶ E → Ω1Q/k ⊗Q E such that for any e ∈ E and q ∈ Q the following Leibniz rule holds:
∇(qe) = (dq) ⊗ e + q∇(e).
Just like the exterior differential operator d, a connection ∇ can be extended canonically to a map, which
we still denote by ∇,
Ω●Q/k ⊗Q E → Ω●+1
Q/k ⊗Q E
such that for any homogeneous element u ∈ Ω●Q/k and e ∈ E
∇(u ⊗ e) = (du) ⊗ e + (−1)∣u∣ u ∧ ∇(e).
Example 3.2. For E = Q, the exterior differential operator d is a connection. More generally, if E = Qr ,
r
Ω●Q/k ⊗Q E ≅ (Ω●Q/k )r and d ⋅ Ir ∶ (Ω●Q/k )r → (Ω●+1
Q/k )

is a connection for Qr .
Every finitely generated projective module E possesses a connection. Given such an E, choose an idempotent e in Mr (Q) for some r such that E = Im(e). Then, from the connection on Qr in the previous example,
we get a connection on E as the following composition:
Ω●Q/k ⊗ E



// Ω● ⊗ Qr
Q/k

d⋅Ir

// Ω●+1 ⊗ Qr
Q/k

1⊗e

// Ω●+1 ⊗ E
Q/k

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Definition 3.3. This connection on E = Im(e) is called the Levi-Civita connection by analogy with the
classical situation in differential geometry.
Definition 3.4. The curvature R of a connection ∇ on a finitely generated Q-module E is defined to be
R ∶= ∇ ○ ∇ ∶ E → Ω2Q/k ⊗Q E.
It can be shown that R is Q-linear.
Proposition 3.5. (Example 4.2.6 of [5])
(1) Let E1 and E2 be projective Q-modules with connections ∇1 and ∇2 , respectively. Then for e1 ∈ E1
and e2 ∈ E2 , we set
∇(e1 ⊕ e2 ) = ∇1 (e1 ) ⊕ ∇2 (e2 ).
This defines a natural connection on the direct sum E1 ⊕ E2 .
(2) In order to define a connection on the tensor product E1 ⊗ E2 , one defines
∇(e1 ⊗ e2 ) = ∇1 (e1 ) ⊗ e2 + e1 ⊗ ∇2 (e2 ).
Note that the second component naturally lands in E1 ⊗Q (Ω1Q/k ⊗Q E2 ), so we need to apply an
isomorphism τ ∶ E1 ⊗Q Ω1Q/k ≅ Ω1Q/k ⊗Q E1 , which sends e1 ⊗ w to w ⊗ e1 to make it into an element
of the target module Ω1Q/k ⊗Q E1 ⊗Q E2 .
Before getting into the next proposition, we want to inform the reader that by exp(R) we mean the series
1+R+ R2! + R3! +⋯+ Rn! +⋯. ∈ ∏n EndQ (E)⊗Q Ω2n
Q/k . In order to do this, we need to make the extra assumption
2

3

n

n+1
that k ⊇ Q. The exterior operator d can be extended to maps ΩnQ/k → ΩQ/k
(for any n ∈ N) by

d(a0 da1 ⋯dan ) = da0 da1 ⋯dan .
Since d(1) = 0 it is immediate that d2 = 0, and the following sequence
Q = Ω0Q/k

d

// Ω1
Q/k

d

// ⋯

d

// Ωn
Q/k

d

// ⋯

is a complex called the de Rham complex of Q over k. The cohomology groups of the de Rham complex are
n
denoted HDR
(Q) and are called the de Rham cohomology of Q over k.

Proposition 3.6. (Proposition 8.1.6 of [7]) The homogeneous component of degree 2n of ch(E, ∇) ∶=
tr(exp(R)) is a cycle in Ω2n
Q/k (of the de Rham complex), where tr stands for the trace map for projective
modules (details in Section ).

CHERN CHARACTER FOR MATRIX FACTORIZATIONS VIA CHERN-WEIL

7

This proposition implies that ch(E, ∇) defines a cohomology class in the de Rham cohomology of Q.
Theorem 3.7. (Theorem-Definition 8.1.7 of [7]) The cohomology class of ch(E, ∇) ∶= tr(exp(R)) is independent of the connection ∇ and defines an element
2n
ch(E) ∈ ∏ HDR
(Q)
n⩾0

which is called the Chern character of the finitely generated projective Q-module E.
Theorem 3.8. (Theorem 8.2.4 of [7])
even
The Chern character induces a ring homomorphism ch ∶ K0 (Q) → HDR
(Q).

4. Main constructions
→ E0 Ð→ E1 )
Given a k-algebra Q, k a Noetherian commutative ring, for a matrix factorization E = (E1 Ð
⎡
⎤
⎢ 0 A⎥
⎢
⎥
⎥), choose connections ∇i ∶ Ei →
of f ∈ Q (so the odd endomorphism of this matrix factorization is d = ⎢
⎢
⎥
⎢B 0 ⎥
⎣
⎦
Ω1Q/k ⊗Q Ei for i = 0, 1. By Proposition 3.5, ∇0 and ∇1 induce a natural connection for the underlying module
A

E = E0 ⊕ E1 of the form

⎡
⎢∇0
⎢
∇=⎢
⎢
⎢0
⎣

B

⎤
0⎥
⎥
⎥.
⎥
∇1 ⎥
⎦

Definition 4.1. ([3]) The Atiyah class of E, written AtE,∇ (or simply just At if there is no confusion), is the
map
∇ ○ d − (1 ⊗ d) ○ ∇ =def AtE,∇ ∶ E → Ω1 [1] ⊗mf E.
As usual, we regard the single module Ω1 as a complex with Ω1 in the degree 0 piece, so Ω1 [1] is the shift
of this complex. Since we will take the Z/2-folding while tensoring a complex with a matrix factorization,
 0) (degree 1 piece on the left). See Remark 2.12 for details
Ω1 [1] is really the matrix factorization (Ω1 ÐÐ
of the tensor product of a module and a matrix factorization.
It’s easy to check that the Atiyah class is a Q-module homomorphism from the Z/2-graded Q-module E
to the Z/2-graded Q-module Ω1 ⊗Q E.
Compositions of Atiyah classes are defined in the following way. For example, by definition, we have
(1 ⊗ At) ○ At ∶ E → Ω1 [1] ⊗mf E → Ω1 [1] ⊗mf Ω1 [1] ⊗mf E, which is defined as
((1 ⊗ ∇) ○ (1 ⊗ d) − (1 ⊗ 1 ⊗ d) ○ (1 ⊗ ∇)) ○ (∇ ○ d − (1 ⊗ d) ○ ∇).

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̃i (for nature numbers i ⩾ 2)
̃2 . Similarly, we can define At
For simplicity, we denote this composition by At
recursively by
̃i ∶= (1 1
̃
i−1 .
At
Ω ⊗⋯Ω1 ⊗ At) ○ At
´ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
i−1
Hence the map
³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ
̃i ∶ E → Ω1 [1] ⊗mf ⋯ ⊗mf Ω1 [1] ⊗mf E
At
i

has i copies of Ω1 [1] in the target.
Definition 4.2. : Define Ati to be the composition:
³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ
∧
E ÐÐ→ Ω1 [1] ⊗mf ⋯ ⊗mf Ω1 [1] ⊗mf E Ð
→ Ωi [i] ⊗mf E.
i

̃i
At

̃1 .
Note that we have At = At1 = At
4.1. Basic construction: the strict morphism ϕ.
Definition 4.3. Define E (1) = ( Q

df ∧

// Ω1 ) ⊗mf E, with Q in degree 0 and Ω1 in degree 1.

Explicitly, E (1) is by definition the following:
E (1) = (

A

E1

⊕

Ω1 ⊗E

E0

//

B

⊕

Ω1 ⊗E

0

//

E1

⊕

Ω1 ⊗E

1

0

)

⎡
⎡
⎤
⎤
⎢ A
⎢B
0 ⎥
0 ⎥
⎢
⎢
⎥
⎥
⎥ and B = ⎢
⎥. For details, see Proposition 2.13. (Note that the Z/2-folding of
with A = ⎢
⎢
⎢
⎥
⎥
⎢df ∧ −B ⎥
⎢df ∧ −A⎥
⎣
⎣
⎦
⎦
( Q

df ∧

// Ω1 ) is the matrix factorization (Ω1 ÐÔ0ÔÐ
 Q)).
df ∧

Consider the following diagram (commutativity will be checked below in Proposition 4.5)

E1

A

// E0

ϕ1

// E1

ϕ0



ϕ1



E1

⊕
Ω1 ⊗E0

where

B

A

//

E0

⊕
Ω1 ⊗E1

⎡
⎤
⎢
⎥
1
⎢
⎥
⎥,
ϕ1 = ⎢
⎢
⎥
⎢∇0 A − (1 ⊗ A)∇1 ⎥
⎣
⎦

We can make the following definition:


B

//

E1

⊕
Ω1 ⊗E0

⎡
⎤
⎢
⎥
1
⎢
⎥
⎥.
ϕ0 = ⎢
⎢
⎥
⎢∇1 B − (1 ⊗ B)∇0 ⎥
⎣
⎦

CHERN CHARACTER FOR MATRIX FACTORIZATIONS VIA CHERN-WEIL

Definition 4.4. Define ϕE,∇ ∶ E → E

(1)

9

⎤
⎡
⎢ 1 ⎥
⎥
⎢
⎥.
to be the morphism ⎢
⎥
⎢
⎢AtE ⎥
⎦
⎣

Proposition 4.5. ϕE,∇ is a strict morphism of matrix factorizations.
Proof. First, let’s check the commutativity for the square on the left; that is A ○ ϕ1 (x) = ϕ0 ○ A(x) for any
x ∈ E1 . It’s enough to look at the second component. Let π2 be the projection to the second component,
then
π2 ○ A ○ ϕ1 (x) = df ∧ x − B(∇0 A − A∇1 )(x)
= df ∧ x − B∇0 A(x) + f ∇1 (x)
= df ∧ x + f ∇1 (x) − B∇0 A(x)
= ∇1 (f ⋅ x) − B∇0 A(x)
= (∇1 B − B∇0 )(A(x))
= π2 ○ ϕ0 ○ A(x)
The commutativity of the right square can be proved in a similar way. Therefore ϕE,∇ is a strict morphism
of matrix factorizations.



