# Twisted Koszul algebras of nonisolated hypersurface singularities

Sangwook Lee  Sangwook Lee: Department of Mathematics and Integrative
Institute of Basic Science
Soongsil University
369 Sangdo-ro, Dongjak-gu, Seoul, Korea<EMAIL_ADDRESS>
###### Abstract.

Given a hypersurface singularity (not necessarily isolated) with a finite
abelian group action, we develop a method to define an explicit product
structure on the twisted Koszul algebra (whose invariant subalgebra is the
orbifold Koszul algebra).

##  1\. Introduction

Hypersurface singularities seem elementary but are still quite interesting
objects due to abundant structures behind them. They have been vigorously
studied in various contexts such as topology, algebraic geometry, symplectic
geometry and representation theory etc.

We can generalize usual hypersurface singularity (which is a polynomial in
this paper) slightly, by considering a finite group action by $G$ on the
affine space together with a function $W$, which is invariant under the
action. Such a pair $(W,G)$ is called an orbifold Landau-Ginzburg model. It is
interesting in its own right, and it also arises naturally in mirror symmetry
in the following manner. Suppose that a symplectic manifold $X$ has a LG
mirror $W:\Bbbk^{n}\to\Bbbk$. If $\tilde{X}$ is a normal covering space of $X$
such that its deck transformation group $G$ is finite abelian, then it is
expected that a mirror of $\tilde{X}$ is given by the orbifold LG model
$(W,{\widehat{G}})$. See [CL20, HJL22] for example.

We will focus on the computation of Hochschild type invariants among several
approaches toward understanding singularities of orbifold LG models. In this
aspect, isolated singularities have been investigated more than nonisolated
ones so far. When the singularity is isolated, we come up with so-called
orbifold Jacobian algebra as desired Hochschild invariant. There are works on
orbifold Jacobian algebras as [CT13, BTW23, HLL20, Shk20]. On the other hand,
when the orbifold singularity is nonisolated its Hochschild invariant (namely
orbifold Koszul algebra) is less understood. Nevertheless, nonisolated
orbifold singularities also naturally arise when we study mirror symmetry of
noncompact manifolds, e.g. [HJL22]. In such a case, the orbifold Koszul
algebra is expected to be isomorphic to the symplectic cohomology of given
noncompact manifold. Since symplectic cohomology is quite difficult to compute
in general, it is desirable to know the structure of orbifold Koszul algebras
as explicit as possible.

In this work, we investigate orbifold Koszul algebras in terms of matrix
factorizations. The original reason for the author to use matrix
factorizations (other than curved algebras as in previous works) is as
follows: in earlier works [CHL17, CHL19] Cho-Hong-Lau discovered symplectic
manifolds with prescribed Lagrangian submanifolds so that we can find their
mirrors and prove (homological) mirror symmetry in geometric way. In this
context we are naturally led to the category of matrix factorizations of the
mirror orbifold LG model, and Hochschild invariants of the category of MFs are
encoded in terms of Floer theory. (The relation between Floer theory and
category of MFs was crucially used also in [HJL22].)

Now we briefly recall the setting and main problem.

######  Definition 1.1.

Let $W\in\Bbbk[x_{1},\cdots,x_{n}]$, and a finite group $G$ act on
$\Bbbk[x_{1},\cdots,x_{n}]$ such that $W$ is invariant under $G$-action. Then
$(W,G)$ is called an orbifold Landau-Ginzburg model.

If an orbifold LG model $(W,G)$ further satisfies that

  * •

$G$ is abelian,

  * •

for $h\in G$, $h\cdot x_{i}=h_{i}x_{i}$ for some $h_{i}\in\Bbbk^{*}$,

then we call $(W,G)$ a diagonal orbifold LG model.

In this paper we will only deal with diagonal orbifold LG models. Now we want
to define an analogue of Chen-Ruan cohomology for a diagonal orbifold LG
model.

$K^{\bullet}(W,G):=\bigoplus_{g\in G}K^{\bullet}(\partial W^{g})\cdot\xi_{g}.$
(1.1)

Here, $K^{*}(\partial W^{g})$ is the Koszul complex of the Jacobian ideal of
the function $W|_{{\textrm{Fix}}(g)}$, and $\xi_{g}$ is a formal generator
which will be precisely defined later. It is rather easy to compute the module
structure of $K^{\bullet}(W,G)$, but it is never clear what is the product on
it. That is why we are led to consider Hochschild cohomology which has a
natural associative algebra structure. We can consider a bimodule over a
(curved) algebra or a category of matrix factorizations, whose Hochschild
cohomology is isomorphic to $K^{\bullet}(W,G)$ as a module. As we mentioned
above, we will use matrix factorizations. The following is the starting point:

######  Theorem A.

Let $(W,G)$ be a diagonal orbifold LG model. Let $\Delta_{1}$ and $\Delta_{h}$
be Koszul matrix factorizations of the diagonal ideal and the $h$-twisted
diagonal ideal respectively (for $h\in G$), of a polynomial

$W\boxminus
W:=W(y_{1},\cdots,y_{n})-W(x_{1},\cdots,x_{n})\in\Bbbk[x_{1},\cdots,x_{n},y_{1},\cdots,y_{n}].$
(1.2)

Then we have a quasi-isomorphism of chain complexes

$(-)_{kos}:\bigoplus_{h\in G}hom_{MF(W\boxminus
W)}(\Delta_{1},\Delta_{h})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}K^{\bullet}(W,G).$

It is known that $\bigoplus_{h\in G}hom_{MF(W\boxminus
W)}(\Delta_{1},\Delta_{h})$ is isomorphic to an endomorphism algebra of a
$(1\times G)$-equivariant matrix factorization of $W\boxminus W$, so we can
define an intrinsic product structure on $K^{\bullet}(W,G)$ via the quasi-
isomorphism. The remaining problem is therefore to construct a quasi-inverse
of $(-)_{kos}$, and it is the main advance of this work.

######  Theorem B.

Let $W\in R=\Bbbk[x_{1},\cdots,x_{n}]$,
$S=\Bbbk[x_{1},\cdots,x_{n},y_{1},\cdots,y_{n}]$ and $(W,G)$ be a diagonal
orbifold LG model. For $i,j\in\\{1,\cdots,n\\}$ and $h\in G$, there exist
polynomials $g_{ji}^{h}\in S$ and $f_{ji}^{h}\in R$ such that for

$\displaystyle\eta_{h}:S[\theta_{1},\cdots,\theta_{n},\partial_{1},\cdots,\partial_{n}]$
$\displaystyle\to
S[\theta_{1},\cdots,\theta_{n},\partial_{1},\cdots,\partial_{n}],$
$\displaystyle\theta_{I}\partial_{J}$
$\displaystyle\mapsto\sum(-1)^{|I|}g^{h}_{ji}\frac{\partial\theta_{I}}{\partial\theta_{i}}\partial_{j}\partial_{J}+\sum
f^{h}_{ji}\frac{\partial^{2}\theta_{I}}{\partial\theta_{j}\partial\theta_{i}}\partial_{J},$

the map $\exp(\eta_{h})$ is a quasi-inverse of $(-)_{kos}$.

Finally, we will illustrate an example which is relevant in mirror symmetry.
This will not only be helpful in understanding the recipe, but also reflect an
interesting geometric feature of our result, in the sense that we can observe
a similarity between an orbifold Koszul algebra and a symplectic cohomology
ring by looking at the explicit algebra structure.

Acknowledgements

This work grew out of the idea from author’s joint work with Hansol Hong and
Hyeongjun Jin and he would like to thank them for useful discussions and
brilliant insights during the collaboration. This work was supported by the
National Research Foundation of Korea (NRF) grant funded by the Korea
government(MSIT)(No. 202117221032) and Basic Science Research Program through
NRF funded by the Ministry of Education (2021R1A6A1A10044154).

##  2\. Preliminaries

Let $\theta_{i}$ and $\partial_{i}$ (for $i=1,\cdots,n$) be formal variables
with $|\theta_{i}|=-1$, $|\partial_{i}|=1$ and

$\theta_{i}\theta_{j}=-\theta_{j}\theta_{i},\;\;\partial_{i}\partial_{j}=-\partial_{j}\partial_{i},\;\;\partial_{i}\theta_{j}=-\theta_{j}\partial_{i}+\delta_{ij}$
(2.1)

Let $I=\\{i_{1},\cdots,i_{k}\\}$ be an ordered subset of $\\{1,\cdots,n\\}$.
We will frequently use following multi-index notations:

$\theta_{I}:=\theta_{i_{1}}\cdots\theta_{i_{k}},\quad\partial_{I}:=\partial_{i_{1}}\cdots\partial_{i_{k}}.$

We recall an elementary definition in terms of above variables.

######  Definition 2.1.

Let $A$ be a commutative ring. For a sequence $(r_{1},\cdots,r_{n})$ of
elements in $A$,

$K^{\bullet}(r_{1},\cdots,r_{n}):=(A\langle\theta_{1},\cdots,\theta_{n}\rangle,\sum_{i}r_{i}\partial_{i})$

is called the Koszul complex of $(r_{1},\cdots,r_{n})$.

The cochain complex $K^{\bullet}(r_{1},\cdots,r_{n})$ is explicitly spelled
out as

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\cdot\theta_{1}\cdots\theta_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sum_{i}r_{i}\partial_{i}}$$\textstyle{\bigoplus\limits_{i_{1}<\cdots<i_{n-1}}A\cdot\theta_{i_{1}}\cdots\theta_{i_{n-1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus\limits_{i}A\cdot\theta_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sum_{i}r_{i}\partial_{i}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

with
$\bigoplus\limits_{i_{1}<\cdots<i_{m}}A\cdot\theta_{i_{1}}\cdots\theta_{i_{m}}=K^{-m}(r_{1},\cdots,r_{n})$.
Koszul complexes can be used to define invariants of singularities as follows.

