Reeb field. Its flow $F_{t}:S^{2n-1}\to S^{2n-1}$ is given in complex
coordinates $z_{j}=x_{j}+{{\bf\it i}}y_{j}$, $j=1,\dots,n$ by

(4.4) $F_{t}(z_{1},\dots,z_{n})=e^{{{\bf\it i}}t}\cdot(z_{1},\dots,z_{n}).$

Since $\dot{F}_{t}({{\bf\it p}})={{\bf\it i}}F_{t}({{\bf\it
p}})=JF_{t}({{\bf\it p}})$, the tangent vector at ${{\bf\it p}}$, coincides
with the value of $R_{\alpha}$ at $F_{t}({{\bf\it p}})$.
2. On ${\mathbb{R}}^{2n+1}$ with coordinates $(x_{1},\dots,x_{n}$, $y_{1},\dots,y_{n},t)$ the standard contact structure is the one with contact form $\alpha:=dt-\sum y_{j}dx_{j}$. Note that $\alpha\wedge(d\alpha)^{n}$ is the volume form on ${\mathbb{R}}^{2n+1}$ and so is indeed non-degenerate. The kernel of $\alpha$ is the field of hyperplanes spanned at $(t,{{\bf\it x}},{{\bf\it y}})$ by the vectors $d/dy_{1},\dots,d/dy_{n}$, $d/dx_{1}+y_{1}\cdot d/dt,\dots,d/dx_{n}+y_{n}\cdot d/dt$. In other words, this is the field of hyperplanes

$(t^{\prime},{{\bf\it x}}^{\prime},{{\bf\it
y}}^{\prime})\mapsto\\{t-(\sum_{j}y^{\prime}_{j})x=0\\}.$

Note that $d\alpha=\sum dx_{j}\wedge dy_{j}$ does not contain $dt$ and so
$d/dt$ is the Reeb vector field. It is everywhere transversal to the field of
hyperplanes.

Since $R_{\alpha}$ is everywhere transversal to the field $\xi$ of
hyperplanes, one has a direct sum splitting
$T_{p}M=(R_{\alpha})_{p}\oplus\xi_{p}$ and this splitting is preserved by the
flow $\phi^{t}$ of the Reeb vector field.

####  4.2.C. Symplectization of a contact manifold

A contact manifold $M$ gives rise to a symplectic manifold, the cylinder

$\mathsf{Cyl}{(M_{\alpha})}=(M\times(-\infty,\infty),\omega),\quad\omega=e^{t}(d\alpha-\alpha\wedge
dt)=d(e^{t}\alpha),$

which is called the _symplectization of $(M,\xi)$_. The Liouville field on it
is the vector field $d/dt$. One also uses the radial coordinate $r=e^{t}$
instead of $t$ and this gives a symplectomorphism with
$(M\times(0,\infty),d(r\alpha))$. The Liouville field becomes $r^{-1}d/dr$ and
the positive end corresponds to $r>1$.

######  Remark 4.9.

The symplectic structure on $\mathsf{Cyl}{(M_{\alpha})}$ is constructed from a
given contact form $\alpha$. As explained in Section 1.3, the contact
structure allows an entire family of contact forms $f\cdot\alpha$ where $f$ is
a positive differentiable function on $M$. However, all of the
symplectizations are symplectomorphic, an explicit symplectomorphism being
iven by

$\phi:\mathsf{Cyl}{(M_{f\alpha})}\to\mathsf{Cyl}{(M_{\alpha})},\quad(x,t)\mapsto(x,t\log
f(x)),$

since $\phi^{*}(e^{t}\alpha)=fe^{t}\alpha=e^{t}(f\alpha)$.

The following relation between contact fields on $(M,\alpha)$ and Hamiltonian
fields on $(M\times(0,\infty),d(r\alpha))$ is very useful for what follows:

######  Lemma 4.10.

Let $H:M\to{\mathbb{R}}$ be a function determining the contact field $X_{H}$
as in Criterion 4.5. Then Hamiltonian field of the function
$\widetilde{H}=r\cdot H$ on $(M\times(0,\infty),d(r\alpha))$ is given by
$X_{\widetilde{H}}(x,r)=X_{H}(x)+d_{Y}H$, where $Y=X_{\alpha}$ is the Reeb
field and $d_{Y}H\in{\mathbb{R}}$ is identified with the tangent vector
$d_{Y}H\cdot d/dr$.