Proposition 4.6. ϕE,∇ is independent of the choice of connections up to homotopy.
′

Proof. Suppose we choose other connections for the Ei , say ∇i ∶ Ei → Ω1Q ⊗Q Ei , i = 0, 1. We show that
ϕ = ϕE,∇ is homotopic to ϕ′ = ϕE,∇′ .
First, ∇i − ∇′i is a morphisms of Q-modules: for any q ∈ Q, x ∈ Ei ,
(∇i − ∇′i )(q ⋅ x) = ∇i (q ⋅ x) − ∇′i (q ⋅ x) = (dq ∧ x + q ⋅ ∇i (x)) − (dq ∧ x + q ⋅ ∇′i (x)) = q ⋅ (∇i − ∇′i )(x).
⎡
⎤
⎤
⎡
⎥
⎢
⎥
⎢
0
0
⎥
⎥
⎢
⎢
⎥ , α1 = ⎢
⎥, which live in the following diagram
Therefore, we can define α0 = ⎢
⎥
⎥
⎢
⎢
⎢(∇0 − ∇′0 )⎥
⎢(∇1 − ∇′1 )⎥
⎦
⎦
⎣
⎣
A

E1

α0


E1

⊕
Ω1 ⊗E0

ss


//

A

B

// E0
ϕ−ϕ′

E0

⊕
Ω1 ⊗E1

α1

// E1

ss
//



B

It’s easy to check that A ○ α0 + α1 ○ B = ϕ − ϕ′ and similarly for the other square.

E1

⊕
Ω1 ⊗E0



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Therefore, we usually drop the ∇ from the notation ϕE,∇ to simply write it as ϕE . When Q is local or if
we take Ei to be free Q-modules, the Atiyah clase is typically like that of the following example.
Example 4.7. For E = (Qn Ð
→ Qn Ð→ Qn ), df = AdB + (dA)B, where dA = (dQ/k (aij )) for the matrix
A

B

A = (aij ). Since ϕ is independent of choice of connection, we use the exterior differential d to construct the

Atiyah class, i.e., ∇i = d for i = 0, 1. First note that we have d ○ A − A ○ d = dA⋅, because (d ○ A − A ○ d)(x) =

d(A ⋅ x) − A ⋅ (dx) = dA ⋅ x + A ⋅ dx − A ⋅ dx = dA ⋅ x. Therefore
⎡
⎢0
⎢
AtE = ⎢
⎢
⎢B
⎣

⎤ ⎡
⎤ ⎡
⎤ ⎡
⎢
⎥ ⎢
⎥ ⎢
A⎥
⎥ ⎢d 0 ⎥ ⎢d 0 ⎥ ⎢ 0
⎥⋅⎢
⎥−⎢
⎥⋅⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎢0 d⎥ ⎢0 d⎥ ⎢B
0⎥
⎦ ⎣
⎦ ⎣
⎦ ⎣

⎤ ⎡
⎤ ⎡
⎢
⎢
A⎥
0
d ○ A − A ○ d⎥
⎥ ⎢
⎥ ⎢ 0
⎥=⎢
⎥=⎢
⎥ ⎢
⎥ ⎢
⎢
⎥ ⎢dB
0⎥
0
⎦ ⎣d ○ B − B ○ d
⎦ ⎣

⎤
dA⎥
⎥
⎥
⎥
0⎥
⎦

hence AtiE =
⎤
⎡
⎤
⎡
i
i
⎢
⎥
⎢³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ·¹
³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ·¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ⎥
¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹ µ
⎥
⎢
⎥
⎢
⎢
⎥
⎢dAdB⋯dAdB
0
dAdB⋯dA⎥
0
⎥
⎢
⎥
⎢
⎥ (i odd)
⎥ (i even) or ⎢
⎢
⎥
⎢
⎥
⎢
⎥
⎢dBdA⋯dB
⎥
⎢
0
0
dBdA⋯dBdA
⎥
⎢´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
⎢
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶⎥
⎥
⎢
⎥
⎢
i
i
⎦
⎣
⎦
⎣
Definition 4.8. Define E (i) ∶= (Q ÐÐ→ Ω1 )⊗i ⊗mf E. Also define morphisms
df ∧

(i)

ϕE ∶= 1⊗i−1 ⊗ ϕE ∶

By our definition, ϕE =

(1)
ϕE .

Note that

(i)
ϕE

2i × 2i identity matrix.

E (i−1) → E (i) .

⎡
⎤
⎢ I2i ⎥
⎢
⎥
⎥, where I2i means the
can be written in the form: ⎢
⎢
⎥
⎢AtE ⋅ I2i ⎥
⎣
⎦
(i)

The following illustrates what we mean by E (i) and ϕE :

E=

E1

E (1) =

E1

A

// E0


A

//

⊕
Ω1 ⊗E0

E0

B

//

⊕
Ω1 ⊗E1

E1

⊕
Ω1 ⊗E0






E1

E0

E1

⊕

Ω1 ⊗E0

⊕

⊕

A

//

Ω1 ⊗E1

⊕

⊕

B

//

Ω1 ⊗E0

⊕

Ω1 ⊗E0

Ω1 ⊗E1

Ω1 ⊗E0

Ω1 ⊗Ω1 ⊗E1

Ω1 ⊗Ω1 ⊗E0

Ω1 ⊗Ω1 ⊗E1


⋮


⋮


⋮

⊕

⋮

// E1




E (2) =

B

⊕

⊕

CHERN CHARACTER FOR MATRIX FACTORIZATIONS VIA CHERN-WEIL

11

and for any e ∈ E,
⎡
⎤
⎡
⎤
⎢ 1 ⎥
⎢ e ⎥
⎢
⎥
⎢
⎥
⎥ (e) = ⎢
⎥
ϕE (e) = ⎢
⎢
⎥
⎢
⎥
⎢AtE ⎥
⎢AtE (e)⎥
⎣
⎦
⎣
⎦

⎤
⎤
⎡
⎢ e ⎥
⎥
⎢
⎥ ⎡
⎤
⎥
⎢
⎥ ⎢
⎥
⎥
e
⎥ ⎢
⎥
⎥ ⎡
⎤ ⎢
⎥
⎢ e ⎥ ⎢AtE (e)⎥ ⎢
1 ⎥
⎥ ⎢
⎥
⎥ ⎢
⎥ ⎢
⎥ = ⎢2AtE (e)⎥
⎥⋅⎢
⎥=⎢
⎥ ⎢
⎥
⎥ ⎢
⎥ ⎢
⎥
⎥ ⎢AtE (e)⎥ ⎢AtE (e)⎥ ⎢
⎥ ⎢ ̃2
⎥
⎥ ⎣
⎦ ⎢
⎢
⎥ ⎢ AtE (e) ⎥
⎥
⎢ ̃2 ⎥ ⎣
⎦
⎥
⎢AtE (e)⎥
AtE ⎥
⎣
⎦
⎦

⎡
⎢ 1
⎢
⎢
⎡
⎤ ⎢
⎢ e ⎥ ⎢
⎥ ⎢
(1) ⎢
ϕE (⎢
⎥) = ⎢
⎢
⎥ ⎢
⎢AtE (e)⎥ ⎢AtE
⎣
⎦ ⎢
⎢
⎢
⎢
⎣

and

⎡
⎢ 1
⎢
⎢
⎢
⎡
⎤ ⎢
⎢
⎢
⎥
e
⎢
⎥ ⎢
⎢
⎥ ⎢
⎥ ⎢
(2) ⎢
ϕE (⎢2AtE (e)⎥) = ⎢
⎢
⎥ ⎢
⎢
⎥ ⎢
⎢ ̃2
⎥ ⎢AtE
⎢ AtE (e) ⎥ ⎢
⎣
⎦ ⎢
⎢
⎢
⎢
⎢
⎢
⎣
(i)

Corollary 4.9. ϕE

1

AtE

⎤
⎡
⎤
⎥
⎢
⎥
e
⎥
⎢
⎥
⎥ ⎡
⎢
⎥
⎤
⎥
⎢
⎥
⎥
⎢2AtE (e)⎥ ⎢
⎥ ⎡
e
⎢
⎥
⎤
⎥ ⎢
⎥ ⎢
⎥
⎥ ⎢
⎥ ⎢
⎢
⎥ ⎢
e
⎥
⎥
⎥
⎢
⎥
̃
2
⎥ ⎢ At (e) ⎥ ⎢3AtE (e)⎥
1 ⎥ ⎢
⎢
⎥
E
⎢
⎥
⎥=⎢
⎥ ⋅ ⎢2At (e)⎥ = ⎢
⎥,
⎥ ⎢
⎥ ⎢
E
⎥
⎥ ⎢
⎥
̃
⎢
⎥ ⎢
2
⎢
⎥
⎥
3
At
(e)
At
(e)
⎥ ⎢
⎢ E
⎥ ⎢ ̃2
E
⎥
⎥
⎢
⎥ ⎢ At (e) ⎥
⎢
⎥
⎥ ⎢
⎥
⎥ ⎣ E
⎢
⎦
̃
̃
3
2
⎢2At (e)⎥ ⎢ At (e) ⎥
⎥
⎥
⎥ ⎣ E
⎢
⎥
E
⎦
⎥
⎢
⎥
⎥
⎢ ̃3
⎥
⎢ AtE (e) ⎥
AtE ⎥
⎦
⎣
⎦