######  Definition 2.2.

Let $(W,G)$ be a diagonal orbifold LG model. For $h\in G$ define

$x_{i}^{h}:=\begin{cases}x_{i}&{\textrm{ if }}h\cdot x_{i}=x_{i}\\\
0&{\textrm{ if }}h\cdot x_{i}\neq x_{i}\end{cases},\quad
W^{h}:=W(x_{1}^{h},\cdots,x_{n}^{h})\in\Bbbk[x_{1}^{h},\cdots,x_{n}^{h}].$

Let

$I_{h}:=\\{i\mid h\cdot x_{i}\neq x_{i}\\},\quad I^{h}:=I_{h}^{c}$

be ordered subsets of $\\{1,\cdots,n\\}$ and define a sequence $\partial
W^{h}:=\big{(}\frac{\partial W^{h}}{\partial x_{i}}:i\in I^{h}\big{)}$ in
$\Bbbk[x_{1}^{h},\cdots,x_{n}^{h}]$. The twisted Koszul complex of $(W,G)$ is
defined as

$K^{\bullet}(W,G):=\bigoplus_{h\in G}K^{\bullet}(\partial
W^{h})\cdot\theta_{I_{h}}.$

Defining $G$-action on variables
$\\{\theta_{1},\cdots,\theta_{n},\partial_{1},\cdots,\partial_{n}\\}$ by

$h\cdot\theta_{i}:=h_{i}^{-1}\theta_{i},\quad
h\cdot\partial_{i}:=h_{i}\partial_{i}$

when $h\cdot x_{i}=h_{i}x_{i}$, we call

$K^{\bullet}(W,G)^{G}=\bigoplus_{h\in G}\big{(}K^{\bullet}(\partial
W^{h})\cdot\theta_{I_{h}}\big{)}^{G}$

the orbifold Koszul complex of $(W,G)$, where $\big{(}K^{\bullet}(\partial
W^{h})\cdot\theta_{I_{h}}\big{)}^{G}$ is the subcomplex of $G$-invariants of
$K^{\bullet}(\partial W^{h})\cdot\theta_{I_{h}}$.

Observe that as each $h$-summand we did not just take the Koszul complex of
Jacobian ideal of $W^{h}$, but shifted it by $\theta_{I_{h}}$. Such a
modification will appear meaningful later.

We recall another important invariant of $W$.

######  Definition 2.3.

A matrix factorization of $W\in R$ is a $\mathbb{Z}/2$-graded projective
$R$-module $P=P_{0}\oplus P_{1}$ together with a morphism $d=(d_{0},d_{1})$ of
degree 1 such that

$d^{2}=W\cdot\mathrm{id}.$

Let $\phi:(P,d)\to(Q,d^{\prime})$ be a morphism of degree $j\in\mathbb{Z}/2$.
Define

$D\phi:=d^{\prime}\circ\phi-(-1)^{|\phi|}\phi\circ d$

and define the composition of morphisms in the usual sense. This defines a dg-
category of matrix factorizations $(MF(W),D,\circ)$.

######  Definition 2.4.

Let $(W,G)$ be a diagonal orbifold LG model and
$S=\Bbbk[x_{1},\cdots,x_{n},y_{1},\cdots,y_{n}]$. Let

$\nabla_{j}W:=\frac{W(y_{1},\cdots,y_{j},x_{j+1},\cdots,x_{n})-W(y_{1},\cdots,y_{j-1},x_{j},\cdots,x_{n})}{y_{j}-x_{j}},$

and let

$\Delta_{h}:=\Big{(}S\langle\theta_{1},\cdots,\theta_{n}\rangle,\sum_{i=1}^{n}\big{(}(y_{i}-hx_{i})\theta_{i}+\nabla_{i}W(h\cdot
x_{1},\cdots,h\cdot x_{n},y_{1},\cdots,y_{n})\cdot\partial_{i}\big{)}\Big{)}$

be a matrix factorization of $W\boxminus W$ (in (1.2)). Here
$S\langle\theta_{1},\cdots,\theta_{n}\rangle$ is $\mathbb{Z}/2$-graded, by
division into even/odd degrees.

Now we introduce the first key step towards our main result.

######  Proposition 2.5.

([HJL22, Proposition 2.6]) We have a quasi-isomorphism of chain complexes

$\bigoplus_{h\in G}hom_{MF(W\boxminus W)}(\Delta_{1},\Delta_{h})\simeq
K^{\bullet}(W,G).$

We will need the following part of the proof of Proposition 2.5. Since
$\Delta_{h}$ is constructed by the resolution of a shifted MCM module
$S/(y_{1}-hx_{1},\cdots,y_{n}-hx_{n})[-n]$ over the hypersurface ring
$S/(W\boxminus W)$, we have quasi-isomorphisms

$\displaystyle\begin{split}hom_{MF(W\boxminus
W)}(\Delta_{1},\Delta_{h})&\simeq
hom_{S}\big{(}\Delta_{1},S/(y_{1}-hx_{1},\cdots,y_{n}-hx_{n})[-n]\big{)}\\\
&\simeq
hom_{S}\big{(}\Delta_{1}[n],S/(y_{1}-hx_{1},\cdots,y_{n}-hx_{n})\big{)}\\\
&\simeq\Delta_{1}^{\vee}[-n]\otimes_{S}\big{(}S/(y_{1}-hx_{1},\cdots,y_{n}-hx_{n})\big{)}.\end{split}$
(2.2)

The latter chain complex in (2.2) is a double complex, and the proof involves
typical spectral sequence argument.

We can spell out a quasi-isomorphism explicitly as follows. Let $\phi\in
hom_{MF(W(y)-W(x))}(\Delta_{1},\Delta_{h})$ be a closed morphism. By (2.2) its
cohomology class $[\phi]$ is completely determined by $pr\circ\phi$, where
$pr:\Delta_{h}\to S\cdot\theta_{1}\cdots\theta_{n}$ is the projection map. We
have a unique expression

$pr\circ\phi=\sum_{J,K}b_{JK}\theta_{J}\theta_{K}\partial_{K^{op}}$ (2.3)

such that $J$ and $K$ are ordered subsets of $\\{1,\cdots,n\\}$, $b_{JK}\in
S$, and $\partial_{K^{op}}:=\partial_{k_{m}}\cdots\partial_{k_{1}}$ whereas
$\theta_{K}=\theta_{k_{1}}\cdots\theta_{k_{m}}$. We point out that $K$ may be
empty. We also define

$(pr\circ\phi)_{+}:=\sum_{J:I_{h}\subset
J}\sum_{K}b_{JK}\theta_{J}\theta_{K}\partial_{K^{op}}.$

For an $S$-module homomorphism $\phi=\sum
f_{IJ}(y_{1},\cdots,y_{n},x_{1},\cdots,x_{n})\theta_{I}\partial_{J}$ we define

$\phi|_{{\textrm{Fix}}(h)}:=\sum
f_{IJ}(x_{1}^{h},\cdots,x_{n}^{h},x_{1}^{h},\cdots,x_{n}^{h})\theta_{I}\partial_{J}.$
(2.4)

With respect to the expression (2.3) and notation (2.4), the quasi-isomorphism

$(-)_{kos}:hom(\Delta_{1},\Delta_{h})\to
K^{\bullet}(\partial_{x_{i}}W^{h}:i\in I^{h})\cdot\theta_{I_{h}}$

can be realized as

$\phi_{kos}=\big{(}\sum_{J:I_{h}\subset
J}\sum_{K}b_{JK}\theta_{J}\big{)}|_{{\textrm{Fix}}(h)}.$

Observe that $\phi_{kos}$ is determined only by $(pr\circ\phi)_{+}$.

##  3\. Classification of closed morphisms of matrix factorizations

This section is devoted to the construction of a quasi-inverse of $(-)_{kos}$.
First, for $0\leq j<i\leq n$ and $h\in G$, define

${\overline{W}}^{h}_{j,i}:=W(y_{1},\cdots,y_{j},x_{j+1},\cdots,x_{i},hx_{i+1},\cdots,hx_{n})\in
S,$
${\widetilde{W}}^{h}_{j,i}:=W(x_{1}^{h},\cdots,x_{j}^{h},x_{j+1},\cdots,x_{i},hx_{i+1},\cdots,hx_{n})\in
R.$

(Recall that $x_{i}^{h}=x_{i}$ if $hx_{i}=x_{i}$ and $x_{i}^{h}=0$ otherwise.)
We also define

${\overline{W}}^{h}_{i,i}:=W(y_{1},\cdots,y_{i},hx_{i+1},\cdots,hx_{n}),$
(3.1)
${\widetilde{W}}^{h}_{i,i}:=W(x_{1}^{h},\cdots,x_{i}^{h},hx_{i+1},\cdots,hx_{n}).$
(3.2)

For $i,j\in\\{1,\cdots,n\\}$ with $j<i$, define

$\displaystyle\begin{split}g^{h}_{ji}&:=\begin{cases}\frac{({\overline{W}}^{h}_{j,i}-{\overline{W}}^{h}_{j-1,i})-({\overline{W}}^{h}_{j,i-1}-{\overline{W}}^{h}_{j-1,i-1})}{(y_{j}-x_{j})(x_{i}-hx_{i})}&{\rm
if\;}i\in I_{h},\\\
\frac{\partial_{i}{\overline{W}}^{h}_{j,i}-\partial_{i}{\overline{W}}^{h}_{j-1,i}}{y_{j}-x_{j}}&{\rm
if\;}i\in I^{h},\end{cases}\\\
f^{h}_{ji}&:=\begin{cases}\frac{({\widetilde{W}}^{h}_{j,i}-{\widetilde{W}}^{h}_{j-1,i})-({\widetilde{W}}^{h}_{j,i-1}-{\widetilde{W}}^{h}_{j-1,i-1})}{(x_{j}-hx_{j})(x_{i}-hx_{i})}&{\rm
if\;}i,j\in I_{h},\\\
\frac{\partial_{i}{\widetilde{W}}^{h}_{j,i}-\partial_{i}{\widetilde{W}}^{h}_{j-1,i}}{x_{j}-hx_{j}}&{\rm
if\;}j\in I_{h}{\rm\;and\;}i\in I^{h},\\\
\frac{(\partial_{j}{\widetilde{W}}^{h}_{j,i}-\partial_{i}{\widetilde{W}}^{h}_{j,i-1})-({\partial_{j}{\widetilde{W}}^{h}_{i-1,i-1}-\partial_{j}{\widetilde{W}}^{h}_{i,i}})}{x_{i}-hx_{i}}&{\rm
if\;}j\in I^{h}{\rm\;and\;}\;i\in I_{h},\\\ 0&{\rm
otherwise}\end{cases}\end{split}$ (3.3)

and

$\displaystyle\begin{split}g^{h}_{ii}&:=\begin{cases}\frac{1}{y_{i}-x_{i}}\cdot\Big{(}\frac{{\overline{W}}^{h}_{i,i}-{\overline{W}}^{h}_{i-1,i-1}}{y_{i}-hx_{i}}-\frac{{\overline{W}}^{h}_{i-1,i}-{\overline{W}}^{h}_{i-1,i-1}}{x_{i}-hx_{i}}\Big{)}&{\rm
if\;}i\in I_{h},\\\
\frac{1}{y_{i}-x_{i}}\cdot\Big{(}\frac{{\overline{W}}^{h}_{i,i}-{\overline{W}}^{h}_{i-1,i-1}}{y_{i}-hx_{i}}-\partial_{i}{\overline{W}}^{h}_{i-1,i}\Big{)}&{\rm
if\;}i\in I^{h}.\end{cases}\end{split}$ (3.4)

Observe that $g^{h}_{ji}=0$ if $j>i$, and $f^{h}_{ji}=0$ if $j\geq i$. Also,
if $i\in I^{h}$ then
$\frac{{\overline{W}}^{h}_{i,i}-{\overline{W}}^{h}_{i-1,i-1}}{y_{i}-hx_{i}}-\partial_{i}{\overline{W}}^{h}_{i-1,i}$
is indeed divisible by $y_{i}-x_{i}$ because it vanishes after we evaluate
$y_{i}=x_{i}$, so $g_{ii}^{h}$ is also well-defined.