###### Proof.

One calculates

$\displaystyle d(r\alpha)(X_{\widetilde{H}})$ $\displaystyle=$
$\displaystyle(dr\wedge\alpha+rd\alpha)(X_{H}+d_{Y}H)$ $\displaystyle=$
$\displaystyle dr\wedge\iota_{X_{H}}\alpha+r\iota_{X_{H}}(d\alpha)+$
$\displaystyle\hskip 50.00008pt(d_{Y}H)\cdot\alpha+r\cdot 0$
$\displaystyle\overset{\eqref{eqn:CF1},\eqref{eqn:CF2}}{=}$ $\displaystyle
Hdr+rdH-(\iota_{Y}dH)\cdot\alpha+(d_{Y}H)\cdot\alpha$ $\displaystyle=$
$\displaystyle d(rH)=d\widetilde{H}.\qed$

In a similar way one shows:

######  Addition 4.11.

The Hamiltonian field of the function $h(r)$ on
$(M\times{\mathbb{R}}_{+},d(r\alpha))$ induces for all $r\in{\mathbb{R}}_{+}$
the field $h^{\prime}(r)R_{\alpha}$ on $M\times r$. In particular, the
coordinate function $r$ induces the Reeb field on $M\times r$.

####  4.2.D. Liouville fields and contact strucures

Symplectic manifolds equipped with Liouville vector field induce a contact
structure on any smooth hypersurface transverse to the field:

######  Proposition 4.12 ([MS17, Prop. 3.57]).

Let $(N,\omega)$ be a symplectic manifold containing a compact hypersurface
$S$ (i.e. a submanifold of $N$ of codimension $1$). Then there exists a
Liouville field $Y$ in a neighborhood of $S$ which is transverse to $S$ if and
only if there exists a contact form $\alpha$ on $S$ such that
$d\alpha=\omega|_{S}$. If $\omega$ is given, in fact $\alpha=i_{Y}\omega$
defines a contact form on every hypersurface transverse to $Y$.

Such $S$ is called a _hypersurface of contact type_. The Liouville flow
$\psi_{t}$, associated to $Y$ maps $S$ to the _positive (negative) side of
$S$_ for positive (negative) time $t$. The contact structure on $S$ depends on
$Y$. Suppose $Y^{\prime}$ is another Liouville field. Then
$d(\iota_{Y^{\prime}-Y}\omega)=0$. In case $b_{1}(N)=0$, this implies
$\iota_{Y^{\prime}-Y}\omega=dH$ for some Hamiltonian function on $N$. In other
words, $Y^{\prime}-Y=X_{H}$, the Hamiltonian vector field associated to $H$ on
$N$. The converse is also clear. Consequently, the collection of Liouville
fields is convex. But then by Gray’s Stability Theorem 4.4 one concludes:

######  Lemma 4.13.

The contact structures on a hypersurface of contact type resulting from the
various Liouville fields are all contactomorphic.

Suppose that there exists a symplectic manifold $(N,\omega)$ containing a
smooth hypersurface $S$ of contact type such that the negative side is
contained in a compact manifold $W\subset N$ with boundary $\partial W=S$. The
resulting contact manifold $(M,\alpha)$, $\omega|_{S}=d\alpha$ is said to be
_symplectically fillable_ and $W$ is a symplectic filling.

######  Examples 4.14.

1\. Cotangent bundles. If $N=T^{*}U$, $U$ a smooth $n$-dimensional manifold,
the Liouville vector field points outwards of the associated ball-bundle
$W=T^{*}_{\leq r}U$. The complement of $W$ is contactomorphic to the
symplectic cylinder cylinder on $S=\partial W$ and so $S$ is symplectically
fillable with $W$.
2\. Milnor fibers. The fibers of the Milnor fibration of an IHS given by a
hypersurface $\\{f({{\bf\it z}})=0\\}$ in ${\mathbb{C}}^{n+1}$ (with
singularity at the origin) all have a symplectic structure induced by the
Kähler structure on ${\mathbb{C}}^{n+1}$ (see Example 1.10(3)).

According to Example 1.11(3) the link of the singularity which is the common
boundary of these fibers, admits a contact structure with contact form
$\lambda=\frac{1}{2}(\sum_{j=1}^{n+1}x_{j}dy_{j}-y_{j}dx_{j})|_{\mathsf{L}_{f}}$
and an outwards pointing Liouville field. The (closed) Milnor fiber then can
be viewed as symplectic filling of the link $(\mathsf{L}_{f},-\lambda)$. The
link is the boundary $\mathsf{L}_{f}\times\\{0\\}$ of a ”positive cylindrical
end” $\mathsf{L}_{f}\times[0,\infty)$ glued to the Milnor fiber (in
$t$-coordinates).

The above examples are Liouville domains:

$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}Y$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}Y$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}Y$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}Y$$W$$\mathsf{Cyl}_{\geq
0}{(S_{\alpha})}$$\partial W={S_{\alpha}}$ Figure 4.1. A Liouville domain $W$
with its symplectic completion $\widehat{W}$.

######  Definition 4.15.

A _Liouville domain_ is a compact symplectic manifold $(W,\omega)$ with
boundary $S=\partial W$ and Liouville field $Y$ defined in a neighborhood of
$S$ and which points outward of $S$. Then $S=\partial W$ is called a
_symplectically convex_ boundary of $W$.