⋯

is a strict morphism of matrix factorizations and it is independent of the choice of

connections (up to homotopy) for any i.
4.2. The map ϕn .
For any nature number n, denote the complex
ndf ∧

Q

// Ω1

(n−1)df ∧

(n−2)df ∧

// Ω2

// ⋯

df ∧

// Ωn

(∗)

by ΩQ,df , where idf ∧ denotes left multiplication by idf (i.e., w1 ∧⋯∧wn ↦ idf ∧w1 ∧⋯∧wn ), for any 0 ⩽ i ⩽ n.
(n)

There is a natural map of chain complexes (Q ÐÐ→ Ω1 )⊗n → ΩQ,df , induced from the natural Q-module
df ∧

(n)

homomorphisms of the following diagram

(Q ⊕ Ω1 )⊗n
◆◆

// (⊕i⩾0 Ωi )⊗n
◆◆
◆◆
&&

∧


⊕i⩾0 Ωi

We again denote this map by ∧.
Proposition 4.10. The map ∧ ∶ (Q ÐÐ→ Ω1 )⊗n → ΩQ,df is a map of complexes.
df ∧

(n)

12

XUAN YU

Proof. The map ∧ is obviously a Q-module homomorphism, so we just need to show ∧ commutes with the
differentials of complexes.
Let us denote the differential in (Q ÐÐ→ Ω1 ) by ∂ and the differential in ΩQ,df by ∂ ′ . Note that for an
df ∧

(n)

element u ∈ Ωm , ∂ ′ u = (n − m)df ∧ u and for v ∈ Q ⊕ Ω1 ,

⎧
⎪
df ∧ v, if ∣v∣ = 0
⎪
⎪
⎪
∂(v) = ⎨
⎪
⎪
⎪
else
⎪
⎩0,
Therefore, for a1 ⊗ ⋯ ⊗ am ∈ Ωm , where ai ∈ Ω1 ,
m

∧∂(a1 ⊗ ⋯ ⊗ am ) = ∧(∑(−1)∣a1 ∣+⋯+∣ai−1 ∣ a1 ⊗ ⋯ ⊗ ∂(ai ) ⊗ ⋯ ⊗ am )
i=1

m

= ∑(−1)∣a1 ∣+⋯+∣ai−1 ∣ a1 ∧ ⋯ ∧ ∂(ai ) ∧ ⋯ ∧ am
i=1

m

= ∑(−1)∣a1 ∣+⋯+∣ai−1 ∣ a1 ∧ ⋯ ∧ (df ∧ ai ) ∧ ⋯ ∧ am
i=1

m

∑

=

i=1 and ∣ai ∣=0

(−1)2(∣a1 ∣+⋯+∣ai−1 ∣) df ∧ a1 ∧ ⋯ ∧ am

= (n − m)df ∧ a1 ∧ ⋯ ∧ am
Also,
∂ ′ ∧ (a1 ⊗ ⋯ ⊗ am )
= ∂ ′ (a1 ∧ ⋯ ∧ am )
= (n − m)df ∧ a1 ∧ ⋯ ∧ am
This completes the proof.



We obviously have:
Corollary 4.11. ∧ ⊗ 1E ∶ (Q ÐÐ→ Ω1 )⊗n ⊗mf E → ΩQ,df ⊗mf E is a strict morphism of matrix factorizations,
df ∧

(n)

for any matrix factorization E.
Definition 4.12. Define ϕnE ∶ E → ΩQ,df ⊗mf E to be the composition (∧ ⊗ 1E ) ○ ϕE
(n)

(n)

(n−1)

○ ϕE

(1)

○ ⋯ ○ ϕE ; i.e.,

ϕnE is the composition of the following chain of strict morphisms
ϕ(1)

ϕ(2)

ϕ(3)

(n)

E ÐÐ→ E (1) ÐÐ→ E (2) ÐÐ→ ⋯ ÐÐ→ E (n) = (Q ÐÐ→ Ω1 )⊗n ⊗mf E ÐÐÐ→ ΩQ,df ⊗mf E
ϕE

df ∧

∧⊗1E

(n)

Corollary 4.13. ϕnE is a strict morphism of matrix factorizations and is independent of choice of connections
up to homotopy.

CHERN CHARACTER FOR MATRIX FACTORIZATIONS VIA CHERN-WEIL
(i)

Proof. We know that each of the ϕE ’s is independent of choice of connections, and thus ϕnE is too.

13



Proposition 4.14. We have (∧ ⊗ 1E ) ○ (1Ω1 [1]⊗(i−1) ⊗ At) = (∧ ⊗ 1E ) ○ (1Ωi−1 ⊗ At) ○ (∧ ⊗ 1E ), for all i; that
is, the following diagram commutes:
³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ
Ω1 [1] ⊗mf ⋯ ⊗mf Ω1 [1] ⊗mf E
i−1

∧⊗1E

// Ωi−1 [i − 1] ⊗mf E

1Ω1 [1]⊗(i−1) ⊗At

1Ωi−1 ⊗At


Ω1 [1] ⊗mf Ω1 [1] ⊗mf ⋯ ⊗mf Ω1 [1] ⊗mf E
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
i


Ωi−1 [i − 1] ⊗mf Ω1 [1] ⊗mf E
❲❲❲❲❲
❲❲❲❲❲
❲❲❲❲❲
∧⊗1E
❲❲❲❲❲
++

∧⊗1E


Ωi [i] ⊗mf E

Proof. We can check this directly. For example, it is obvious for i = 1.



For the sake of simplicity, we will drop all the 1 ⊗ ⋯ ⊗ 1 if there is no confusion from now on. For example,
for the above proposition, we will in fact write it as (∧ ⊗ 1E ) ○ At = (∧ ⊗ 1E ) ○ At ○ (∧ ⊗ 1E ).
Similarly, we have the following corollary.
Corollary 4.15. (∧ ⊗ 1E ) ○ ϕ(i) = (∧ ⊗ 1E ) ○ ϕ(i) ○ (∧ ⊗ 1E ), for any i. That is, we have the following
commutative diagram:
ϕ(i)

E (i−1)

// E (i)

∧⊗1E


(i−1)
ΩQ,df

Proof. Note that

⊗mf E
ϕ

(∧ ⊗ 1E ) ○ ϕ

(i)

(i)

∧⊗1E

// Ω(i) ⊗mf E
Q,df
4
4
✐✐✐
✐
✐
✐
✐
✐✐✐✐
✐✐✐∧⊗1
✐
✐
✐
E
✐

// Ω(i−1) ⊗mf (Q → Ω1 ) ⊗mf E
Q,df

⎡
⎤
⎢ I2i ⎥
⎢
⎥
⎥
= (∧ ⊗ 1E ) ○ ⎢
⎢
⎥
⎢At ⋅ I2i ⎥
⎣
⎦
⎤
⎡
⎥
⎢
I2i
⎥
⎢
⎥
=⎢
⎥
⎢
⎢(∧ ⊗ 1E ) ○ At ⋅ I2i ⎥
⎦
⎣
⎡
⎤
⎥
⎢
I2i
⎥
⎢
⎥
=⎢
⎥
⎢
⎢(∧ ⊗ 1E ) ○ At ○ (∧ ⊗ 1E ) ⋅ I2i ⎥
⎦
⎣

14

XUAN YU

⎡
⎤
⎢ I2i ⎥
⎢
⎥
⎥ ○ (∧ ⊗ 1E )
= (∧ ⊗ 1E ) ○ ⎢
⎢
⎥
⎢At ⋅ I2i ⎥
⎣
⎦
= (∧ ⊗ 1E ) ○ ϕ(i) ○ (∧ ⊗ 1E ).



n
n
(n)
Corollary 4.16. We have ϕn = ∑ ( )Ati ∈ ΩQ,df ⊗mf E.
i
i=0

n−1

Proof. The base case is clear, now by induction, say ϕn−1 = ∑ (
i=0

n−1
)Ati , then
i

ϕn = (∧ ⊗ 1E ) ○ ϕ(n) ○ ϕ(n−1) ○ ⋯ ○ ϕ
= (∧ ⊗ 1E ) ○ ϕ(n) ○ (∧ ⊗ 1E ) ○ ϕ(n−1) ○ ⋯ ○ ϕ
= (∧ ⊗ 1E ) ○ ϕ(n) ○ ϕn−1
n−1
n−1
= (∧ ⊗ 1E ) ○ ϕ(n) ○ ∑ (
)Ati
i
i=0

⎡
⎤
⎢
⎥ n−1 n − 1
I2n
⎢
⎥
⎥○ ∑ (
=⎢
)Ati
⎢
⎥ i=0
i
⎢(∧ ⊗ 1E ) ○ At ⋅ I2n ⎥
⎣
⎦
n−1

= ∑(
i=0

n−1
n−1
n−1
)Ati + At( ∑ (
)Ati )
i
i
i=0

n
n
= ∑ ( )Ati .
i=0 i



̃n .
4.3. The map ϕ
For any nature number n, define (Ω● , df, n) to be the complex:
df ∧

Q

// Ω1

df ∧

df ∧

// Ω2

// ⋯

df ∧

// Ωn .