For $S=k[x_{1},\cdots,x_{n},y_{1},\cdots,y_{n}]$, define an $S$-linear map

$\displaystyle\eta_{h}:S[\theta_{1},\cdots,\theta_{n},\partial_{1},\cdots,\partial_{n}]$
$\displaystyle\to
S[\theta_{1},\cdots,\theta_{n},\partial_{1},\cdots,\partial_{n}],$
$\displaystyle\theta_{I}\partial_{J}$
$\displaystyle\mapsto\sum(-1)^{|I|}g^{h}_{ji}\frac{\partial\theta_{I}}{\partial\theta_{i}}\partial_{j}\partial_{J}+\sum
f^{h}_{ji}\frac{\partial^{2}\theta_{I}}{\partial\theta_{j}\partial\theta_{i}}\partial_{J}.$
(3.5)

We also define a map

$\exp(\eta_{h}):=1+\eta_{h}+\frac{\eta_{h}^{2}}{2!}+\cdots\;\;:S[\theta_{1},\cdots,\theta_{n},\partial_{1},\cdots,\partial_{n}]\to
S[\theta_{1},\cdots,\theta_{n},\partial_{1},\cdots,\partial_{n}].$ (3.6)

(3.6) is a finite sum due to graded commutativity of $\theta$ and $\partial$,
hence well-defined.

######  Theorem 3.1.

Let $h\in G$, and let $\displaystyle\sum_{|I|=r}a_{I}\theta_{I}\theta_{I_{h}}$
be a cocycle of the Koszul complex of $(\partial_{i}W^{h}:i\in I^{h})$ where
$a_{I}\in k[x_{1}^{h},\cdots,x_{n}^{h}]$ is identified as an element of $S$ by
natural inclusion. Then $\exp(\eta_{h})\big{(}\sum
a_{I}\theta_{I}\theta_{I_{h}}\big{)}$ is a closed element in
$hom_{MF(W(y)-W(x))}(\Delta_{1},\Delta_{h})$.

###### Proof.

Throughout the proof, we omit all upper indices $h$, i.e.

$g_{ji}=g^{h}_{ji},\;\;f_{ji}=f^{h}_{ji},\;\;{\overline{W}}={\overline{W}}^{h},\;\;{\widetilde{W}}={\widetilde{W}}^{h}.$

Let $f\theta_{I}\partial_{J}\in hom_{MF(W(y)-W(x))}(\Delta_{1},\Delta_{h})$.
For the differential $D$ of the chain complex
$hom_{MF(W(y)-W(x))}(\Delta_{1},\Delta_{h})$, decompose
$D={D^{\uparrow}}+{D^{\downarrow}}$ as follows:

$\displaystyle{D^{\uparrow}}(f\theta_{I}\partial_{J}):=$
$\displaystyle\sum(y_{i}-hx_{i})\theta_{i}\cdot
f\theta_{I}\partial_{J}+(-1)^{|I|+|J|+1}\sum
f\theta_{I}\partial_{J}\cdot(y_{i}-x_{i})\theta_{i},$
$\displaystyle{D^{\downarrow}}(f\theta_{I}\partial_{J}):=$
$\displaystyle\sum\nabla_{i}W(hx,y)\partial_{i}\cdot
f\theta_{I}\partial_{J}+(-1)^{|I|+|J|+1}\sum
f\theta_{I}\partial_{J}\cdot\nabla_{i}W(x,y)\partial_{i}.$

######  Step 1.

We have

$\displaystyle{D^{\uparrow}}\big{(}\sum_{I}a_{I}\theta_{I}\theta_{I_{h}}\big{)}$
$\displaystyle=$ $\displaystyle\sum_{1\leq i\leq
n}\sum_{I}(y_{i}-hx_{i})\theta_{i}\cdot
a_{I}\theta_{I}\theta_{I_{h}}+\sum_{1\leq i\leq
n}\sum_{I}(-1)^{|I|+|I_{h}|+1}a_{I}\theta_{I}\theta_{I_{h}}\cdot(y_{i}-x_{i})\theta_{i}$
$\displaystyle=$ $\displaystyle\sum_{i\in
I^{h}}\sum_{I}(y_{i}-hx_{i})\theta_{i}\cdot
a_{I}\theta_{I}\theta_{I_{h}}+\sum_{i\in
I^{h}}\sum_{I}(-1)^{|I|+|I_{h}|+1}a_{I}\theta_{I}\theta_{I_{h}}\cdot(y_{i}-x_{i})\theta_{i}$
$\displaystyle=$ $\displaystyle\sum_{i\in
I^{h}}\sum_{I}(y_{i}-x_{i})\theta_{i}\cdot
a_{I}\theta_{I}\theta_{I_{h}}-\sum_{i\in
I^{h}}\sum_{I}(y_{i}-x_{i})\theta_{i}\cdot a_{I}\theta_{I}\theta_{I_{h}}$
$\displaystyle=$ $\displaystyle 0.$

######  Step 2.

Now we spell out
${D^{\downarrow}}\Big{(}\frac{\eta_{h}^{k}}{k!}\big{(}a_{I}\theta_{I}\theta_{I_{h}}\big{)}\Big{)}$
and
${D^{\uparrow}}\Big{(}\frac{\eta_{h}^{k+1}}{(k+1)!}\big{(}a_{I}\theta_{I}\theta_{I_{h}}\big{)}\Big{)}$
for each $I\subset I^{h}$.

By graded commutativity (see [Lee22] for more detail), we have

$\frac{\eta_{h}^{k}}{k!}(a_{I}\theta_{I}\theta_{I_{h}})=\sum_{\stackrel{{\scriptstyle
l+m=k}}{{1\leq i_{\bullet},j_{\bullet}\leq
n}}}(-1)^{\epsilon_{l}}g_{j_{1}i_{1}}\cdots
g_{j_{l}i_{l}}f_{i_{l+1}i_{l+2}}\cdots f_{i_{l+2m-1}i_{l+2m}}\cdot
a_{I}\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j_{1}}\cdots\partial_{j_{l}},$
(3.7)

with the sign given by $\epsilon_{l}:=l(|I|+|I_{h}|)-\frac{l(l-1)}{2}.$ The
following identities will be frequently (and implicitly) used in the sequel:

$(-1)^{|I|+|I_{h}|+\epsilon_{l}+l}=(-1)^{\epsilon_{l+1}},\quad(-1)^{\epsilon_{l+2}}=(-1)^{\epsilon_{l}+1}.$

We introduce a notation

$\Phi_{(j_{1},i_{1}),\cdots,(j_{l},i_{l})}^{(i_{l+1},i_{l+2}),\cdots,(i_{l+2m-1},i_{l+2m})}:=g_{j_{1}i_{1}}\cdots
g_{j_{l}i_{l}}f_{i_{l+1}i_{l+2}}\cdots f_{i_{l+2m-1}i_{l+2m}}.$

Then we can write

$\displaystyle{D^{\downarrow}}\Big{(}\frac{\eta_{h}^{k}}{k!}\big{(}a_{I}\theta_{I}\theta_{I_{h}}\big{)}\Big{)}$
$\displaystyle=$ $\displaystyle\sum_{\stackrel{{\scriptstyle 1\leq i\leq
n}}{{l+m=k,1\leq i_{\bullet},j_{\bullet}\leq
n}}}\nabla_{i}W(hx,y)\partial_{i}\cdot(-1)^{\epsilon_{l}}\Phi_{(j_{1},i_{1}),\cdots,(j_{l},i_{l})}^{(i_{l+1},i_{l+2}),\cdots,(i_{l+2m-1},i_{l+2m})}\cdot
a_{I}\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j_{1}}\cdots\partial_{j_{l}}$
$\displaystyle+\sum_{\stackrel{{\scriptstyle 1\leq j\leq n}}{{l+m=k,1\leq
i_{\bullet},j_{\bullet}\leq
n}}}(-1)^{|I|+|I_{h}|+1+\epsilon_{l}}\Phi_{(j_{1},i_{1}),\cdots,(j_{l},i_{l})}^{(i_{l+1},i_{l+2}),\cdots,(i_{l+2m-1},i_{l+2m})}\cdot
a_{I}\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j_{1}}\cdots\partial_{j_{l}}\cdot(\nabla_{j}W)\partial_{j}$
$\displaystyle=$ $\displaystyle\sum_{\stackrel{{\scriptstyle 1\leq j\leq
n}}{{l+m=k,1\leq i_{\bullet},j_{\bullet}\leq
n}}}(-1)^{|I|+|I_{h}|+\epsilon_{l}}(\nabla_{j}W(hx,y)-\nabla_{j}W)\cdot\Phi_{(j_{1},i_{1}),\cdots,(j_{l},i_{l})}^{(i_{l+1},i_{l+2}),\cdots,(i_{l+2m-1},i_{l+2m})}\cdot
a_{I}\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j_{1}}\cdots\partial_{j_{l}}\partial_{j}$
$\displaystyle+\sum_{\stackrel{{\scriptstyle 1\leq i\leq n}}{{l+m=k,1\leq
i_{\bullet},j_{\bullet}\leq
n}}}(-1)^{\epsilon_{l}}\nabla_{i}W(hx,y)\cdot\Phi_{(j_{1},i_{1}),\cdots,(j_{l},i_{l})}^{(i_{l+1},i_{l+2}),\cdots,(i_{l+2m-1},i_{l+2m})}\cdot
a_{I}\frac{\partial^{l+2m+1}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i}\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j_{1}}\cdots\partial_{j_{l}}$
$\displaystyle=$ $\displaystyle\sum_{\stackrel{{\scriptstyle 1\leq j\leq
n}}{{l+m=k,1\leq i_{\bullet},j_{\bullet}\leq
n}}}(-1)^{|I|+|I_{h}|+\epsilon_{l}+l}(\nabla_{j}W(hx,y)-\nabla_{j}W)\cdot\Phi_{(j_{1},i_{1}),\cdots,(j_{l},i_{l})}^{(i_{l+1},i_{l+2}),\cdots,(i_{l+2m-1},i_{l+2m})}\cdot
a_{I}\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j}\partial_{j_{1}}\cdots\partial_{j_{l}}$
$\displaystyle+\sum_{\stackrel{{\scriptstyle 1\leq i\leq n}}{{l+m=k,1\leq
i_{\bullet},j_{\bullet}\leq
n}}}(-1)^{\epsilon_{l}}\nabla_{i}W(hx,y)\cdot\Phi_{(j_{1},i_{1}),\cdots,(j_{l},i_{l})}^{(i_{l+1},i_{l+2}),\cdots,(i_{l+2m-1},i_{l+2m})}\cdot
a_{I}\frac{\partial^{l+2m+1}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i}\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j_{1}}\cdots\partial_{j_{l}}$
$\displaystyle=$ $\displaystyle\sum_{\stackrel{{\scriptstyle 1\leq j\leq
n}}{{l+m=k,1\leq i_{\bullet},j_{\bullet}\leq
n}}}(-1)^{\epsilon_{l+1}}(\nabla_{j}W(hx,y)-\nabla_{j}W)\cdot\Phi_{(j_{1},i_{1}),\cdots,(j_{l},i_{l})}^{(i_{l+1},i_{l+2}),\cdots,(i_{l+2m-1},i_{l+2m})}\cdot
a_{I}\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j}\partial_{j_{1}}\cdots\partial_{j_{l}}$
$\displaystyle+\sum_{\stackrel{{\scriptstyle 1\leq i\leq n}}{{l+m=k,1\leq
i_{\bullet},j_{\bullet}\leq
n}}}(-1)^{\epsilon_{l}}\nabla_{i}W(hx,y)\cdot\Phi_{(j_{1},i_{1}),\cdots,(j_{l},i_{l})}^{(i_{l+1},i_{l+2}),\cdots,(i_{l+2m-1},i_{l+2m})}\cdot
a_{I}\frac{\partial^{l+2m+1}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i}\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j_{1}}\cdots\partial_{j_{l}}$
$\displaystyle=$ $\displaystyle\sum_{\stackrel{{\scriptstyle 1\leq j\leq
n}}{{l+m=k,1\leq i_{\bullet},j_{\bullet}\leq
n}}}(-1)^{\epsilon_{l+1}}(\nabla_{j}W(hx,y)-\nabla_{j}W)\cdot\Phi_{(j_{1},i_{1}),\cdots,(j_{l},i_{l})}^{(i_{l+1},i_{l+2}),\cdots,(i_{l+2m-1},i_{l+2m})}\cdot
a_{I}\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j}\partial_{j_{1}}\cdots\partial_{j_{l}}$
$\displaystyle+\sum_{\stackrel{{\scriptstyle 1\leq i,j\leq n}}{{l+m=k,1\leq
i_{\bullet},j_{\bullet}\leq
n}}}(-1)^{\epsilon_{l+1}}\nabla_{i}W(hx,y)\cdot\Phi_{(j,i_{l+2}),(j_{1},i_{1}),\cdots,(j_{l},i_{l})}^{(i_{l+3},i_{l+4}),\cdots,(i_{l+2m-1},i_{l+2m})}$
$\displaystyle\qquad\qquad\qquad\qquad\cdot
a_{I}\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i}\partial\theta_{i_{l+2}}\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l}}\partial\theta_{i_{l+3}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j}\partial_{j_{1}}\cdots\partial_{j_{l}}.$