The flow of the Liouville field gives a suitable neighborhood $U$ of $S$ in
$W$ a cylinder-like structure, say

$G:S\times[-\delta,0]\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\sim\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}U\subset W.$

Hence $(S\times[-\delta,0],G^{*}(e^{t}\cdot d\alpha))$ then becomes a compact
subset of $\mathsf{Cyl}{(W_{\alpha})}$. So using $G$ the domain $W$ can be
glued along $S\times\\{0\\}$ to the positive cylindrical end
$\mathsf{Cyl}_{\geq 0}{(S_{\alpha})}=S_{\alpha}\times[0,\infty)$ (in
$t$-coordinates) which by definition gives $\widehat{W}$, the _symplectic
completion_ of $W$ illustrated in Figure 4.1.

######  Remark 4.16.

By Remark 4.9 two contact forms on $S$ giving the same contact structure on
the cylindrical ends give symplectomorphic cylindrical ends. For each
$\epsilon>0$ the symplectomorphism restricted to $t\geq\epsilon$ extends to a
neighborhood of $S$ in the symplectic filling by replacing $f(x)$ for
$0<t<\epsilon$ by a positive function $g(x,t)$ with $\lim_{t\to 0}g(x,t)=1$
and $\lim_{t\to\epsilon}g(x,t)=f(x)$. In particular $\omega$ does not change
on $W$ under the symplectomorphism.

So the contact form $\alpha$ on $S\times\\{0\\}$ is induced by the Liouvillle
field, but the contact form on $S\times\\{t\\}$, $t\geq\epsilon$ can be
supposed to be equal to $f\cdot\alpha$, $f$ any positive function on $S$ and
so is not necessarily induced by a Liouville field.

On a Liouville domain $(W,\omega)$ the Liouville vector field $Y$ preserves
$\omega$ and points outwards from its boundary $S=\partial W$ while the Reeb
field for $\alpha=\iota_{Y}\omega|_{S}$ is tangent to $S$ but is not contained
in the contact field $\xi$. So one gets a direct sum splitting

(4.5) $T_{p}W=(R_{\alpha})_{p}\oplus\xi_{p}\oplus Y_{p},\quad p\in S=\partial
W.$

Observe that $\omega$ restricts non-degenerately to the span of
$(R_{\alpha})_{p}$ and $Y_{p}$, since
$\omega(Y,R_{\alpha})=\alpha(R_{\alpha})=1$. A periodic flow of $R_{\alpha}$
induces a flow of the contact field $\xi$ which preserves its symplectic
structure. So a trivialization of $TW$ along a closed orbit of the flow
induced by $R_{\alpha}$ preserves $\xi$ induces a curve in the symplectic
group $\psi:[0,T]\to\operatorname{\mathsf{S}p}({2n-2})$ starting at
$I_{2n-2}$. If $\gamma$ is a periodic orbit of period $T$, then $\psi(T)$ is
called the _linearized return map_.

######  Example 4.17 (The standard sphere $S^{2n-1}$).

This is a continuation of the calculations of Example 4.8.1. Observe that the
contact field $\xi$ of $S^{2n-1}$ at the point $e_{1}$ is given by the $2n-2$
tangent vectors $e_{3},\dots,e_{2n}$. Rephrased in terms of the complex basis
$\\{e_{1},e_{3},\dots,e_{2n-1}\\}$, this subspace can be written
$\xi_{e_{1}}={\mathbb{C}}e_{3}+{\mathbb{C}}e_{5}+\cdots+{\mathbb{C}}e_{2n-1}$.
The tangent map (or linearization) of the Reeb flow $F_{t}:{{\bf\it x}}\mapsto
e^{{{\bf\it i}}t}{{\bf\it x}}$ is the linear map
$\psi(t):{\mathbb{C}}^{n}\to{\mathbb{C}}^{n}$ given by multiplication by
$e^{{{\bf\it i}}t}$ and its restriction to the subspace $\xi_{e^{{{\bf\it
i}}t}e_{1}}\subset T_{e^{{{\bf\it i}}t}}S^{2n-1}$ is likewise multiplication
by $e^{{{\bf\it i}}t}$. This is a symplectic matrix as it should, and the
linearized return map is its value at $2\pi$ which is the identity.

###  4.3.  Strongly Milnor fillable links

These notes are mainly devoted to the symplectic and algebraic geometry of
isolated hypersurface singularities and for those the link is the boundary of
the Milnor fiber. For isolated singularities of varieties that cannot be
embedded as hypersurfaces in ${\mathbb{C}}^{N}$ there is an alternative
filling. To explain this, let $X\subset{\mathbb{C}}^{N}$ be an algebraic
subvariety and assume that $(X,x)$ is an isolated normal singularity. If
$S(x,\delta)$ is the euclidean sphere in ${\mathbb{C}}^{N}$ with center $x$
and small enough radius $\delta$, the intersection $L_{x}=X\cap S(x,\delta)$
is diffeomorphic to the link of $x$ in $X$. If $\omega=d\lambda$ is the
standard Kähler form on ${\mathbb{C}}^{N}$ (see Eqn. (1.4)), then $\lambda$
gives $L_{x}$ a contact structure.