From now on, we assume in addition that k ⊃ Q. Under this assumption, there is an isomorphism of
complexes ΩQ,df → (Ω● , df, n) defined by
(n)

ndf ∧

Q

// Ω2

df ∧


// Ω1

(n−2)df ∧

// ⋯

df ∧

// Ωn

1
n(n−1)

1
n

=


Q

(n−1)df ∧

// Ω1

df ∧


// Ω2

1
n!

df ∧

// ⋯

df ∧


// Ωn

CHERN CHARACTER FOR MATRIX FACTORIZATIONS VIA CHERN-WEIL

15

̃n ; i.e.,
We compose ϕn with the above isomorphism to get a morphism which now called ϕ
̃n ∶ E
ϕ

// (Ω● , df, n) ⊗mf E

̃n :
By Proposition 4.16, we have an expression for ϕ
n

̃n = ∑
ϕ

1 i
At
i!
i=0

5. Chern character for matrix factorizations and its basic properties

For a k-algebra Q, where k is a commutative unital ring that contains Q, we constructed (for any given
̃n ∶ E → (Ω● , df, n) ⊗mf E in last section. Now, we can
n ∈ N) a strict morphism of matrix factorizations ϕ
define a Chern character for matrix factorizations.
5.1. Supertrace.
Let Q be any commutative ring and M a finitely generated projective Q-module. Let M ∗ = HomQ (M, Q)
be the dual of M . Consider the two maps
EndQ (M ) oo

ξ

M ∗ ⊗Q M

ǫ

// Q

given by ξ(α ⊗ n)(m) = α(m)n, with m, n ∈ M, α ∈ M ∗ and by ǫ(α ⊗ n) = α(n) respectively. If M is a
finitely generated projective Q-module, then ξ is an isomorphism and the composite ǫ ○ ξ −1 is the standard
trace map: ǫ ○ ξ −1 = tr. Suppose that M is free of finite rank over Q, with Q a k-algebra. Then a Q-linear
map M → M ⊗Q Ω∗Q/k , upon choice of basis, is a matrix with coefficients in Ω∗Q/k , that is, an element of
EndQ (M ) ⊗Q Ω∗Q/k . Since a projective module is a direct summand of a free module, the same is true when
M is projective, i.e.,
HomQ (M, M ⊗Q Ω∗ ) ≅ EndQ (M ) ⊗Q Ω∗ .
Therefore, when M is a matrix factorization, the underlying module M is projective so Ati can be viewed
as an element of EndQ (M ) ⊗Q ΩiQ/k .
Definition 5.1. Given a Z/2-graded finitely generated projective Q-module M and an endomorphism T of
M of degree 0, using that EndQ (M )0 = EndQ (M0 ⊕ M1 )0 = EndQ (M0 ) ⊕ EndQ (M1 ), define the supertrace
to be
str(T ) ∶= tr(T0 ) − tr(T1 ) ∈ Q
where T = T0 ⊕ T1 with Ti ∈ EndQ (Mi ), i = 0, 1.

16

XUAN YU

Proposition 5.2.
(1) If α, β ∶ E → E are strict morphisms of matrix factorization and α is homotopic to β, then str(α) =
str(β).
(2) str is an invariant under cyclic permutations, i.e.,
str(α1 ○ ⋯ ○ αn ) = str(ασ(1) ○ ⋯ ○ ασ(n) )
for σ a cyclic permutation of n elements.
 E0 )
Proof. Say E = (E1 ÐÔÐ
A

B

(1) There are Q-module homomorphisms x ∶ E0 → E1 and y ∶ E1 → E0 such that Ax + yB = α0 − β0 and
By + xA = α1 − β1 . So
(Ax + yB) − (By + xA) = (α0 − β0 ) − (α1 − β1 )
(Ax − xA) − (yB − By) = (α0 − α1 ) − (β0 − β1 )
tr(Ax − xA) − tr(yB − By) = tr(α0 − α1 ) − tr(β0 − β1 )
(tr(Ax) − tr(xA)) − (tr(yB) − tr(By)) = 0 = str(α) − str(β)
str(α) = str(β).
(2) This is obvious since tr is an invariant under cyclic permutations.

5.2. Chern character.
Before making the definition of the Chern character of matrix factorizations, let’s first recall the definition
of a smooth algebra and prove a few propositions necessary for the definition.
Definition 5.3.

(1) If k is an algebraically closed field, then Q is smooth of relative dimension d if it is

of finite type, its dimension is d, and the module Ω1Q/k of differentials is a finitely generated locally
free Q-module of rank d.
(2) Let k be an arbitrary field, k its algebraic closure. Then Q is smooth of relative dimension d if Q ⊗k k
is smooth of relative dimension d over k.
(3) Let θ ∶ Q → Q′ be a ring map, then θ is smooth of relative dimension d if it is flat, finitely presented,
and for all primes p of Q, the fibre ring k(p) ⊗Q Q′ is smooth of relative dimension d over k(p), where
k(p) is the residue field at p.

CHERN CHARACTER FOR MATRIX FACTORIZATIONS VIA CHERN-WEIL

17

Proposition 5.4. Suppose Q is a smooth k-algebra of relative dimension d where k is a commutative unital
̃d ) = str(ϕ
̃
d+1 ) = ⋯.
ring that contains Q. Then str(ϕ
Proof. We have str(Atd+1 ) ∈ Ωd+1 ⊗mf E = 0. Therefore
̃
d+1 ) =
str(ϕ
∑

d+1
i=0

1
1
1
str(Ati ) = 1 + str(At) + ⋯ + str(Atd ) +
str(Atd+1 )
i!
d!
(d + 1)!
= 1 + str(At) + ⋯ +

1
str(Atd ) + 0
d!

̃d )
= str(ϕ

̃d ) = str(ϕ
̃
̃d ) = str(ϕ̃
d+1 ) and hence str(ϕ
d+i ) for any i ⩾ 1.
i.e., str(ϕ



→ E0 Ð→ E1 ) ∈ M F (Q, f ), df ∧ str(AtiE ) = 0 in
Proposition 5.5. Given any matrix factorization E = (E1 Ð
B

A

Ωi+1 for any i. If i is an odd integer, str(AtiE ) = 0.

Proof. For the underlying finitely generated projective Q-module E = E0 ⊕ E1 of E, the trace homomorphism
tr is End(E) ⊗ Ω● → Ω● , so str(AtiE ) ∈ Ωi since it’s the differentce of two elements in Ωi .
It’s enough to check this locally, so we adopt the notations used in Example 4.7. In particular, since the
map ΩiQ/k → Ωi+1
is injective, it suffices to check df ∧ str(AtiE ) = 0 after inverting f . Therefore we may
Q[ 1 ]/k
f

assume A and B are invertible matrices with entries in Q[ f1 ]. Notice that str(AtiE ) = 0 when i is odd so we

just need to show this when i is an even integer.
Since A is invertible, B = f ⋅ A−1 , therefore
dB = df ⋅ A−1 + f ⋅ dA−1
since A−1 ⋅ A = I we have dA−1 = −A−1 dA ⋅ A−1 hence
dB = df ⋅ A−1 − f A−1 dA ⋅ A−1
Also, since df = dA ⋅ B + A ⋅ dB and say i = 2l,
³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ
= 2tr(df dAdBdAdB⋯dAdB)
i

df

∧ str(AtiE )

= 2tr(df dA(df ⋅ A−1 − f A−1 dA ⋅ A−1 )⋯dA(df ⋅ A−1 − f A−1 dA ⋅ A−1 ))
= 2tr(df dA(−f A−1 dA ⋅ A−1 )⋯dA(−f A−1 dA ⋅ A−1 ))
= (−1)l 2tr(dA ⋅ A−1 ⋯dA ⋅ A−1 ) = 0
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
even numbers

18

XUAN YU

We have the last equality because switching matrices of odd forms is only going to introduces a sign. Also,
the product (dA ⋅ A−1 )even = (dA ⋅ A−1 )odd ⋯(dA ⋅ A−1 )odd stays the same no matter how you switch, therefore
it has to be zero.



By the above proposition, we know that str(AtiE ) vanishes when i is odd and is a cycle (of the complex

(Ω● , df, n)) when i is even, so it defines an element of the homology.