For $K=\\{k_{1},\cdots,k_{i}\\}$ and $L=\\{l_{1},\cdots,l_{j}\\}$ which are
ordered subsets of $\\{1,\cdots,n\\}$, we denote

$\theta_{K}:=\theta_{k_{1}}\cdots\theta_{k_{i}},\quad\partial_{L}:=\partial_{l_{1}}\cdots\partial_{l_{j}}.$

For $\Psi=\sum\limits_{K,L}a_{KL}\theta_{K}\partial_{L}\in
S[\theta_{1},\cdots,\theta_{n},\partial_{1},\cdots,\partial_{n}],$ denote the
summand $a_{KL}\theta_{K}\partial_{L}$ by
$\langle\Psi,\theta_{K}\partial_{L}\rangle$. Be careful that
$\langle\Psi,\theta_{K}\partial_{L}\rangle$ is not just $a_{KL}$.

So we compute
$\langle{D^{\downarrow}}\Big{(}\frac{\eta_{h}^{k}}{k!}(a_{I}\theta_{I}\theta_{I_{h}})\Big{)},\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j_{0}}\partial_{j_{1}}\cdots\partial_{j_{l}}\rangle$
for fixed indices $i_{1},\cdots,i_{l+2m}$ and $j_{0},\cdots,j_{l}$, where
$l+m=k$. We have

$\displaystyle\Big{\langle}{D^{\downarrow}}\Big{(}\frac{\eta_{h}^{k}}{k!}(a_{I}\theta_{I}\theta_{I_{h}})\Big{)},\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j_{0}}\partial_{j_{1}}\cdots\partial_{j_{l}}\Big{\rangle}$
(3.8) $\displaystyle=$
$\displaystyle\sum_{\stackrel{{\scriptstyle\sigma\in{\textrm{Perm}}(\\{j_{0},\cdots,j_{l}\\})}}{{\tau\in{\textrm{Perm}}(\\{i_{1},\cdots,i_{l+2m}\\})}}}(-1)^{\epsilon_{l+1}}\Big{(}(\nabla_{\sigma(j_{0})}W(hx,y)-\nabla_{\sigma(j_{0})}W)\cdot\Phi_{(\sigma(j_{1}),\tau(i_{1})),\cdots,(\sigma(j_{l}),\tau(i_{l}))}^{(\tau(i_{l+1}),\tau(i_{l+2})),\cdots,(\tau(i_{l+2m-1}),\tau(i_{l+2m}))}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\nabla_{\tau(i_{l+1})}W(hx,y)\cdot\Phi_{(\sigma(j_{0}),\tau(i_{l+2})),(\sigma(j_{1}),\tau(i_{1})),\cdots,(\sigma(j_{l}),\tau(i_{l}))}^{(\tau(i_{l+3}),\tau(i_{l+4})),\cdots,(\tau(i_{l+2m-1}),\tau(i_{l+2m}))}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\cdot
a_{I}\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{\tau(i_{1})}\cdots\partial\theta_{\tau(i_{l+2m})}}\partial_{\sigma(j_{0})}\partial_{\sigma(j_{1})}\cdots\partial_{\sigma(j_{l})}$

where ${\textrm{Perm}}(A)$ is the set of permutations of $A$.

Next, we will compute
$\langle{D^{\uparrow}}\Big{(}\frac{\eta_{h}^{k+1}}{(k+1)!}\big{(}a_{I}\theta_{I}\theta_{I_{h}}\big{)}\Big{)},\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j_{0}}\partial_{j_{1}}\cdots\partial_{j_{l}}\rangle$
accordingly.

$\displaystyle{D^{\uparrow}}\Big{(}\frac{\eta_{h}^{k+1}}{(k+1)!}\big{(}a_{I}\theta_{I}\theta_{I_{h}}\big{)}\Big{)}$
$\displaystyle=$ $\displaystyle\sum_{\stackrel{{\scriptstyle 1\leq j\leq
n}}{{l+m=k,1\leq i_{\bullet},j_{\bullet}\leq
n}}}(-1)^{\epsilon_{l+1}}(y_{i}-hx_{i})\theta_{i}\cdot
g_{j_{0}i_{0}}\Phi_{(j_{1},i_{1}),\cdots,(j_{l},i_{l})}^{(i_{l+1},i_{l+2}),\cdots,(i_{l+2m-1},i_{l+2m})}\cdot
a_{I}\frac{\partial^{l+2m+1}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i_{0}}\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j_{0}}\partial_{j_{1}}\cdots\partial_{j_{l}}$
$\displaystyle+\sum_{\stackrel{{\scriptstyle 1\leq j\leq n}}{{l+m=k,1\leq
i_{\bullet},j_{\bullet}\leq
n}}}(-1)^{\epsilon_{l+2}+|I|+|I_{h}|}(y_{i}-x_{i})g_{ri_{l+1}}\Phi_{(j_{0},i_{i_{l+2}}),(j_{1},i_{1}),\cdots,(j_{l},i_{l})}^{(i_{l+3},i_{l+4}),\cdots,(i_{l+2m-1},i_{l+2m})}$
$\displaystyle\qquad\qquad\cdot
a_{I}\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i_{l+1}}\partial\theta_{i_{l+2}}\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l}}\partial\theta_{i_{l+3}}\cdots\partial\theta_{i_{l+2m}}}(\partial_{r}\partial_{j_{0}}\partial_{j_{1}}\cdots\partial_{j_{l}})\cdot\theta_{i},$