The idea is to regard a tubular neighborhood of the exceptional set in a
resolution of singularities of $x$ as a substitute for the Milnor filling. By
Hironaka’s results recalled in Section 2.1, there is a ”good” embedded
resolution

$\sigma:(\widetilde{X},E)\to(X,x)$

for which $E=\cup_{i\in I}E_{i}$ is a hypersurface whose components $E_{i}$
form a normal crossing divisor. Since $\sigma$ is a resolution of the
singularity at $x$, the inverse image under $\sigma$ of the link embeds
diffeomorphically in $\widetilde{X}-E$ and the link (with its contact
structure) admits a special type of filling in $\widetilde{X}$, called a
_strong Milnor filling_ in the following sense:

######  Theorem 4.18.

Let $B(x,\delta)$ be the euclidean ball in ${\mathbb{C}}^{N}$ with center $x$
and radius $\delta$ and set

$W=\sigma^{-1}(X\cap B(x,\delta)).$

Then for small enough $\delta$, its boundary $\partial W$, is diffeomorphic to
the link of $x$ in $X$ and $W$ is a symplectic filling of the link.

###### Outline of the Proof.

Clearly, for $\delta$ small enough, $\partial W$ is contactomorphic to the
link $L_{x}=\mathsf{L}_{X,x}$ with contact structure induced from the standard
contact structure on the $2N-1$-sphere.

A single blow up $p:\widetilde{\mathbb{C}}^{N}\to{\mathbb{C}}^{N}$ in the
point $x$ has a Kähler metric of the form
$\eta:=p^{*}\omega+\epsilon\cdot\tau$, where $\omega=d\lambda$ is the standard
Kähler form on ${\mathbb{C}}^{N}$ (see Eqn. (1.4)) and $\tau$ is a closed
$(1,1)$-form which is strictly positive along the fibers of $E=p^{-1}x\to x$
and zero outside a compact neighborhood of $x$. For a proof see e.g [Voi02,
Prop. 3.24]. Assuming for simplicity that one blow up resolves $X$, then $W$
is a submanifold of $\widetilde{\mathbb{C}}^{N}$ and $\eta$ restricts to a
Kähler form on $W$. The form $\eta$ restricts to $p^{*}\omega$ near the
boundary, i.e., near the link of the singularity, provided $\epsilon$ is small
enough. But since $\omega=d\lambda$ for some $1$-form $\lambda$, on the link
one has $\eta=d\alpha$ where $\alpha=p^{*}\lambda$. This is the contact form
defining the contact structure on the link. The general case is slightly more
complicated. Details are left to the reader. ∎

######  Remark 4.19.

There is an alternative procedure due to McLean as explained in the proof of
[McL16, Lemma 5.25]. This approach is better suited to make a comparison
between minimal discrepancy and symplectic phenomena near the contact
boundary. The alternative construction gives a contact form on the link which
is isotopic to the classical contact structure on the link considered above.
So, by Gray’s stability theorem, Theorem 4.4, the two contact structures are
contactomorphic.

##  Chapter 5 Hamiltonian and Reeb dynamics, symplectic cohomology

### Introduction

The main goal of this chapter is to give a basic idea of symplectic
cohomology. This requires to introduce (in Sections 5.1–5.2) the
Conley–Zehnder index of a periodic orbit of a Hamiltonian flow. After the
definition of symplectic cohomology in Section 5.3 (assuming some deep results
in global analysis), I shall

  * •

extract contact invariants from symplectic cohomology for a certain type of
boundary of a Liouville domain, namely a so-called dynamically convex
boundary.

  * •

give an overview of McLean’s results which relate the algebraic notion of
minimal discrepancy and the symplectic notion of highest minimal index of
periodic orbits of the Reeb flow(=Hamilton flow restricted to the link).
Applied to the link of a cDV singularity, these results will be shown to
determine whether the singularity is canonical or terminal (Theorem 5.19).
Another important result for 3-dimensional singularities is the
characterization of smoothness in terms of contact invariants (cf. Corollary
5.22).