Now we are ready to give our definition of the Chern character. Assume that Q is now a smooth k-algebra
of relative dimension n with k a commutative unital ring that contains Q.
Definition 5.6. We define the Chern character of E ∈ M F (Q, f ) to be
n ker(Ω2i Ð
1
Ð→ Ω2i+1 )
str(AtiE ) ∈ H0 ((Ω● , df, n)Z/2 ) = ⊕
.
df ∧
i=0 Im(Ω2i−1 Ð
i=0 i!
Ð→ Ω2i )
df ∧

n

̃n ) = ∑
ch(E) ∶= str(ϕ
E

Recall that (Ω● , df, n) is the complex Q

df ∧

df ∧

// Ω1

// Ω2

df ∧

df ∧

// ⋯

// Ωn and (Ω● , df, n)Z/2

is the Z/2-folding of this complex. Also, notice that by Proposition 5.5, str(Ati∣ E) = 0 when i is an odd integer.
1
Therefore the Chern character is in fact ch(E) = ∑
str(At2i
E ).
i⩾0 (2i)!
Our definition of the Chern is the same as the one in Platt [8]. For the special case when Q = k[[x1 , ⋯, xn ]]
and f ∈ Q an isolated singularity, see the following example.
Example 5.7. Let f ∈ Q = k[[x1 , ⋯, xn ]] be an isolated singularity at the origin (that is, the localizations at

 Qr ) a matrix factorization
every prime except the maximal ideal m = (x1 , ⋯, xn ) is regular), E = (Qr ÐÔÐ
B
⎡
⎤
⎢ 0 A⎥
⎢
⎥
⎥. E is a free Q-module, so as before, we can choose the exterior differential d to be the
with d = ⎢
⎢
⎥
⎢B 0 ⎥
⎣
⎦
⎡
⎤
⎢ 0 dA⎥
⎢
⎥
connections and we get that AtE = ⎢
⎥. Also, notice that in this situation (Ω● , df, n) is exact except in
⎢
⎥
⎢dB 0 ⎥
⎣
⎦
n
1
1
position n (the dimension of Q) [11]. Therefore we have that ch(E) = ∑ str(AtiE ) = str(dAdB⋯dAdB),
i!
n!
i=0
which agrees with the Chern character obtained by [6], [8] and [10]. It differs by a sign with the ones in [1],
A

[2], [3] and [9].
→ E0′ Ð
→ E1′ )in MF(Q,f ),
→ E0 Ð→ E1 ) and E ′ = (E1′ Ð
Proposition 5.8. Given matrix factoriztions E = (E1 Ð
A

B

C

a strict morphism β ∶ E ′ → E, then the following diagram commute up to homotopy
ϕ
̃
E

E′

df ∧
// (Q Ð
Ð→ Ω1 ) ⊗mf E ′
1⊗β

β


E

ϕ
̃
E′


df ∧
// (Q Ð
Ð→ Ω1 ) ⊗mf E

S

CHERN CHARACTER FOR MATRIX FACTORIZATIONS VIA CHERN-WEIL

19

⎤
⎡
⎢ 1 ⎥
⎥
⎢
⎥ = 1 + AtE .
Proof. Recall that ϕ
̃E = ⎢
⎥
⎢
⎢AtE ⎥
⎦
⎣
Choose connections ∇i and

∇′i ,

⎡
⎤
⎢
⎥
0
⎢
⎥
then we can construct module homomorphisms ψ0 = ⎢
⎥ , ψ1 =
′ ⎥
⎢
⎢(∇0 β − (1 ⊗ β)∇0 )⎥
⎣
⎦

⎡
⎤
⎢
⎥
0
⎢
⎥
⎢
⎥, which lives in the diagram
′
⎢
⎥
⎢(∇1 β − (1 ⊗ β)∇1 )⎥
⎣
⎦
C

′

E1

D

// E0′

// E1′

ψ0

E1

⊕

Ω1 ⊗E0

ϕ
̃
ϕE ′
E ○β−(1⊗β)○̃



ss
//

⊕

ss
//

Ω1 ⊗E1

A



ψ1

E0

E1

⊕

Ω1 ⊗E0

B

⎤
⎡
⎢
⎥
0
⎢
⎥
⎥.
where ϕ
̃E ○ β − (1 ⊗ β) ○ ϕ̃
E ′ is the matrix ⎢
⎢
⎥
⎢AtE ○ β − (1 ⊗ β) ○ AtE ′ ⎥
⎣
⎦
First of all ψ0 and ψ1 are indeed module homomorphism:
ψ0 (q ⋅ x) = ∇0 β(q ⋅ x) − (1 ⊗ β)∇0 (q ⋅ x)
′

= ∇0 (q ⋅ β(x)) − (1 ⊗ β)(dq ∧ x + q ⋅ ∇0 (x))
′

= dq ∧ β(x) + q ⋅ ∇0 β(x) − dq ∧ β(x) − (1 ⊗ β)(q ⋅ ∇0 (x))
′

= q ⋅ ∇0 β(x) − q ⋅ (1 ⊗ β)(∇0 (x))
′

= q(∇0 β − (1 ⊗ β)∇0 )(x)
′

= q ⋅ ψ0 (x)
The same argument shows that ψ1 is also a module homomorphism.
We want to show that ψ0 and ψ1 give us a homotopy. For the degree 0 part, we need to show (̃
ϕE ○ β −

(1 ⊗ β) ○ ϕ̃
E ′ )0 = A ○ ψ0 + ψ1 ○ D.

⎤
⎡
⎢ A
0 ⎥
⎥
⎢
⎥ and elements are 2 × 1 column vectors, the equality in the first row is easy so
Recall that A = ⎢
⎥
⎢
⎢df ∧ −B ⎥
⎦
⎣
we just need to check the equality for the second row. Hence,
(AtE ○ β − (1 ⊗ β) ○ AtcE ′ )0

20

XUAN YU

= (∇1 B − B∇0 )β − (1 ⊗ β)(∇1 D − D∇0 )
′

′

= ∇1 Bβ − B∇0 β − (1 ⊗ β)∇1 D + (1 ⊗ β)D∇0
′

′

= ∇1 βD − B∇0 β − (1 ⊗ β)∇1 D + B(1 ⊗ β)∇0
′

′

= (∇1 β − (1 ⊗ β)∇1 )D − B(∇0 β − (1 ⊗ β)∇0 )
′

′

= ψ1 ○ D + A ○ ψ0
In the above calculation, we use Bβ = βD and (1 ⊗ β)D = B(1 ⊗ β) by the fact that β is a strict morphism
of matrix factorizations, i.e., the following commutative diagram:

′

E1

C

β

// E ′
0

D

β


E1

A


// E0

// E ′
1
β

B


// E1


n .
̃n ○ β ∼ (1 ⊗ 1 ⊗ ⋯ ⊗ 1 ⊗ β) ○ ϕ̃
Corollary 5.9. Under the same hypothesis as in Proposition 5.8, we have ϕ
E
E′

Proof. Indeed,
̃
̃
n−1 ○ ϕ
n−1 ○ (1 ⊗ β) ○ ϕ
̃n ○ β = ϕ
ϕ
̃E ○ β ∼ ϕ
̃
E′
E
E
E
then by induction
n .
̃n ○ β ∼ (1 ⊗ 1 ⊗ ⋯ ⊗ 1 ⊗ β) ○ ϕ̃
ϕ
E
E′


Corollary 5.10. We have ch(E) = ch(E ′ ) for homotopy equivalent matrix factorizations E and E ′ of M F (Q, f ).
α

 E ′ , such that α ○ β ∼ 1E ′ and β ○ α ∼ 1E , by Corollary 5.9
Proof. Say we have E ÐÔÐ
β

n ○α
̃n ○ β ○ α ∼ (1 ⊗ 1 ⊗ ⋯ ⊗ 1 ⊗ β) ○ ϕ̃
ϕ
E
E′

Therefore, by Proposition 5.2
̃n )
str(ϕ
E
̃n ○ β ○ α)
=str(ϕ
E
n ○ α)
=str((1⊗n ⊗ β) ○ ϕ̃
E′
n )
=str(α ○ (1⊗n ⊗ β) ○ ϕ̃
E′

CHERN CHARACTER FOR MATRIX FACTORIZATIONS VIA CHERN-WEIL

21

n )
=str(ϕ̃
E′

This gives
ch(E) = ch(E ′ ).


Theorem 5.11. Given any distinguished triangle P Ð
→ Q → cone(θ) → P[1] in [M F (Q, f )], we have
θ

(∗)

ch(Q) = ch(P) + ch(cone(θ))

Proof. We will prove this theorem by explicit calculation of the Chern character.
First, it’s enough to check equality for the even components, as discussed in the proof of Proposition 5.5.
n
n
̃n ) = str(∑ 1 AtiE ) = ∑ 1 str(AtiE ) for any matrix factorization E, so it’s enough
By definition ch(E) = str(ϕ
E
i=0 i!
i=0 i!
2i
2i
to provestr(At2i
)
=
str(At
)
+
str(At
),
for
all even integers 2i between 1 and n.
Q
P
cone(θ)
Say P = (P1 Ð
→ P0 Ð→ P1 ) and Q = (Q1 Ð
→ Q0 Ð→ Q1 ), the mapping cone is
A

B

C

D

cone(θ) = ( Q1 ⊕ P0 oo

⎡
⎢
⎢C
⎢
⎢
⎢
⎢
⎢0
⎢
⎣

⎤
⎥
⎥
⎥
⎥
⎥
⎥
−B ⎥⎥⎦

θ0

//
⎡
⎢
⎢D
⎢
⎢
⎢
⎢
⎢0
⎢
⎣

⎤
⎥
θ1 ⎥⎥
⎥
⎥
⎥
−A⎥⎦⎥

Q0 ⊕ P1 )

Q
Q
P
Choose any connections ∇P
0 and ∇1 for P, similarly ∇0 and ∇1 for Q. We have induced connections for

cone(θ):

cone(θ)
∇1

⎡ Q
⎢∇0
⎢
=⎢
⎢
⎢
⎣

⎤
⎥
⎥
⎥
P⎥
∇1 ⎥
⎦

cone(θ)
∇0

⎡ Q
⎢∇1
⎢
=⎢
⎢
⎢
⎣

⎤
⎥
⎥
⎥
P⎥
∇0 ⎥
⎦

Since the Chern character is independent of choice of connections, we use these to compute the Atiyah
class Atcone(θ) for cone(θ), which is just
⎡ Q
⎢∇
⎢ 1
⎢
⎢
⎢
⎢
Atcone(θ) = ⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣

∇P
0
∇Q
0

⎤ ⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥⋅⎢
⎥ ⎢
⎥ ⎢D
⎥ ⎢
⎥ ⎢
P⎥ ⎢
∇1 ⎥ ⎢
⎦ ⎣

C

θ1
−A

⎤ ⎡
⎢
θ0 ⎥
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎢
−B ⎥
⎥ ⎢
⎥−⎢
⎥ ⎢
⎥ ⎢D
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣

C

θ1
−A

⎤ ⎡ Q
⎢
θ0 ⎥
⎥ ⎢∇1
⎥ ⎢
⎥ ⎢
⎢
−B ⎥
⎥ ⎢
⎥⋅⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣

∇P
0
∇Q
0

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
P⎥
∇1 ⎥
⎦

22

XUAN YU

where

⎡
⎢
⎢
⎢
⎢
⎢
⎢
=⎢
⎢
⎢Y
⎢
⎢
⎢
⎢
⎣

X

∗
W

⎤
∗⎥
⎥
⎥
⎥
Z⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦

Q
X = (∇Q
1 C − C∇0 )
Q
Y = (∇Q
0 D − D∇1 )
P
Z = (B∇P
1 − ∇0 B)
P
W = (A∇P
0 − ∇1 A).

Hence

Therefore,

⎡
⎢XY
⎢
⎢
⎢
⎢
⎢
2
Atcone(θ) = ⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣

⎡
⎢(XY )i
⎢
⎢
⎢
⎢
⎢
2i
Atcone(θ) = ⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
for any even integer 2i between 1 and n.

∗
ZW
YX

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
∗ ⎥
⎥
⎥
⎥
W Z⎥
⎦

∗
(ZW )i

(Y X)i

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
∗ ⎥
⎥
⎥
i⎥
(W Z) ⎥
⎦

This gives that
str(At2i
cone(θ) )
⎡
⎢(XY )i
⎢
= tr ⎢
⎢
⎢
⎣

⎤
⎡
⎥
⎢(Y X)i
⎥
⎢
⎥ − tr ⎢
⎢
i⎥
⎢
(ZW ) ⎥
⎦
⎣
∗

⎤
⎥
⎥
⎥
i⎥
(W Z) ⎥
⎦
∗

= 2tr((XY )i ) − 2tr((W Z)i ).
hence
2i
i
i
i
str(At2i
P ) + str(Atcone(θ) ) = 2tr((W Z) ) + 2tr((XY ) ) − 2tr((W Z) )

CHERN CHARACTER FOR MATRIX FACTORIZATIONS VIA CHERN-WEIL

= 2tr((XY )i ) = str(At2i
Q ).

23



5.3. Grothendieck group.
Recall that the Grothendieck group K0 (T ) of a triangulated category T is the free abelian group generated

by isomorphism classes for objects of T , modulo the relations [X] + [Z] = [Y ] for distinguished triangles
X → Y → Z → X[1].
Corollary 5.12. The Chern character induces a map from K0 ([M F (Q, f )]) to H0 ((Ω⋅ , df, n)Z/2 ).
Proof. Any distinguished triangle is isomorphic (in the homotopy category) to a triangle of the form of
Theorem 5.11. Now apply Corollary 5.10 and Theorem 5.11



Now we will prove that the Chern character is a ring homomorphism.
Lemma 5.13. ⊕ K0 ([M F (Q, f )]) is a ring via [E]f ⋅ [F ]g ∶= [E ⊗mf F ]f +g .
f ∈Q

Proof. First we have to show that the above multiplication is well-defined.
Since we know from the definition that − ⊗mf − preserves homotopy equivalences of matrix factorizations.
For any given E ≃ E ′ and F ≃ F ′ , we have E ⊗mf F ≃ E ′ ⊗mf F ′ , so the tensor product is well-defined on
the free abelian group generated by isomorphism classes of matrix factorizations; we denote this group by
⊕ Z([M F (Q, f )]).

f ∈Q

Now, let’s show that ⊕ Z([M F (Q, f )]) is a commutative ring under the above multiplication.
f ∈Q

(1) (E)f ⋅ (F )g ∈ ⊕ Z([M F (Q, f )])
f ∈Q

(2) E ⊗mf F ≅ F ⊗mf E hence (E)f ⋅ (F )g = (F )g ⋅ (E)f

(3) ((E)f ⋅ (F )g ) ⋅ (G)h = (E ⊗mf F )f +g ⋅ (G)h = ((E ⊗mf F ) ⊗mf G)f +g+h .

Also, (E)f ⋅ ((F )g ⋅ (G)h ) = (E)f ⋅ ((F ⊗mf G)g+h ) = (E ⊗mf (F ⊗mf G))f +g+h , where (E)f means

the isomorphism class of the matrix factorization E ∈ M F (Q, f ). Hence the above shows that the
multiplication is associative.
 Q) ∈ M F (Q, 0) such that (1)0 ⋅ (E)f = (E)f .
(4) There is an identity 1 = (0 ÐÔÐ
0

0

Indeed, 1 ⊗mf E equals to
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢−1 ⊗ e
⎢
0
⎣

⎤

1 ⊗ e1 ⎥⎥⎥
⎥
⎥
⎥
⎥
⎥
⎦

 0 ⊕ E0 .
0 ⊕ E1 ÐÔÔÔÔÔÔÔÔÔÔÐ
⎤
⎡
⎥
⎢
⎢
−1 ⊗ e1 ⎥⎥
⎢
⎢
⎢
⎢
⎢1 ⊗ e
⎢
0
⎣

⎥
⎥
⎥
⎥
⎥
⎦

24

XUAN YU

Therefore we have an isomorphism of 1 ⊗mf E and E, i.e., (1)0 ⋅ (E)f = (E)f .

Similarly, we also have (E)f ⋅ (1)0 = (E)f .

(5) The last thing to check is the distribution law. In particular, I will show that
((E)f + (F )f ) ⋅ (G)g = (E)f ⋅ (G)g + (F )f ⋅ (G)f
for any E, F ∈ M F (Q, f ) and G ∈ M F (Q, g).
In fact, we have
((E)f + (F )f ) ⋅ (G)g = (E ⊕ F )f ⋅ (G)g

= ((E ⊕ F ) ⊗mf G)f +g

= ((E ⊗mf G) ⊕ (F ⊗mf G))f +g

= (E ⊗mf G)f +g + (F ⊗mf G)f +g
= (E)f ⋅ (G)g + (F )f ⋅ (G)g

Similarly, we can prove that (G)g ⋅ ((E)f + (F )f ) = (G)g ⋅ (E)f + (G)g ⋅ (F )f .
The above shows that the isomorphism classes of all matrix factorizations is a monoid under
− ⊗mf −, so ⊕ Z([M F (Q, f )]) is in fact a commutative ring.
f ∈Q

Finally, to show that this multiplication is well-defined on the quotient group, it’s enough to prove
that the subgroup
{[Q] − [P] − [W] ∶ P → Q → W → P[1] a distinguished triangle}
is an ideal of ⊕ Z([M F (Q, f )]). This amounts to the following fact: tensor product is a triangulated
f ∈Q

functor [13].

Definition 5.14. : Define Kf (Q) ∶= ⊕ K0 ([M F (Q, if )]); this is in fact a subring of ⊕ K0 ([M F (Q, if )]).
f ∈Q

i∈Z⩾0

Proof. Given any [a], [b] ∈ Kf (Q), [a] + [b] ∈ Kf (Q); [a] ⋅ [b] ∈ Kf (Q); [1] ∈ Kf (Q); [−a] ∈ Kf (Q) (since

[−a] = [a[1]] ∈ Kf (Q).)



Lemma 5.15. ⊕ H0 ((Ω⋅ , df, n)Z/2 ) is a commutative ring via ∧.
f ∈Q

Proof. First, assume n is an odd integer. A similar proof works when n is even. Recall that (Ω⋅ , df, n) is the
following complex:
df ∧

Q

// Ω1

df ∧

// Ω2

df ∧

// ⋯

df ∧

// Ωn .

CHERN CHARACTER FOR MATRIX FACTORIZATIONS VIA CHERN-WEIL

25

1
 E0 ) where
Therefore the Z/2-folding (Ω⋅ , df, n)Z/2 is the matrix factorization E = (E1 ÐÔÔÐ

D

D0

E1 = Ω1 ⊕ Ω3 ⊕ ⋯ ⊕ Ωn ,

and

⎡
⎢ 0
0
⎢
⎢
⎢
⎢df ∧
0
⎢
⎢
⎢
D1 = ⎢ 0
df ∧
⎢
⎢
⎢
⎢
⋯
⎢
⎢
⎢ 0
0
⎢
⎣

Hence H0 ((Ω⋅ , df, n)Z/2 ) =

⋯
⋯
⋯
⋯
⋯

E0 = Q ⊕ Ω2 ⊕ ⋯ ⊕ Ωn−1

⎤
0 ⎥
⎥
⎥
⎥
0 ⎥
⎥
⎥
⎥
0 ⎥
⎥
⎥
⎥
⎥
⎥
⎥
df ∧⎥
⎥ n+1 n+1
⎦ 2 × 2

⎡
⎢df ∧
0
⎢
⎢
⎢
⎢ 0
df ∧
⎢
D0 = ⎢
⎢
⎢
⋯
⎢
⎢
⎢
0
⎢ 0
⎣

⎤
0 ⎥
⎥
⎥
⎥
⋯
0 ⎥
⎥
⎥
⎥
⎥
⋯
⎥
⎥
⎥
⋯ df ∧⎥ n+1 n+1
⎦ 2 × 2
⋯

kerD0
is a ring by properties of the wedge product. For example, for any
ImD1

a, b, c ∈ ⊕ Ωi , we have
i even

(a ∧ b) ∧ c = a ∧ (b ∧ c).