hence

$\displaystyle\Big{\langle}{D^{\uparrow}}\Big{(}\frac{\eta_{h}^{k+1}}{(k+1)!}\big{(}a_{I}\theta_{I}\theta_{I_{h}}\big{)}\Big{)},\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j_{0}}\partial_{j_{1}}\cdots\partial_{j_{l}}\Big{\rangle}$
(3.9) $\displaystyle=$
$\displaystyle\sum_{\stackrel{{\scriptstyle\sigma\in{\textrm{Perm}}(\\{j_{0},\cdots,j_{l}\\})}}{{\tau\in{\textrm{Perm}}(\\{i_{1},\cdots,i_{l+2m}\\})}}}\Big{(}\sum_{1\leq
i\leq
n}(-1)^{\epsilon_{l+1}}(x_{i}-hx_{i})g_{\sigma(j_{0}),i}\cdot\Phi_{(\sigma(j_{1}),\tau(i_{1})),\cdots,(\sigma(j_{l}),\tau(i_{l}))}^{(\tau(i_{l+1}),\tau(i_{l+2})),\cdots,(\tau(i_{l+2m-1}),\tau(i_{l+2m}))}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\cdot
a_{I}\theta_{i}\frac{\partial^{l+2m+1}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i}\partial\theta_{\tau(i_{1})}\cdots\partial\theta_{\tau(i_{l+2m})}}\partial_{\sigma(j_{0})}\partial_{\sigma(j_{1})}\cdots\partial_{\sigma(j_{l})}$
$\displaystyle\qquad\qquad\qquad\qquad+\sum_{j>\tau(i_{l+1})}(-1)^{\epsilon_{l+1}}(x_{j}-hx_{j})f_{\tau(i_{l+1}),j}\cdot\Phi_{(\sigma(j_{0}),\tau(i_{l+2})),(\sigma(j_{1},\tau(i_{1})),\cdots,(\sigma(j_{l}),\tau(i_{l}))}^{(\tau(i_{l+3}),\tau(i_{l+4})),\cdots,(\tau(i_{l+2m-1}),\tau(i_{l+2m}))}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\cdot
a_{I}\theta_{j}\frac{\partial^{l+2m+1}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{\tau(i_{l+1})}\partial\theta_{j}\partial\theta_{\tau(i_{l+2})}\partial\theta_{\tau(i_{1})}\cdots\partial\theta_{\tau(i_{l})}\partial\theta_{\tau(i_{l+3})}\cdots\partial\theta_{\tau(i_{l+2m})}}\partial_{\sigma(j_{0})}\partial_{\sigma(j_{1})}\cdots\partial_{\sigma(j_{l})}$
$\displaystyle\qquad\qquad\qquad\qquad+\sum_{k<\tau(i_{l+1})}(-1)^{\epsilon_{l+1}}(x_{k}-hx_{k})f_{k,\tau(i_{l+1})}\cdot\Phi_{(\sigma(j_{0}),\tau(i_{l+2})),(\sigma(j_{1},\tau(i_{1})),\cdots,(\sigma(j_{l}),\tau(i_{l}))}^{(\tau(i_{l+3}),\tau(i_{l+4})),\cdots,(\tau(i_{l+2m-1}),\tau(i_{l+2m}))}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\cdot
a_{I}\theta_{k}\frac{\partial^{l+2m+1}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{k}\partial\theta_{\tau(i_{l+1})}\partial\theta_{\tau(i_{l+2})}\partial\theta_{\tau(i_{1})}\cdots\partial\theta_{\tau(i_{l})}\partial\theta_{\tau(i_{l+3})}\cdots\partial\theta_{\tau(i_{l+2m})}}\partial_{\sigma(j_{0})}\partial_{\sigma(j_{1})}\cdots\partial_{\sigma(j_{l})}$
$\displaystyle\qquad\qquad\qquad\qquad+\sum_{r\neq
j_{0},\cdots,j_{n}}(-1)^{\epsilon_{l+2}+|I|+|I_{h}|+l+2}(y_{r}-x_{r})g_{r,\tau(i_{l+1})}\cdot\Phi_{(\sigma(j_{0}),\tau(i_{l+2})),(\sigma(j_{1},\tau(i_{1})),\cdots,(\sigma(j_{l}),\tau(i_{l}))}^{(\tau(i_{l+3}),\tau(i_{l+4})),\cdots,(\tau(i_{l+2m-1}),\tau(i_{l+2m}))}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\cdot
a_{I}\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{\tau(i_{l+1})}\partial\theta_{\tau(i_{l+2})}\partial\theta_{\tau(i_{1})}\cdots\partial\theta_{\tau(i_{l})}\partial\theta_{\tau(i_{l+3})}\cdots\partial\theta_{\tau(i_{l+2m})}}\Big{(}\frac{\partial\theta_{r}}{\partial\theta_{r}}\Big{)}\partial_{\sigma(j_{0})}\partial_{\sigma(j_{1})}\cdots\partial_{\sigma(j_{l})}\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{\stackrel{{\scriptstyle\sigma\in{\textrm{Perm}}(\\{j_{0},\cdots,j_{l}\\})}}{{\tau\in{\textrm{Perm}}(\\{i_{1},\cdots,i_{l+2m}\\})}}}\Big{[}\sum_{i\neq
i_{1},\cdots,i_{l+2m}}(-1)^{\epsilon_{l+1}}(x_{i}-hx_{i})g_{\sigma(j_{0}),i}\cdot\Phi_{(\sigma(j_{1}),\tau(i_{1})),\cdots,(\sigma(j_{l}),\tau(i_{l}))}^{(\tau(i_{l+1}),\tau(i_{l+2})),\cdots,(\tau(i_{l+2m-1}),\tau(i_{l+2m}))}$
$\displaystyle\qquad\qquad\qquad\qquad+\Big{(}\sum_{r\neq
j_{0},\cdots,j_{l}}(-1)^{\epsilon_{l+3}}(y_{r}-x_{r})g_{r,\tau(i_{l+1})}+\sum_{j\neq
i_{1},\cdots,i_{l+2m}}(-1)^{\epsilon_{l+1}+1}(x_{j}-hx_{j})f_{\tau(i_{l+1}),j}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\sum_{k\neq
i_{1},\cdots,i_{l+2m}}(-1)^{\epsilon_{l+1}}(x_{k}-hx_{k})f_{k,\tau(i_{l+1})}\Big{)}\cdot\Phi_{(\sigma(j_{0}),\tau(i_{2})),(\sigma(j_{1}),\tau(i_{1})),\cdots,(\sigma(j_{l}),\tau(i_{l}))}^{(\tau(i_{l+3}),\tau(i_{l+4})),\cdots,(\tau(i_{l+2m-1}),\tau(i_{l+2m}))}\Big{]}$
$\displaystyle\qquad\qquad\qquad\qquad\cdot
a_{I}\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{\tau(i_{1})}\cdots\partial\theta_{\tau(i_{l+2m})}}\partial_{\sigma(j_{0})}\partial_{\sigma(j_{1})}\cdots\partial_{\sigma(j_{l})}.$

At the end of the computation, we needed to exclude some summands.
Nevertheless, it can be shown that if we add those ”dummy” terms to (3.9),
then the additional dummy terms does not affect the whole computation, because
of the cancellation in pairs among newly added dummy terms (we refer readers
to [Lee22] for the detail). In summary, we have

$\displaystyle\Big{\langle}{D^{\uparrow}}\Big{(}\frac{\eta_{h}^{k+1}}{(k+1)!}\big{(}a_{I}\theta_{I}\theta_{I_{h}}\big{)}\Big{)},\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j_{0}}\partial_{j_{1}}\cdots\partial_{j_{l}}\Big{\rangle}$
(3.10) $\displaystyle=$
$\displaystyle\sum_{\stackrel{{\scriptstyle\sigma\in{\textrm{Perm}}(\\{j_{0},\cdots,j_{l}\\})}}{{\tau\in{\textrm{Perm}}(\\{i_{1},\cdots,i_{l+2m}\\})}}}(-1)^{\epsilon_{l+1}}\Big{[}\sum_{i\geq\sigma(j_{0})}(x_{i}-hx_{i})g_{\sigma(j_{0}),i}\cdot\Phi_{(\sigma(j_{1}),\tau(i_{1})),\cdots,(\sigma(j_{l}),\tau(i_{l}))}^{(\tau(i_{l+1}),\tau(i_{l+2})),\cdots,(\tau(i_{l+2m-1}),\tau(i_{l+2m}))}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\Big{(}-\sum_{r\leq\tau(i_{l+1})}(y_{r}-x_{r})g_{r,\tau(i_{l+1})}-\sum_{j>\tau(i_{l+1})}(x_{j}-hx_{j})f_{\tau(i_{l+1}),j}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\sum_{k<\tau(i_{l+1})}(x_{k}-hx_{k})f_{k,\tau(i_{l+1})}\Big{)}\cdot\Phi_{(\sigma(j_{0}),\tau(i_{l+2})),(\sigma(j_{1}),\tau(i_{1})),\cdots,(\sigma(j_{l}),\tau(i_{l}))}^{(\tau(i_{l+3}),\tau(i_{l+4})),\cdots,(\tau(i_{l+2m-1}),\tau(i_{l+2m}))}\Big{]}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\cdot
a_{I}\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{\tau(i_{1})}\cdots\partial\theta_{\tau(i_{l+2m})}}\partial_{\sigma(j_{0})}\partial_{\sigma(j_{1})}\cdots\partial_{\sigma(j_{l})}.$

######  Step 3.

We show

$\displaystyle\sum_{i:i\geq j}(x_{i}-hx_{i})g_{ji}$
$\displaystyle=\nabla_{j}W-\nabla_{j}W(hx,y)$ (3.11)

and

$\displaystyle\sum_{j:j\leq
i}(y_{j}-x_{j})g_{ji}+\sum_{j:j>i}(x_{j}-hx_{j})f_{ij}-\sum_{k:k<i}(x_{k}-hx_{k})f_{ki}$
$\displaystyle=\nabla_{i}W(hx,y)-\partial_{i}W^{h}.$ (3.12)

(By definition of $W^{h}$, $\partial_{i}W^{h}=0$ if $i\notin I^{h}$.)

If $j\in I_{h}$, then by direct computation we have

$(x_{j}-hx_{j})g_{jj}=-\frac{{\overline{W}}_{j,j}-{\overline{W}}_{j-1,j-1}}{y_{j}-hx_{j}}+\frac{{\overline{W}}_{j,j}-{\overline{W}}_{j-1,j}}{y_{j}-x_{j}}.$

It still holds for $j\in I^{h}$, namely the right hand side is zero, because
in this case

$\frac{{\overline{W}}_{j,j}-{\overline{W}}_{j-1,j-1}}{y_{j}-hx_{j}}=\frac{{\overline{W}}_{j,j}-{\overline{W}}_{j-1,j}}{y_{j}-x_{j}}.$

For $j<i\leq n$, we can show

$(x_{i}-hx_{i})g_{ji}=\frac{({\overline{W}}_{j,i}-{\overline{W}}_{j-1,i})-({\overline{W}}_{j,i-1}-{\overline{W}}_{j-1,i-1})}{y_{j}-x_{j}}$

as follows: for $i\in I_{h}$, it is straightforward by definition of $g_{ji}$.
If $i\in I^{h}$, then the LHS is zero, and the RHS is also zero due to
${\overline{W}}_{j,i}={\overline{W}}_{j,i-1}$ and
${\overline{W}}_{j-1,i}={\overline{W}}_{j-1,i-1}$. Hence

$\sum_{i:i\geq
j}(x_{i}-hx_{i})g_{ji}=-\frac{{\overline{W}}_{j,j}-{\overline{W}}_{j-1,j-1}}{y_{j}-hx_{j}}+\frac{{\overline{W}}_{j,n}-{\overline{W}}_{j-1,n}}{y_{j}-x_{j}}.$

Now it is straightforward from (3.1) that

$\frac{{\overline{W}}_{j,j}-{\overline{W}}_{j-1,j-1}}{y_{j}-hx_{j}}=\nabla_{j}W(hx,y),\quad\frac{{\overline{W}}_{j,n}-{\overline{W}}_{j-1,n}}{y_{j}-x_{j}}=\nabla_{j}W$

and (3.11) is proved.