###  5.1. The Maslov index

This index is an an integer associated to a _loop_ in the symplectic group
based at the identity. It is a (based) homotopy invariant. To explain the
definition, let $(V,\omega)$ be an even dimensional real vector space $V$
equipped with a non-degenerate skew form $\omega$. By definition the
symplectic group is given by

$\operatorname{\mathsf{S}p}({V}):=\\{T\in\operatorname{\mathsf{G}L}({V})\mid\omega(Tx,Ty)=\omega(x,y)\text{
for all }x,y\in V\\}.$

A symplectic basis $\\{e_{1},\dots,e_{n},f_{1},\dots,f_{n}\\}$ for $V$ is one
for which $\omega(e_{i},e_{j})=0$, $\omega(f_{i},f_{j})=0$ and
$\omega(e_{i},f_{j})=-\omega(f_{j},e_{i})=\delta_{ij}$, $i,j=1,\dots,n$. In
other words, this is a basis in which $\omega$ is represented by the matrix

$J_{n}=\begin{pmatrix}0_{n}&\text{\rm id}_{n}\\\ -\text{\rm
id}_{n}&0_{n}\end{pmatrix}.$

In such a basis the symplectic transformations are given by the symplectic
matrices $\\{M\in{\mathbb{R}}_{2n\times
2n}\mid{M}^{{\mathsf{T}}}J_{n}M=J_{n}\\}$. These have polar decomposition
given by

$M=PQ,\quad
P=(MM^{T})^{\frac{1}{2}},\,Q=P^{-\frac{1}{2}}M\in\operatorname{\mathsf{O}}({2n})\cap\operatorname{\mathsf{S}p}({2n}).$

Hence $Q$ can be written as $Q=\begin{pmatrix}X&-Y\\\ Y&X\end{pmatrix}$ which
leads to a homomorphism

$\rho:\operatorname{\mathsf{S}p}({2n})\to
S^{1}\subset{\mathbb{C}},\quad\rho(M)=\det(X+{{\bf\it i}}Y).$

So a path $\psi:I=[0,1]\to\operatorname{\mathsf{S}p}({2n})$ projects to a path
$\rho\raise 1.0pt\hbox{{$\scriptstyle\circ$}}\psi$ in the circle. Closed paths
based at $\psi(0)=I_{2n}$ then give closed path on $S^{1}$. Such a loop has a
winding number which defines the _Maslov index_ of the loop. It does not
depend on the chosen symplectic basis and it is invariant under homotopies
preserving the base point. It also is additive under concatenation of loops
which shows that
$\pi_{1}(\operatorname{\mathsf{S}p}({V}),I_{2n})\simeq{\mathbb{Z}}$.

For non-closed paths whose end point is a matrix with no eigenvalues equal to
$1$ the above definition can be modified. The idea is to extend the path in a
careful way so that $\rho$ gives a closed path on the circle. The choice of
path makes use of the the _Maslov cycle_ , the hypersurface
$\Sigma\subset\operatorname{\mathsf{S}p}({2n})$ consisting of matrices with
eigenvalue $1$, i.e.,

$\Sigma=\\{M\in\operatorname{\mathsf{S}p}({2n})\mid\det(M-I)=0\\}.$

To explain the procedure, first note that $\Sigma$ divides
$\operatorname{\mathsf{S}p}({2n})$ in two connected components determined by
the sign of $\det(M-I)$. One chooses matrices, say $K^{\pm}$ in each of these
components with the property that its square under $\rho$ projects to $1\in
S^{1}$. For instance, one may take $K^{+}=\begin{pmatrix}-I_{n}&0_{n}\\\
0_{n}&-I_{n}\end{pmatrix}$, respectively $K^{-}=Q$, $Q=\begin{pmatrix}D&0\\\
0&D^{-1}\end{pmatrix}$ with $D=\text{diag}(a,-1.\dots,-1)$, $a>0,a\not=1$. By
assumption the endpoint $\psi(1)$ of the path belongs to a connected component
of $\operatorname{\mathsf{S}p}({2n})-\Sigma$ and so this point can be
connected within this component to the appropriate point $K^{\pm}$ yielding a
path $\widetilde{\psi}:[0,2]\to\operatorname{\mathsf{S}p}({2n})$. So now
$t\mapsto\rho^{2}(\widetilde{\psi}(t))$ is loop on the circle. It winding
number is the searched for Maslov index for non-closed paths,

(5.1) $\mu(\psi)=\deg(t\mapsto\rho^{2}(\widetilde{\psi}(t)))\in{\mathbb{Z}}.$

Note that this excludes the case where $\psi$ itself is a loop at $I_{2n}$.
The definition of $\mu$ can be extended to all paths in such a way that its
value does not change under homotopies leaving endpoints fixed. This property
together with a few more characterizes the $\mu$-invariant, the obvious one
being the additivity under concatenation. See [RS93] where it is also shown
([RS93, Remarks 4.10 and 5.3–5.5]) that for loops at $I_{2n}$ one gets twice
the original Maslov index and that it coincides with the above defined
$\mu$-index for non-closed paths.