It is not hard to see that elements on the two sides of the above equation determine the same element in
homology. The same holds for other conditions to make a set into a ring. It is commutative since we are
dealing only with even former (in general a ∧ b = (−1)ij b ∧ a for a ∈ Ωi and b ∈ Ωj ; therefore, i, j even means
a ∧ b = b ∧ a). From this we see that ⊕ H0 ((Ω⋅ , df, n)Z/2 ) is a commutative ring.
f ∈Q

It is clear that ⊕ H0 ((Ω⋅ , idf, n)Z/2 ) is a subring of ⊕ H0 ((Ω⋅ , df, n)Z/2 ).
i∈Z⩾0



f ∈Q

Theorem 5.16. Given two matrix factorizations E ∈ [M F (Q, f )] and F ∈ [M F (Q, g)], we have the following
commutative diagram
n
n ⊗ϕ
̃
̃
ϕ
F
E

// ((Ω● , df, n) ⊗mf E) ⊗mf ((Ω● , dg, n) ⊗mf F )
E ⊗mf F ❨❨❨❨
❨❨❨❨❨❨
❨❨❨❨❨❨
❨❨❨❨❨❨
❨❨❨❨❨❨
∧○(1⊗τ ⊗1
❨❨❨❨❨❨
̃
ϕn

,,
E⊗mf F
(Ω● , df + dg, n) ⊗mf (E ⊗mf F )
where τ ∶ E ⊗mf (Ω● , df, n) → (Ω● , df, n) ⊗mf E is the isomorphism τ (a ⊗ b) = (−1)∣a∣∣b∣ b ⊗ a.
Remark 5.17. The above diagram makes sense, since the relative dimension of Q over k is n. After changing
the position, wedging things together, terms with degree higher than n vanish.
Proof. (of Theorem5.16) First, by Proposition 3.5, for the underlying modules E and F , if we choose a
connection ∇E for E and ∇F for F , then there is a natural connection for the tensor product: ∇E ⊗1+1⊗∇F .
Also, the differential for the tensor product of two matrix factorizations is given by dE⊗mf F (e ⊗ f ) = dE (e) ⊗
f + (−1)∣e∣ e ⊗ dF (f ), where e ∈ E and f ∈ F . After a careful calculation, we have that
AtE⊗mf F (e ⊗ f ) = AtE (e) ⊗ f + (−1)∣e∣ e ⊗ AtF (f )

(∗)

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i.e., AtE⊗mf F = AtE ⊗ 1 + τ (1 ⊗ AtF ), where τ is the map in the statement of the theorem.
n
Another observation we want to make before looking into ϕ̃
E⊗mf F is that

∧ ○ (AtE ⊗ 1) ○ τ (1 ⊗ AtF ) = ∧ ○ τ (1 ⊗ AtF ) ○ (AtE ⊗ 1).
In fact, we have
∧ ○ (AtE ⊗ 1) ○ τ (1 ⊗ AtF )(e ⊗ f ) = ∧ ○ (AtE ⊗ 1) ○ τ (1 ⊗ AtF )((−1)∣e∣ σ(e ⊗ AtF (f )))
where σ is the same as τ but doesn’t introduce a sign. Say for simplicity that AtF (f ) = u ⊗ f ′ and AtE (e) =
w ⊗ e′ (these should really be sums of simple tensors, nonetheless, the idea is the same and the case for simple
tensors is more clear), then the above is
∧ ○ (AtE ⊗ 1) ○ ((−1)∣e∣ u ⊗ e ⊗ f ′ )
= (−1)∣e∣ ∧ (u ⊗ AtE (e) ⊗ f ′ )
= (−1)∣e∣ ∧ (u ⊗ w ⊗ e′ ⊗ f ′ )
= (−1)∣e∣ ⋅ (u ∧ w ⊗ e′ ⊗ f ′ )
= −(−1)∣e∣ ⋅ (w ∧ u ⊗ e′ ⊗ f ′ )
= −(−1)∣e∣ ∧ σ(AtE (e) ⊗ AtF (f )).
For ∧ ○ τ (1 ⊗ AtF ) ○ (AtE ⊗ 1)(e ⊗ f ), we have
∧ ○ τ (1 ⊗ AtF ) ○ (AtE ⊗ 1)(e ⊗ f )
= ∧ ○ τ (1 ⊗ AtF ) ○ (AtE (e) ⊗ f )
= ∧ ○ τ (1 ⊗ AtF ) ○ (w ⊗ e′ ⊗ f )
= ∧ ○ τ (w ⊗ e′ ⊗ AtF (f ))
= ∧ ○ τ (w ⊗ e′ ⊗ u ⊗ f ′ )
= (−1)∣e ∣ ∧ (w ⊗ u ⊗ e′ ⊗ f ′ )
′

= (−1)∣e∣+1 (w ∧ u ⊗ e′ ⊗ f ′ )
= −(−1)∣e∣ ∧ σ(AtE(e) ⊗ AtF (f )).

CHERN CHARACTER FOR MATRIX FACTORIZATIONS VIA CHERN-WEIL

27

Therefore, the tow compositions of the operators AtE ⊗ 1 and τ (1 ⊗ AtF ) are the same after ∧ and more

importantly we get −(−1)∣e∣ ∧ σ(AtE (e) ⊗ AtF (f )) applying to the element e ⊗ f . Then it is not hard to see
that

⎧
⎪
∧σ(AtkE (e) ⊗ AtsF (f ))
if one of k, s is even
⎪
⎪
⎪
∧(AtE ⊗ 1) ○ (τ ()1 ⊗ AtF ) = ⎨
⎪
∣e∣
k
s
⎪
⎪
⎪
⎩−(−1) ∧ σ(AtE (e) ⊗ AtF (f )) if both k and s are odd
k

s

n
We can compute ϕ̃
E⊗mf F by formula (∗), the degree ith piece is (remember the notation ∼ indicates we

have already applied ∧ to the Atiyah class)
1
1 i
At
= (AtE ⊗ 1 + τ (1 ⊗ AtF ))i
i! E⊗mf F i!
i
1
⎧
k
s
⎪
if one of k, s is even
⎪
∑ ( ) ∧ σ(AtE ⊗ AtF )
⎪
⎪
i!
k
⎪
⎪ k+s=i
=⎨
⎪
1
i
⎪
k
s
⎪
−(−1)∣e∣
⎪
∑ ( ) ∧ σ(AtE ⊗ AtF ) if both k and s are odd
⎪
⎪
i!
k
⎩
k+s=i
1
⎧
⎪
∧ σ(AtkE ⊗ AtsF )
if one of k, s is even
⎪
∑
⎪
⎪
k!s!
⎪
⎪k+s=i
=⎨
⎪
1
⎪−(−1)∣e∣
⎪
∧ σ(AtkE ⊗ AtsF ) if both k and s are odd
⎪
∑
⎪
⎪
⎩
k+s=i k!s!

n is
̃n ⊗ ϕ
̃
Meanwhile, the ith component for ϕ
∑
E
F

1
(AtkE ⊗ AtsF ). Therefore, say AtkE (e) = w′ ⊗ e and
k+s=i k!s!
AtsE (f ) = u′ ⊗ f with w′ ∈ Ωk and u′ ∈ Ωs (hence ∣e∣ = ∣e∣ + 1 if k is odd and ∣e)∣ = ∣e∣ if k is even), we have
1
(AtkE (e) ⊗ AtsF (f )))
k+s=i k!s!

(∧ ○ (1 ⊗ τ ⊗ 1))( ∑

1
∧ σ(AtkE (e) ⊗ AtsF (f ))
k+s=i k!s!

= (−1)∣e∣⋅s ⋅ ( ∑

1
⎧
⎪
∧ σ(AtkE (e) ⊗ AtsF (f )) if both k, s are odd
−(−1)∣e∣ ⋅ ∑
⎪
⎪
⎪
⎪
k+s=i k!s!
⎪
=⎨
⎪
1
⎪
⎪
⋅ ∧σ(AtkE (e) ⊗ AtsF (f ))
otherwise
⎪
∑
⎪
⎪
k!s!
⎩k+s=i
This completes the proof of the theorem.



Corollary 5.18. The Chern character ch ∶ Kf (Q) → ⊕ H0 ((Ω⋅ , idf, n)Z/2 ) is a ring homomorphism, i.e.,
i∈Z⩾0

ch([E] ⋅ [F ]) = ch([E])ch([F ])
n
n
̃n ̃
Proof. Theorem 5.16 tells us that ϕ̃
E⊗mf F = (∧ ○ (1 ⊗ τ ⊗ 1)) ○ (ϕE ⊗ ϕF ). The corollary follows by applying

str to this equation.