It remains to show (3.12). First we assume $i\in I_{h}$. By direct computation
we have

$-\sum_{j=1}^{i}(y_{j}-x_{j})g_{ji}=-\nabla_{i}W(hx,y)+\frac{{\overline{W}}_{0,i}-{\overline{W}}_{0,i-1}}{x_{i}-hx_{i}}$

and

$\displaystyle\sum_{k=1}^{i-1}(x_{k}-hx_{k})f_{ki}-\sum_{j=i+1}^{n}(x_{j}-hx_{j})f_{ij}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{i-1}\frac{({\widetilde{W}}_{k,i}-{\widetilde{W}}_{k-1,i})-({\widetilde{W}}_{k,i-1}-{\widetilde{W}}_{k-1,i-1})}{x_{i}-hx_{i}}-\sum_{j=i+1}^{n}\frac{({\widetilde{W}}_{i,j}-{\widetilde{W}}_{i-1,j})-({\widetilde{W}}_{i,j-1}-{\widetilde{W}}_{i-1,j-1})}{x_{i}-hx_{i}}$
$\displaystyle=$
$\displaystyle\frac{({\widetilde{W}}_{i-1,i}-{\widetilde{W}}_{0,i})-({\widetilde{W}}_{i-1,i-1}-{\widetilde{W}}_{0,i-1})+({\widetilde{W}}_{i-1,n}-{\widetilde{W}}_{i-1,i})-({\widetilde{W}}_{i,n}-{\widetilde{W}}_{i,i})}{x_{i}-hx_{i}}$
$\displaystyle=$
$\displaystyle\frac{{\widetilde{W}}_{0,i-1}-{\widetilde{W}}_{0,i}-{\widetilde{W}}_{i-1,i-1}+{\widetilde{W}}_{i-1,n}-{\widetilde{W}}_{i,n}+{\widetilde{W}}_{i,i}}{x_{i}-hx_{i}}$
$\displaystyle=$
$\displaystyle-\frac{{\overline{W}}_{0,i}-{\overline{W}}_{0,i-1}}{x_{i}-hx_{i}}.$
(3.13)

The last identity is due to ${\overline{W}}_{0,i}={\widetilde{W}}_{0,i}$ and
${\widetilde{W}}_{i,i}={\widetilde{W}}_{i,n}$. Therefore, if $i\in I_{h}$,
then

$\sum_{j=1}^{n}(y_{j}-x_{j})g_{ji}+\sum_{j:j>i}(x_{j}-hx_{j})f_{ij}-\sum_{k:k<i}(x_{k}-hx_{k})f_{ki}=\nabla_{i}W(hx,y).$

Next let $i\in I^{h}$. We have

$-\sum_{j=1}^{i}(y_{j}-x_{j})g_{ji}=-\nabla_{i}W(hx,y)+\partial_{i}\big{(}W(x_{1},\cdots,x_{i},hx_{i+1},\cdots,hx_{n})\big{)},$

and

$\displaystyle\sum_{k=1}^{i-1}(x_{k}-hx_{k})f_{ki}-\sum_{j=i+1}^{n}(x_{j}-hx_{j})f_{ij}$
$\displaystyle=$
$\displaystyle\partial_{i}\big{(}W(x_{1}^{h},\cdots,x_{i-1}^{h},x_{i},hx_{i+1},\cdots,hx_{n})\big{)}-\partial_{i}\big{(}W(x_{1},\cdots,x_{i},hx_{i+1},\cdots,hx_{n})\big{)}$
$\displaystyle+\partial_{i}\big{(}W(x_{1}^{h},\cdots,x_{i}^{h},hx_{i+1},\cdots,hx_{n})\big{)}-\partial_{i}\big{(}W(x_{1}^{h},\cdots,x_{i}^{h},x_{i+1},\cdots,x_{n})\big{)}$
$\displaystyle+\partial_{i}\big{(}W(x_{1}^{h},\cdots,x_{n}^{h})\big{)}-\partial_{i}\big{(}W(x_{1}^{h},\cdots,x_{i}^{h},hx_{i+1},\cdots,hx_{n})\big{)}$
$\displaystyle=$
$\displaystyle\partial_{i}W^{h}-\partial_{i}\big{(}W(x_{1},\cdots,x_{i},hx_{i+1},\cdots,hx_{n})\big{)}$

where the last identity is due to
${\widetilde{W}}_{i,i}={\widetilde{W}}_{i,n}$ and $x_{i}=hx_{i}$. We conclude
that for $i\in I^{h}$

$\sum_{j=1}^{n}(y_{j}-x_{j})g_{ji}+\sum_{j:j>i}(x_{j}-hx_{j})f_{ij}-\sum_{k:k<i}(x_{k}-hx_{k})f_{ki}=\nabla_{i}W(hx,y)-\partial_{i}W^{h}$

and we finish the proof of (3.12).

######  Step 4.

We rewrite (3.10) using (3.11) and (3.12).

$\displaystyle\Big{\langle}{D^{\uparrow}}\Big{(}\frac{\eta_{h}^{k+1}}{(k+1)!}\big{(}a_{I}\theta_{I}\theta_{I_{h}}\big{)}\Big{)},\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j_{0}}\partial_{j_{1}}\cdots\partial_{j_{l}}\Big{\rangle}$
$\displaystyle=$
$\displaystyle\sum_{\stackrel{{\scriptstyle\sigma\in{\textrm{Perm}}(\\{j_{0},\cdots,j_{l}\\})}}{{\tau\in{\textrm{Perm}}(\\{i_{1},\cdots,i_{l+2m}\\})}}}(-1)^{\epsilon_{l+1}}\Big{(}(\nabla_{\sigma(j_{0})}W-\nabla_{\sigma(j_{0})}W(hx,y))\cdot\Phi_{(\sigma(j_{1}),\tau(i_{1})),\cdots,(\sigma(j_{l}),\tau(i_{l}))}^{(\tau(i_{l+1}),\tau(i_{l+2})),\cdots,(\tau(i_{l+2m-1}),\tau(i_{l+2m}))}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\big{(}\partial_{\tau(i_{l+1})}W^{h}-\nabla_{\tau(i_{l+1})}W(hx,y)\big{)}\cdot\Phi_{(\sigma(j_{0}),\tau(i_{l+2})),(\sigma(j_{1}),\tau(i_{1})),\cdots,(\sigma(j_{l}),\tau(i_{l}))}^{(\tau(i_{l+3}),\tau(i_{l+4})),\cdots,(\tau(i_{l+2m-1}),\tau(i_{l+2m}))}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\cdot
a_{I}\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{\tau(i_{1})}\cdots\partial\theta_{\tau(i_{l+2m})}}\partial_{\sigma(j_{0})}\partial_{\sigma(j_{1})}\cdots\partial_{\sigma(j_{l})}.$

Combining (3.8),

$\displaystyle\Big{\langle}{D^{\downarrow}}\Big{(}\frac{\eta_{h}^{k}}{k!}(a_{I}\theta_{I}\theta_{I_{h}})\Big{)}+{D^{\uparrow}}\Big{(}\frac{\eta_{h}^{k+1}}{(k+1)!}\big{(}a_{I}\theta_{I}\theta_{I_{h}}\big{)}\Big{)},\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i_{1}}\cdots\partial\theta_{i_{l+2m}}}\partial_{j_{0}}\partial_{j_{1}}\cdots\partial_{j_{l}}\Big{\rangle}$
$\displaystyle=$
$\displaystyle\sum_{\stackrel{{\scriptstyle\sigma\in{\textrm{Perm}}(\\{j_{0},\cdots,j_{l}\\})}}{{\tau\in{\textrm{Perm}}(\\{i_{1},\cdots,i_{l+2m}\\})}}}(-1)^{\epsilon_{l+1}}\partial_{\tau(i_{l+1})}W^{h}\cdot\Phi_{(\sigma(j_{0}),\tau(i_{l+2})),(\sigma(j_{1}),\tau(i_{1})),\cdots,(\sigma(j_{l}),\tau(i_{l}))}^{(\tau(i_{l+3}),\tau(i_{l+4})),\cdots,(\tau(i_{l+2m-1}),\tau(i_{l+2m}))}$
$\displaystyle\qquad\qquad\qquad\qquad\cdot
a_{I}\frac{\partial^{l+2m}(\theta_{I}\theta_{I_{h}})}{\partial\theta_{\tau(i_{1})}\cdots\partial\theta_{\tau(i_{l+2m})}}\partial_{\sigma(j_{0})}\partial_{\sigma(j_{1})}\cdots\partial_{\sigma(j_{l})}.$

Rearranging indices, we have

$\displaystyle{D^{\downarrow}}\Big{(}\frac{\eta_{h}^{k}}{k!}\big{(}\sum_{I}a_{I}\theta_{I}\theta_{I_{h}}\big{)}\Big{)}+{D^{\uparrow}}\Big{(}\frac{\eta_{h}^{k+1}}{(k+1)!}\big{(}\sum_{I}a_{I}\theta_{I}\theta_{I_{h}}\big{)}\Big{)}$
$\displaystyle=$ $\displaystyle\sum_{\stackrel{{\scriptstyle l+m=k}}{{1\leq
i_{\bullet},j_{\bullet},k_{\bullet}\leq
n}}}(-1)^{\epsilon_{l+1}+1}\Phi_{(j_{0},i_{0}),\cdots,(j_{l},i_{l})}^{(k_{1},k_{2}),\cdots,(k_{2m-3},k_{2m-2})}\frac{\partial^{l+m-1}}{\partial\theta_{i_{0}}\cdots\partial\theta_{i_{l}}\partial\theta_{k_{1}}\cdots\partial\theta_{k_{2m-2}}}\Big{(}\sum_{\stackrel{{\scriptstyle|I|=r}}{{i\in
I^{h}}}}\partial_{i}W^{h}\cdot
a_{I}\frac{\partial(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i}}\Big{)}\partial_{j_{0}}\cdots\partial_{j_{l}},$

but since $\sum_{|I|=r}a_{I}\theta_{I}\theta_{I_{h}}$ is a cocycle of the
Koszul complex of $(\partial_{i}W^{h}:i\in I^{h})$, we have

$\sum_{\stackrel{{\scriptstyle|I|=r}}{{i\in I^{h}}}}\partial_{i}W^{h}\cdot
a_{I}\frac{\partial(\theta_{I}\theta_{I_{h}})}{\partial\theta_{i}}=0.$

Therefore,

${D^{\downarrow}}\Big{(}\frac{\eta_{h}^{k}}{k!}\big{(}\sum_{I}a_{I}\theta_{I}\theta_{I_{h}}\big{)}\Big{)}+{D^{\uparrow}}\Big{(}\frac{\eta_{h}^{k+1}}{(k+1)!}\big{(}\sum_{I}a_{I}\theta_{I}\theta_{I_{h}}\big{)}\Big{)}=0$

and our assertion follows. ∎

It remains to show that $\exp(\eta_{h})$ indeed gives a quasi-inverse of
$(-)_{kos}$.