In particular there is freedom to move the path in such a way tha the Maslov
index becomes susceptible to calculation, for example by deforming it to
become non-degenerate in the following sense.

######  Definition 5.1.

A path $\psi:[0,T]\to\operatorname{\mathsf{S}p}({2n})$ starting at $I_{2n}$ is
said to be _non-degenerate_ if

  * •

$\psi$ meets $\Sigma$ transversally;

  * •

at an intersection point the so-called crossing quadratic form (see below) is
non-degenerate.

The _crossing quadratic form_ at an intersection point $\psi(t_{0})\in\Sigma$
is the quadratic form $Q(\psi,t_{0})$ on $(V,\omega)$ which defined by

$Q(\psi,t_{0})(v)=\omega(v,d\psi/dt|_{t_{0}}(v)).$

If the crossing form is non-degenerate it can be diagonalized over
${\mathbb{R}}$ such that diagonal entries are non-zero. Denote its signature
($\\#$ of positive enties - $\\#$ of negative entries) by $\text{sgn}(p)$. It
turns out that the Maslov index of any non-degenerate path
$\psi:[0,1]\to\operatorname{\mathsf{S}p}({2n})$, closed or non-closed is
expressible in terms of these signatures:

(5.2) $\mu(\psi)=\frac{1}{2}\text{sgn }\psi(0)+\frac{1}{2}\text{sgn
}\psi(1)+\sum_{0<t_{*}<1}\text{sgn }\psi(t_{*}).$

This is a half-integer and an integer for paths with end points not on the
Maslov cycle or for loops. See e.g. e [MS17, p. 45–47] or in [RS93] for a
proof. The following example shows that for non-degenerate paths the
$\mu$-index can be calculated in a straightforward fashion. See also Example
5.5.

######  Example 5.2.

Consider the path $\psi(t)=\begin{pmatrix}\cos(2\pi a\cdot t)&\sin(2\pi a\cdot
t)\\\ -\sin(2\pi a\cdot t)&\cos(2\pi a\cdot t)\end{pmatrix}$, $t\in[0,1]$,
$a\in{\mathbb{Q}}^{+}$. This path intersects $\Sigma$ when
$at\in{\mathbb{Z}}$. If $a$ is not an integer this is the case for
$t_{*}=(\lfloor a\rfloor-k)/a$, $k=0,\dots,\lfloor a\rfloor$. Since
$\dot{\psi}(t_{*})=J=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$, one finds
$\omega(v,Jv)=v\cdot v$, a form of index $2$. Hence $\mu(\psi)=1+2\lfloor
a\rfloor$. The path $\psi$ is a loop in case $a$ is an integer, say $a=m$, and
then $\mu(\psi)=1+2(m-1)+1=2m$. So this is twice the original Maslov index.

###  5.2. The Conley–Zehnder index

One applies the preceding construction first of all in the setting of a
symplectic manifold $N$ equipped with a smooth function $H$. The flow of the
Hamiltonian vector field $X_{H}$ induces a path in the tangent bundle of $N$
along any integral curve ${{\bf\it x}}(t)$. The tangent bundle admits a fiber-
wise symplectic structure and since the flow preserves the symplectic
structure, it induces a path of symplectic frames ${{\bf\it F}}(t)$. Assuming
that ${{\bf\it x}}(t)$ is a closed path, say ${{\bf\it x}}(0)={{\bf\it
x}}(1)$, one makes an important

Assumption: There is an orientation preserving trivialization
$\sigma:TN|_{{{\bf\it x}}([0,1])}\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\sim\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}[0,1]\times{\mathbb{R}}^{2n},2n=\dim
N$.

Hence the standard basis for ${\mathbb{R}}^{2n}$ gives the standard frame
${{\bf\it E}}(t)$ along ${{\bf\it x}}(t)$ and one may assume that ${{\bf\it
F}}(0)={{\bf\it E}}(0)$. Then ${{\bf\it F}}(t)=\psi(t){{\bf\it E}}(t)$,
$\psi(t)$ a path of symplectic matrices starting at $I_{2n}$, which is called
the _path in $\operatorname{\mathsf{S}p}({2n})$ induced by the trivialization
$\sigma$_.

The above assumption holds if there is a $2$-disc spanning the closed path
${{\bf\it x}}$ so that ${{\bf\it x}}$ is contractible. This will automatically
be the case if the manifold $N$ is simply connected, for example if $N$ is a
Milnor fiber of some isolated hypersurface singularity.

######  Definition 5.3.

Let ${{\bf\it x}}:[0,1]\to N$ be a smooth closed integral curve of the
Hamiltonian flow associated to $H$ and let
$\psi:[0,1]\to\operatorname{\mathsf{S}p}({2n})$ be the path of symplectic
matrices induced by the trivialization $\sigma$ (which has been assumed to
exist). The _Conley–Zehnder index_
$\operatorname{\mu_{\mathsf{C}Z}}({H,{{\bf\it x}}})$ of ${{\bf\it x}}$ is
equal to the index $\mu(\psi)$ (see (5.1)).