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5.4. Functoriality.
Consider a k-algebra homomorphism ϕ ∶ R → S that sends f ∈ R to g ∈ S. For any matrix factorization
 E0 ) ∈ M F (R, f ), there is then a naturally induced matrix factorization
E = (E1 ÐÔÐ
A

B

 E0 ⊗R S) ∈ M F (S, g).
E ⊗mf S = (E1 ⊗R S ÐÔÔÐ
A⊗1

B⊗1

It is obvious that E1 ⊗R S and E0 ⊗R S are finitely generated projective S-modules. Also, we do have
(B ⊗ 1) ○ (A ⊗ 1) = (A ⊗ 1) ○ (B ⊗ 1) = g ⋅ id: in fact, (B ⊗ 1) ○ (A ⊗ 1)(e1 ⊗ s) = (f ⋅ e1 ) ⊗ s, but since we are
talking about S-modules, (f ⋅ e1 ) ⊗ s = e1 ⊗ (ϕ(f ) ⋅ s) = e1 ⊗ (g ⋅ s) = g ⋅ (e1 ⊗ s).

Definition 5.19. For a k-algebra homomorphism ϕ as above, define a functor ϕ∗ ∶ M F (R, f ) → M F (S, g)

that sends E to ϕ∗ (E) ∶= E ⊗mf S and a strict morphism α = (α0 , α1 ) ∶ E → F to a strict morphism
ϕ∗ (α) ∶= (α0 ⊗ 1, α1 ⊗ 1) ∶ ϕ∗ (E) → ϕ∗ (F ).

The functor ϕ∗ is well-defined on the homotopy category of matrix factorizations. Also we can talk about
ϕ∗ (E) = E ⊗R S for a finitely generated R-module E by regarding E as a matrix factorization of zero. In

particular, there is a natural map µ ∶ Ω1R/k ⊗R S = ϕ∗ (Ω1R/k ) → Ω1S/k which sends dR/k (r) ⊗ s to s ⋅ dS/k (ϕ(r)).
Let us prove a lemma before getting into the statement about functoriality.

Lemma 5.20. For ϕ as above and any finitely generated projective R-module E, there is a naturally induced
connection ∇E⊗R S ∶= µ(∇E ⊗ 1) + σ(1 ⊗ dS/k ) for the S-module ϕ∗ (E), i.e.,
∇E⊗R S ∶ ϕ∗ (E) ∶= E ⊗R S → Ω1S/k ⊗S (E ⊗R S) ≅ Ω1S/k ⊗R E
Proof. First, notice that we have the following two compositions:
E ⊗k S ÐÐÐ→ (Ω1R/k ⊗R E) ⊗k S ≅ Ω1R/k ⊗R (E ⊗k S) ≅ (Ω1R/k ⊗R S) ⊗k E Ð
→ Ω1S/k ⊗k E
∇E ⊗1

µ

and
1⊗dS/k

σ

E ⊗k S ÐÐÐÐ→ E ⊗R Ω1S/k Ð
→ Ω1S/k ⊗k E
Let’s denote the some of the above two compositions by ∇E⊗k S . It is obvious that they are both k-linear,
one can also show that ∇E⊗k S is in fact R-linear by checking directly. Hence we get an induced map:
E ⊗k S


s
E ⊗R S

∇E⊗k S

s

s

s

// Ω1 ⊗k E
S/k
s99

// // Ω1 ⊗R E
S/k

CHERN CHARACTER FOR MATRIX FACTORIZATIONS VIA CHERN-WEIL

29

which we denote by ∇E⊗R S .
Now the only thing left to check is that ∇E⊗R S satisfies the Leibniz rule, i.e.,
∇E⊗R S (s ⋅ (e ⊗ s′ )) = dS/k (s) ⊗ (e ⊗ s′ ) + s ⋅ ∇E⊗R S (e ⊗ s′ ),
for any e ∈ E, s, s′ ∈ S.
Let’s prove it using the same technique as in the proof of Theorem 5.16. Say ∇E (e) = dR/k (r) ⊗ e′ ∈
Ω1R/k ⊗R E,
∇E⊗R S (s ⋅ (e ⊗ s′ )) = ∇E⊗R S (e ⊗ ss′ )
= µ(∇E (e) ⊗ ss′ ) + σ(e ⊗ dS/k (ss′ ))
= µ(dR/k (r) ⊗ e′ ⊗ ss′ ) + (dS/k (s) ⋅ s′ + s ⋅ dS/k (s′ )) ⊗ e
= ss′ ⋅ dS/k (ϕ(r)) ⊗ e′ + (dS/k (s) ⋅ s′ + s ⋅ dS/k (s′ )) ⊗ e.
Meanwhile,
dS/k (s) ⊗ (e ⊗ s′ ) + s ⋅ ∇E⊗R S (e ⊗ s′ ) = s′ ⋅ dS/k (s) ⊗ e + s ⋅ (µ(∇E ⊗ 1)(e ⊗ s′ ) + σ(1 ⊗ dS/k )(e ⊗ s′ ))
= s′ ⋅ dS/k (s) ⊗ e + s ⋅ (µ(∇E (e) ⊗ s′ ) + dS/k (s′ ) ⊗ e)
= s′ ⋅ dS/k (s) ⊗ e + s ⋅ dS/k (s′ ) ⊗ e + s ⋅ µ(dR/k (r) ⊗ e′ ⊗ s′ )
= s′ ⋅ dS/k (s) ⊗ e + s ⋅ dS/k (s′ ) ⊗ e + s ⋅ (s′ dS/k (ϕ(r)) ⊗ e′ ).

For simplicity (and to make future calculations easier), we follow the usual convention of denoting ∇E⊗R S
by ∇E ⊗ 1 + 1 ⊗ dS/k . Now we can state and prove
Proposition 5.21 (Functoriality). Under the above hypotheses and the extra condition that both R and S
are smooth k-algebras with the same relative dimension n, we have ϕ∗ ○ ch = ch ○ ϕ∗ .
Proof. By our formula for the Chern character, it’s enough to show ϕ∗ (At(E)) = At(ϕ∗ (E)) for a matrix
factorization E.
By Lemma 5.20, choose the nature connection ∇E ⊗ 1 + 1 ⊗ dS/k for E ⊗R S, so the Atiyah class of
ϕ∗ (E) = E ⊗mf S is
⎡
⎢∇E ⊗ 1 + 1 ⊗ dS/k
⎢
⎢
⎢
⎢
⎣

⎤⎡
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎢
∇E ⊗ 1 + 1 ⊗ dS/k ⎥
⎦ ⎣B ⊗ 1

⎤
A ⊗ 1⎥
⎥
⎥−
⎥
⎥
⎦

30

XUAN YU

⎤⎡
⎡
⎢
⎢
A ⊗ 1⎥
⎢
⎥ ⎢∇E ⊗ 1 + 1 ⊗ dS/k
⎢
⎥⎢
⎢
⎥⎢
⎢B ⊗ 1
⎥⎢
⎣
⎦⎣

The Atiyah class of E is

⎤
⎥
⎥
⎥
⎥
∇E ⊗ 1 + 1 ⊗ dS/k ⎥
⎦

⎡
⎤
⎢
(∇E A − A∇E ) ⊗ 1⎥
⎢
⎥
=⎢
⎥
⎢
⎥
⎢(∇E B − B∇E ) ⊗ 1
⎥
⎣
⎦
⎡
⎢
⎢
⎢
⎢
⎢∇E B − B∇E
⎣

⎤
∇E A − A∇E ⎥
⎥
⎥
⎥
⎥
⎦

Now is obvious from the definition of ϕ∗ on strict morphisms that ϕ∗ (At(E)) = At(ϕ∗ (E)).



Acknowledgements
This is part of the author’s PhD thesis written under the supervision of Professor Mark Walker at the
University of Nebraska-Lincoln. I would like to thank him for his help and advice of this article.
References
[1] N. Carqueville and D. Murfet. Adjunctions and defects in Landau-Ginzburg models. arXiv preprint arXiv:1208.1481, 2012.
[2] T. Dyckerhoff and D. Murfet. The Kapustin-Li formula revisited. Arxiv preprint ArXiv:1004.0687, 2010.
[3] T. Dyckerhoff and D. Murfet. Pushing forward matrix factorisations. Arxiv preprint arXiv:1102.2957, 2011.
[4] D. Eisenbud. Homological algebra on a complete intersection, with an application to group representations. Trans. Amer.
Math. Soc, 260, 1980.
[5] D. Huybrechts. Complex geometry: An introduction. Springer Verlag, 2005.
[6] A. Kapustin and Y. Li. Topological correlators in Landau-Ginzburg models with boundaries. Advances in Theoretical and
Mathematical Physics, 7(4):727–749, 2003.
[7] J.L. Loday. Cyclic homology, volume 301. Springer Verlag, 1998.
[8] D. Platt. Chern Character for Global Matrix Factorizations. arXiv preprint arXiv:1209.5686, 2012.
[9] A. Polishchuk and A. Vaintrob. Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations. Arxiv
preprint ArXiv:1002.2116, 2010.
[10] E. Segal. The closed state space of affine Landau-Ginzburg B-models. Arxiv preprint ArXiv:0904.1339, 2009.
[11] Mark E Walker. Supports, smoothness and Hochschild homology for matrix factorizations. In preparation.
[12] Y. Yoshino. Tensor products of matrix factorizations. Nagoya Mathematical Journal, 152:39–56, 1998.
[13] Xuan Yu. Geometric Study of the Category of Matrix Factorizations. PhD dissertation, 2013.

Department of Mathematics, University of Nebraska-Lincoln, Lincoln NE 68588 USA
Current address: Departement Wiskunde-Informatica, Universiteit Antwerpen, Middelheimlaan 1, 2020 Antwerpen, Belgium
E-mail address: xuan.yu@uantwerpen.be