######  Theorem 3.2.

Let $\gamma$ be a cocycle of $K^{\bullet}(\partial_{x_{i}}W^{h}:i\in
I^{h})\cdot\theta_{I_{h}}$. Then we have the following identity of Koszul
cohomology classes:

$\big{[}\big{(}\exp(\eta_{h})(\gamma)\big{)}_{kos}\big{]}=[\gamma].$

###### Proof.

Without loss of generality let $\gamma\in K^{-s}(\partial_{x_{i}}W^{h}:i\in
I^{h})\cdot\theta_{I_{h}}$, so

$\displaystyle\gamma=\sum_{|I|=s}c_{I}\theta_{I}\theta_{I_{h}}.$

Decompose $\eta_{h}=\eta_{h}^{1,1}+\eta_{h}^{2,0}$ where

$\eta_{h}^{1,1}(\theta_{I}\partial_{J})=\sum(-1)^{|I|}g^{h}_{ji}\frac{\partial\theta_{I}}{\partial\theta_{i}}\partial_{j}\partial_{J},\quad\eta_{h}^{2,0}(\theta_{I}\partial_{J})=\sum
f^{h}_{ji}\frac{\partial^{2}\theta_{I}}{\partial\theta_{j}\partial\theta_{i}}\partial_{J}.$

Since the operations $\eta_{h}^{1,1}$ and $\eta_{h}^{2,0}$ commute, we have

$\exp(\eta_{h})(\gamma)=\sum_{n}\sum_{k+l=n}\frac{(\eta_{h}^{1,1})^{k}}{k!}\cdot\frac{(\eta_{h}^{2,0})^{l}}{l!}(\gamma).$
(3.14)

Since $f^{h}_{ji}\neq 0$ only if $i\in I_{h}$ or $j\in I_{h}$, we deduce that

$\big{(}pr\circ\big{(}\exp(\eta_{h})(\gamma)\big{)}\big{)}_{+}=\big{(}pr\circ\big{(}\exp(\eta_{h}^{1,1})(\gamma)\big{)}\big{)}_{+}$

and the right hand side is explicitly

$\big{(}pr\circ\big{(}\exp(\eta_{h}^{1,1})(\gamma)\big{)}\big{)}_{+}=\gamma+\sum_{k\geq
1}\sum_{\stackrel{{\scriptstyle i_{\bullet},j_{\bullet}\in
I,}}{{\\{i_{1},\cdots,i_{k}\\}\cap\\{j_{1},\cdots,j_{k}\\}=\emptyset}}}\pm
g^{h}_{j_{1}i_{1}}\cdots g^{h}_{j_{k}i_{k}}\cdot
c_{I}\frac{\partial^{k}\theta_{I}}{\partial\theta_{i_{1}}\cdots\partial\theta_{i_{k}}}\theta_{I_{h}}\cdot\partial_{j_{1}}\cdots\partial_{j_{k}}.$
(3.15)

(We did not specify signs in the summation because it will not be needed.)

Now we introduce some degree notations. Given a Koszul complex element
$\alpha\in K^{\bullet}(\partial_{x_{i}}W^{h}:i\in I^{h})\cdot\theta_{I_{h}}$,
let us denote

$\alpha=\sum_{m}\alpha^{m}$

where $\alpha^{m}\in K^{m}(\partial_{x_{i}}W^{h}:i\in
I^{h})\cdot\theta_{I_{h}}$.

When $\phi=\exp(\eta_{h})(\gamma)$, we obtain

$\phi_{kos}=\sum_{k\geq 0}\phi_{kos}^{-s+2k}$

where

$\displaystyle\phi_{kos}^{-s+2k}=$
$\displaystyle\Big{(}\sum_{\stackrel{{\scriptstyle i_{\bullet},j_{\bullet}\in
I,}}{{\\{i_{1},\cdots,i_{k}\\}\cap\\{j_{1},\cdots,j_{k}\\}=\emptyset}}}\pm
g^{h}_{j_{1}i_{1}}\cdots g^{h}_{j_{k}i_{k}}\cdot
c_{I}\frac{\partial^{2k}\theta_{I}}{\partial\theta_{i_{1}}\cdots\partial\theta_{i_{k}}\partial_{j_{1}}\cdots\partial_{j_{k}}}\theta_{I_{h}}\Big{)}|_{{\textrm{Fix}}(h)}.$
(3.16)

Clearly $\phi_{kos}^{-s}=\gamma$. Let us show that $\phi_{kos}^{-s+2k}$ is
homologous to zero for $k>0$. Consider the following decomposition

$\phi=\sum_{k\geq 0}\phi^{-s+2k}$

where each component $\phi^{-s+2k}$ corresponds to $\phi_{kos}^{-s+2k}$ by the
quasi-isomorphism $(-)_{kos}$. Two morphisms $\phi^{-s+2k}$ and
$\big{(}\exp(\eta_{h})(\phi_{kos}^{-s+2k})\big{)}^{-s+2k}$ of matrix
factorizations are homotopic, because their corresponding Koszul cocycles are
identical. Consider these morphisms restricted to the submodule $S\subset
S\langle\theta_{1},\cdots,\theta_{n}\rangle$ in the following diagram.

$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{1}}$$\scriptstyle{\phi^{-s+2k}}$$\scriptstyle{(\exp(\eta_{h})(\phi_{kos}^{-s+2k}))^{-s+2k}}$$\scriptstyle{\sum_{i}(y_{i}-x_{i})\theta_{i}}$$\scriptstyle{s_{2}}$$\textstyle{\bigoplus_{i}S\cdot\theta_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{3}}$$\textstyle{\bigoplus\limits_{|J|=s-2k,i\in
J\sqcup I_{h}}S\cdot\theta_{J\sqcup
I_{h}-\\{i\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sum_{i}(y_{i}-hx_{i})\theta_{i}}$$\textstyle{\bigoplus\limits_{|J|=s-2k}S\cdot\theta_{J}\theta_{I_{h}}}$$\textstyle{\bigoplus\limits_{|J^{\prime}|=s-2k+1}S\cdot\theta_{J^{\prime}}\theta_{I_{h}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sum_{i\in
I^{h}}\nabla_{i}W(hx,y)\partial_{i}}$

The restriction of $\phi^{-s+2k}$ to $S$ is $0$ for $k>0$, because any
component of $\phi$ which does not consist of contraction operators
$\partial_{1},\cdots,\partial_{n}$ is differentiated at least once by
$\frac{\partial}{\partial\theta_{i}}$ for some $i\in I_{h}$. On the other
hand, $\big{(}\exp(\eta_{h})(\phi_{kos}^{-s+2k})\big{)}^{-s+2k}$ is restricted
to $\phi_{kos}^{-s+2k}$ which is identified as a module map from $S$. By
homotopy relation we have

$\phi_{kos}^{-s+2k}=\big{(}\sum_{i}(y_{i}-hx_{i})\theta_{i}\big{)}\cdot
s_{1}+\big{(}\sum_{i\in I^{h}}\nabla_{i}W(hx,y)\partial_{i}\big{)}\cdot
s_{2}+(-1)^{|s_{3}|+1}s_{3}\cdot\sum_{i}(y_{i}-x_{i})\theta_{i}.$

Since $\phi_{kos}^{-s+2k}$ depends only on $x_{1}^{h},\cdots,x_{n}^{h}$, we
have

$\phi_{kos}^{-s+2k}=\Big{(}\big{(}\sum_{i}(y_{i}-hx_{i})\theta_{i}\big{)}\cdot
s_{1}+\big{(}\sum_{i\in I^{h}}\nabla_{i}W(hx,y)\partial_{i}\big{)}\cdot
s_{2}+(-1)^{|s_{3}|+1}s_{3}\cdot\sum_{i}(y_{i}-x_{i})\theta_{i}\Big{)}|_{{\textrm{Fix}}(h)}.$

It is evident that
$(y_{i}-hx_{i})\theta_{i}|_{{\textrm{Fix}}(h)}=(y_{i}-x_{i})\theta_{i}|_{{\textrm{Fix}}(h)}=0$,
so

$\phi_{kos}^{-s+2k}=\Big{(}\big{(}\sum_{i\in
I^{h}}\nabla_{i}W(hx,y)\partial_{i}\big{)}\cdot
s_{2}\Big{)}|_{{\textrm{Fix}}(h)}.$

By definition of $\nabla_{i}W$, if $i\in I^{h}$ then

$(\nabla_{i}W)(x_{1}^{h},\cdots,x_{n}^{h},x_{1}^{h},\cdots,x_{n}^{h})=\partial_{x_{i}}W^{h},$

so $\phi_{kos}^{-s+2k}$ is in the image of $\sum_{i\in
I^{h}}\partial_{x_{i}}W^{h}\cdot\partial_{i}$, hence is a coboundary element.
∎

There was a noteworthy fact in the middle of the proof. This will be useful in
explicit examples.

######  Proposition 3.3.

Let $\phi=\sum_{I}a_{I}\theta_{I}+\sum\limits_{\stackrel{{\scriptstyle
J,K}}{{K\neq\emptyset}}}b_{JK}\theta_{J}\partial_{K}$ be a closed morphism in
$hom(\Delta_{1},\Delta_{h})$. Then

$\phi_{kos}=(\sum_{I}a_{I}\theta_{I})_{kos}.$

######  Definition 3.4.

Let $(W,G)$ be an orbifold LG model (with diagonal $G$-action). Then

$\cup_{kos}:K^{\bullet}(W,G)\otimes K^{\bullet}(W,G)\to K^{\bullet}(W,G),$

defined for $\alpha\in K^{\bullet}(W,h)$ and $\beta\in
K^{\bullet}(W,h^{\prime})$ as

$\alpha\cup_{kos}\beta:=\big{(}h^{\prime}_{*}(\exp(\eta_{h})(\alpha))\cdot\exp(\eta_{h^{\prime}})(\beta)\big{)}_{kos},$
(3.17)

induces an associative algebra structure on $H^{\bullet}(K^{\bullet}(W,G))$,
where for $\phi=f(x,y)\theta_{I}\partial_{J}\in hom(\Delta_{1},\Delta_{h})$,

$h^{\prime}_{*}\phi:=f(x,h^{\prime-1}y)\cdot\rho(h^{\prime-1})(\theta_{I}\partial_{J})\in
hom(\Delta_{h^{\prime}},\Delta_{h^{\prime}h}).$ (3.18)

The role of (3.18) is clear: we cannot compose morphisms
$\exp(\eta_{h})(\alpha)$ and $\exp(\eta_{h^{\prime}})(\beta)$ as themselves
due to the discrepancy of domain and codomain, so we need to translate the
latter morphism properly. See [CL20] for more explanation.

We want to emphasize that every step involved in (3.17) only requires simple
calculus of differential operators and can be made as an automated process
(for example in a computer program if needed), though at first glance it looks
not so intrinsic due to the detour to the category of matrix factorizations.
We need to remark is that the product structure on Koszul cohomology of an
orbifold LG model was also defined in [Shk20] via transferring the product
structure on a bar complex for Hochschild cohomology of a curved algebra. The
advance of our work from previous ones is that our product structure
$\cup_{kos}$ is more directly described in nonisolated cases.