This index does not depend on the chosen trivialization $\sigma$ and it is a
homotopy invariant (for paths leaving begin and endpoints fixed). If the path
$\psi$ is non-degenerate, it is given by (5.2). In case $\psi$ is a loop, this
integer is twice the Maslov index.

######  Examples 5.4.

1. By Theorem 1.3, the tangent bundle of the Milnor fiber is trivializable. Hence in the Conley–Zehnder indices for $1$-periodic orbits of the Hamilton flow are well defined.
2\. Liouville domains. The Reeb flow on the boundary $S$ of a Liouville
domains $W$ and on the slices $M_{r}=\\{r=\text{constant}\\}$ of its
cylindrical end in the completion $\widehat{W}$ can be compared with the
Hamiltonian flow on $W$ for special Hamiltonians which on the cylindrical end
are of the form $h(r)$ and so are constant along $M_{r}$. By Corollary 4.11
the Hamiltonian flow coincides with the Reeb flow (but the ”speed” may be
different). Recalling the splitting (4.5):

$T_{p}W=(R_{\alpha})_{p}\oplus\xi_{p}\oplus Y_{p},\quad p\in S,$

note that the vector subspace $\xi_{p}$ is preserved by the Reeb flow
$R_{\alpha}$. It is a symplectic subspace since $\omega$ restricts non-
degenerately to it. Indeed, it does so on its (symplectic) orthogonal
complement, i.e., the span of $Y$ and $R_{\alpha}$, as observed just after
(4.5). Observe however that, like the Reeb vector field, also the splitting
depends on the chosen contact form $\alpha$.

Under the assumption that $TW|_{S}$ can be trivialized, this implies that one
can define a Conley-Zehnder index for a closed path ${{\bf\it x}}$ of the Reeb
flow as the Maslov index of the associated path in
$\operatorname{\mathsf{S}p}({2n})$ obtained by following the induced flow in
the tangent bundle $TW$ along ${{\bf\it x}}$. This path starts at the identity
and ends at a matrix describing the linearized return map defined in § 4.2.B
in the situation of contact fields.

One can also consider periodic orbits $\gamma$ of the Reeb flow on an
$2n-1$-dimensional contact manifold $(M,\xi)$. The contact field $\xi$ is
preserved by the Reeb flow and if the tangent bundle of $M$ can be trivialized
along $\gamma$, the linearized flow restricted to $\xi$ induces a path in
$\operatorname{\mathsf{S}p}({2n-2})$ and so has a Conley–Zehnder index.

######  Example 5.5 (The standard sphere $S^{2n-1}$ revisited).

Example 4.17 exhibits a Reeb orbit on $S^{2n-1}$ for which the lifted path
ends on the Maslov cycle. The Maslov index of the orbit starting at
$(1,0,\dots,0)$ is equal to $n-1$ since the path $t\mapsto\det(e^{{{\bf\it
i}}t}\cdot I_{n-1})$, $t\in[0,2\pi$], has winding number $(n-1)$. The
Conley–Zehnder index, which is the Maslov index for paths then equals
$2(n-1)$, as observed previously. Note that the path is a non-degenerate loop.

Consider the perturbed contact structure with contact form
$\sum_{j}a_{j}(x_{j}dy_{j}-y_{j}dx_{j})$ on $S^{2n-1}$, where the
$a_{j}\in{\mathbb{Q}}^{+}$ are linearly independent over ${\mathbb{Q}}$. Of
course the contact field is not the same as the standard contact field (this
is only the case if all the $a_{i}$ are the same), but it is homotopic to the
standard contact structure. Its Reeb field in complex coordinates at $\
(p_{1},\dots,p_{n})$ is given by by $(({{\bf\it i}}t/a_{1})\cdot
p_{1},\dots,({{\bf\it i}}t/a_{n})\cdot p_{n})$ with flow $F_{t}(e^{{{\bf\it
i}}t/a_{1}}z_{1},\dots,e^{{{\bf\it i}}t/a_{n}}z_{n})$. Since the periods $2\pi
a_{j}$ are independent over ${\mathbb{Q}}$, the only way to get a periodic
Reeb orbit occurs when all but one coordinate equals zero which gives $n$ of
these through each basis vector $e_{j}$ and with period $2\pi a_{j}$.