##  4\. An example in mirror symmetry

Consider a polynomial
$W=x_{1}^{2}+x_{2}^{2}+x_{1}x_{2}x_{3}\in\mathbb{C}[x_{1},x_{2},x_{3}]$ with
$\mathbb{Z}/2$-action

$\rho\cdot x_{1}=-x_{1},\;\rho\cdot x_{2}=-x_{2},\;\rho\cdot x_{3}=x_{3}$

($\rho$ is the nonidentity element in $\mathbb{Z}/2$). We want to remark that
$W$ is a mirror of the orbifold sphere $\mathbb{P}^{1}_{2,2,\infty}$ which is
the $\mathbb{Z}/2$-quotient of $T^{*}S^{1}$ (see [CCJ22]). Therefore,
$\mathbb{Z}/2$-equivariant theory of $W$ is expected to be deeply related to
the symplectic geometry of $T^{*}S^{1}$. In this section let us compute the
orbifold Koszul algebra structure of $(W,\mathbb{Z}/2)$ and compare it with
the symplectic cohomology of $T^{*}S^{1}$. Recall that
$SH^{*}(T^{*}S^{1})\cong\mathbb{C}[x,x^{-1},\theta]$ where $\theta$ is a
degree 1 element with $\theta^{2}=0$ (see [Pas19] for example).

First, the usual Koszul complex $K^{\bullet}(\partial W)$ is

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(4.1)

where
$d_{W}=(2x_{1}+x_{2}x_{3})\partial_{1}+(2x_{2}+x_{1}x_{3})\partial_{2}+x_{1}x_{2}\partial_{3}$.
To avoid confusion we will distinguish a cocycle $\alpha$ of (4.1) by
$\alpha^{1}$.

Since $W^{\rho}=W(x_{1}^{\rho},x_{2}^{\rho},x_{3}^{\rho})=0$, the
$\rho$-twisted Koszul complex $K^{\bullet}(\partial
W^{\rho})\cdot\theta_{I_{\rho}}$ is

$0\to\mathbb{C}[x_{3}]\cdot\theta_{1}\theta_{2}\theta_{3}\stackrel{{\scriptstyle
0}}{{\longrightarrow}}\mathbb{C}[x_{3}]\cdot\theta_{1}\theta_{2}\to 0.$

Again, we will denote a cocycle $\beta$ of this complex by $\beta^{\rho}$.

Let us compute $\exp(\eta_{\rho})((\theta_{1}\theta_{2})^{\rho})$ as follows.
First,

$\displaystyle{\overline{W}}^{\rho}_{0,0}$ $\displaystyle=W(\rho x_{1},\rho
x_{2},\rho x_{3})=x_{1}^{2}+x_{2}^{2}+x_{1}x_{2}x_{3},$
$\displaystyle{\overline{W}}^{\rho}_{0,1}=W(x_{1},\rho x_{2},\rho
x_{3})=x_{1}^{2}+x_{2}^{2}-x_{1}x_{2}x_{3},$
$\displaystyle{\overline{W}}^{\rho}_{0,2}$ $\displaystyle=W(x_{1},x_{2},\rho
x_{3})=x_{1}^{2}+x_{2}^{2}+x_{1}x_{2}x_{3},$
$\displaystyle{\overline{W}}^{\rho}_{0,3}=W(x_{1},x_{2},x_{3})=x_{1}^{2}+x_{2}^{2}+x_{1}x_{2}x_{3},$
$\displaystyle{\overline{W}}^{\rho}_{1,1}$ $\displaystyle=W(y_{1},\rho
x_{2},\rho x_{3})=y_{1}^{2}+x_{2}^{2}-x_{1}x_{2}x_{3},$
$\displaystyle{\overline{W}}^{\rho}_{1,2}=W(y_{1},x_{2},\rho
x_{3})=y_{1}^{2}+x_{2}^{2}+x_{1}x_{2}x_{3},$
$\displaystyle{\overline{W}}^{\rho}_{1,3}$
$\displaystyle=W(y_{1},x_{2},x_{3})=y_{1}^{2}+x_{2}^{2}+x_{1}x_{2}x_{3},$
$\displaystyle{\overline{W}}^{\rho}_{2,2}=W(y_{1},y_{2},\rho
x_{3})=y_{1}^{2}+y_{2}^{2}+y_{1}y_{2}x_{3},$
$\displaystyle{\overline{W}}^{\rho}_{2,3}$
$\displaystyle=W(y_{1},y_{2},x_{3})=y_{1}^{2}+y_{2}^{2}+y_{1}y_{2}x_{3},$
$\displaystyle{\overline{W}}^{\rho}_{3,3}=W(y_{1},y_{2},y_{3})=y_{1}^{2}+y_{2}^{2}+y_{1}y_{2}y_{3},$

and likewise,

$\displaystyle{\widetilde{W}}^{\rho}_{0,0}$ $\displaystyle=W(\rho x_{1},\rho
x_{2},\rho x_{3})=x_{1}^{2}+x_{2}^{2}+x_{1}x_{2}x_{3},$
$\displaystyle{\widetilde{W}}^{\rho}_{0,1}=W(x_{1},\rho x_{2},\rho
x_{3})=x_{1}^{2}+x_{2}^{2}-x_{1}x_{2}x_{3},$
$\displaystyle{\widetilde{W}}^{\rho}_{0,2}$
$\displaystyle={\widetilde{W}}^{\rho}_{0,3}=x_{1}^{2}+x_{2}^{2}+x_{1}x_{2}x_{3},$
$\displaystyle{\widetilde{W}}^{\rho}_{1,1}={\widetilde{W}}^{\rho}_{1,2}={\widetilde{W}}^{\rho}_{1,3}=x_{2}^{2},$
$\displaystyle{\widetilde{W}}^{\rho}_{2,2}$
$\displaystyle={\widetilde{W}}^{\rho}_{2,3}={\widetilde{W}}^{\rho}_{3,3}=0.$

Actually, we do not need ${\overline{W}}^{\rho}_{j,3}$ and
${\widetilde{W}}^{\rho}_{j,3}$ if we only compute
$\exp(\eta_{\rho})((\theta_{1}\theta_{2})^{\rho})$ (they are needed for
$\exp(\eta_{\rho})((\theta_{1}\theta_{2}\theta_{3})^{\rho})$ whose computation
will be omitted). Plugging ${\overline{W}}^{\rho}_{j,i}$ and
${\widetilde{W}}^{\rho}_{j,i}$ into (3.3) and (3.4), we have

$\displaystyle g^{\rho}_{11}=1,\quad g^{\rho}_{12}=x_{3},\quad
g^{\rho}_{13}=x_{2},\quad g^{\rho}_{22}=1,\quad g^{\rho}_{23}=y_{1},\quad
g^{\rho}_{33}=0,$ $\displaystyle f^{\rho}_{12}=-\frac{x_{3}}{2},\quad
f^{\rho}_{13}=-\frac{x_{2}}{2},\quad f^{\rho}_{23}=0.$

By (3.5) and (3.6) it directly follows that

$\exp(\eta_{\rho})((\theta_{1}\theta_{2})^{\rho})=\theta_{1}\theta_{2}+\theta_{2}\partial_{1}-x_{3}\theta_{1}\partial_{1}-\theta_{1}\partial_{2}+\partial_{1}\partial_{2}+\frac{x_{3}}{2}.$

Consequently

$\displaystyle\rho_{*}\big{(}\exp(\eta_{\rho})((\theta_{1}\theta_{2})^{\rho})\big{)}\cdot\exp(\eta_{\rho})((\theta_{1}\theta_{2})^{\rho})$
$\displaystyle=$
$\displaystyle(\theta_{1}\theta_{2}+\theta_{2}\partial_{1}-x_{3}\theta_{1}\partial_{1}-\theta_{1}\partial_{2}+\partial_{1}\partial_{2}+\frac{x_{3}}{2})\cdot(\theta_{1}\theta_{2}+\theta_{2}\partial_{1}-x_{3}\theta_{1}\partial_{1}-\theta_{1}\partial_{2}+\partial_{1}\partial_{2}+\frac{x_{3}}{2})$
$\displaystyle=$
$\displaystyle\frac{x_{3}^{2}}{4}-1+c_{1}\cdot\partial_{1}+c_{2}\cdot\partial_{2}+c_{12}\cdot\partial_{1}\partial_{2}$

where $c_{1},c_{2},c_{12}\in S\langle\theta_{1},\theta_{2},\theta_{3}\rangle$.
By Proposition 3.3, we have

$\big{(}\rho_{*}\big{(}\exp(\eta_{\rho})((\theta_{1}\theta_{2})^{\rho})\big{)}\cdot\exp(\eta_{\rho})((\theta_{1}\theta_{2})^{\rho})\big{)}_{kos}=\frac{x_{3}^{2}}{4}-1.$
(4.2)

Likewise, we can deduce that

$\big{(}\rho_{*}\big{(}\exp(\eta_{\rho})((\theta_{1}\theta_{2})^{\rho})\big{)}\cdot\exp(\eta_{\rho})((\theta_{1}\theta_{2}\theta_{3})^{\rho})\big{)}_{kos}=\frac{x_{2}}{2}\theta_{1}-\frac{x_{2}x_{3}}{4}\theta_{2}+\big{(}\frac{x_{3}^{2}}{4}-1\big{)}\theta_{3}.$
(4.3)

Above multiplications (4.2) and (4.3) on $\rho$-sector are the most nontrivial
ones. Besides them, commutativity of $\cup_{kos}$ on
$K^{\bullet}(W,\mathbb{Z}/2)^{\mathbb{Z}/2}$ are readily checked. To compare
$K^{\bullet}(W,\mathbb{Z}/2)^{\mathbb{Z}/2}$ with
$\mathbb{C}[x,x^{-1},\theta]$ we need a nonconstant element which is a unit.
Let $\gamma_{+},\gamma_{-}\in K^{\bullet}(W,\mathbb{Z}/2)^{\mathbb{Z}/2}$ be
defined by

$\gamma_{+}:=\big{(}\frac{x_{3}}{2}\big{)}^{1}+(\theta_{1}\theta_{2})^{\rho},\quad\gamma_{-}:=\big{(}\frac{x_{3}}{2}\big{)}^{1}-(\theta_{1}\theta_{2})^{\rho}.$

By commutativity we have $\gamma_{+}\cup_{kos}\gamma_{-}=1.$ This explains a
similarity of $K^{\bullet}(W,\mathbb{Z}/2)^{\mathbb{Z}/2}$ with
$\mathbb{C}[x,x^{-1},\theta]$.

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