The linearized flow is represented in the standard basis by the diagonal
matrix $\psi(t)=(e^{{{\bf\it i}}t/a_{1}},\dots,e^{{{\bf\it i}}t/a_{n}})$. Let
me consider the flow starting at $e_{1}=(1,0,\dots,0)$. As in Example 4.17 one
finds that the restriction to the contact field along this orbit is described
by the path of a complex diagonal matrix $(d_{2}(t),\dots,\dots,d_{n}(t))$,
$d_{k}(t)=e^{{{\bf\it i}}ta_{1}/a_{k}}$, $k=2,\dots,n$, where the time has
been rescaled to be in the interval $[0,1]$. The index of this path can be
calculated using (5.2). The crossings occur when for some $k\geq 2$ one has
$t_{*}\in 2\pi(a_{k}/a_{1}){\mathbb{Z}}$ and also $0\leq t_{*}\leq 2\pi$.
Since $a_{1}/a_{k}$ is not an integer, this is the case if $t_{*}=(\lfloor
a_{1}/a_{k}\rfloor-j)(a_{k}/a_{1})\cdot 2\pi$, $j=0,\dots,\lfloor
a_{1}/a_{k}\rfloor$. As in Example 5.2, one finds that at each of these
crossings (except for $j=0$) the index equals $2$ while for $j=0$ the
contribution equals $(n-1)$. Hence, the Conley–Zehnder index equals
$n-1+2\sum_{k=2}^{n-1}\lfloor a_{1}/a_{k}\rfloor$. Similarly, for the other
closed orbits at $e_{k}$, one finds $\gamma_{k}=n-1+2\sum_{k\not=j}\lfloor
a_{j}/a_{k}\rfloor$, $j=3,\dots,n$. Suppose $a_{1}<a_{2}\cdots<a_{n}$, one has
$\lfloor a_{j}/a_{n}\rfloor=0$ for $j=1,\dots,n-1$ and so $\gamma_{n}=(n-1)$
which is the minimal index for any such flow. Higher minimal indices are only
possible if there is a non-trivial relation with ${\mathbb{Q}}$-coefficients
among the $a_{j}$ and then $2(n-1)$ is the highest possible minimal index,
realized by the standard flow.

###  5.3. Symplectic Cohomology of a Liouville domain $(W,\omega)$

In this section it is assumed that $W$ is a parallellizable Liouville domain,
i.e $TW$ is trivializable.

####  5.3.A. Interlude: Morse (co)homology

Floer (co)homology which underlies the concept of symplectic cohomology is an
extension of Morse homology. Since the definition of Floer homology is quite
involved, the construction of Morse homology helps to understand Floer
homology. As a reference for details I advise the illuminating lecture notes
[Hut02].

The setting of symplectic geometry is now changed: one starts with a
differentiable manifold $M$ equipped with a differentiable function
$f:M\to{\mathbb{R}}$ with isolated critical points, i.e., points where $df=0$.
The Hessian $H_{p}(f)$ at a critical point $p$ is the quadratic form given in
local coordinates $x_{1},\dots,x_{m}$ by the matrix of the second order
partials $\partial^{2}f/\partial x_{i}\partial x_{j}$ at $p$. This is
independent of the choice of coordinates. If $H_{p}(f)$ is non-degenerate, it
is a diagonalizable matrix with, say $h_{+}(p)$ positive and $h_{-}(p)$
negative eigenvalues. In other words, locally at $p$, coordinates
$y_{1},\dots,y_{m}$ can be found such that the function $f$ can be written as

$f(y_{1},\dots,y_{m})=f(p)+y_{1}^{2}+\cdots
y_{i}^{2}-(y_{i+1}^{2}+\dots+y^{2}_{m}),\quad i=h_{+}(p),m-i=h_{-}(p).$

If all the Hessians are non-degenerate, $f$ is called a _Morse function_ , and
$h_{-}(p)$ is the _Morse index_ at $p$. The $i$-th Morse chain group is given
by

$C_{i}^{\rm Morse}M=\bigoplus_{p}{\mathbb{Z}}\cdot p,\quad p\text{ critical
point with }h_{-}(p)=i.$

In order to define the Morse _complex_ relating the Morse chain groups, one
first chooses a metric $g$ making it possible to define the negative gradient
vector field $-\nabla(f)$ and its flow $\Psi_{s}:M\to M$, i.e.
$\Psi_{0}=\text{\rm id}$ and $d\Psi_{s}/ds=-\nabla(f)$. Next, to every
critical point $p$ of a Morse function $f$ one associates two submanifolds,
$N^{\pm}_{p}=\\{q\in M\mid\lim_{s\to\pm\infty}\Psi_{s}(q)=p\\}$, the ascending
and descending submanifolds at $p$. One can show that $N^{\pm}_{p}$ is an
embedded disc of dimension $h_{\pm}(p)$. If for all critical points of the
Morse function the ascending and descending submanifolds at $p$ are
transversal, one calls $(f,g)$ a _Morse–Smale datum_. Given a Morse function
$f$, the pair $(f,g)$ is Morse–Smale for generic metrics $g$. Assuming this,
for a pair $(p,q)$ of critical points, one sets

${\mathcal{A}}(p,q)=N^{-}_{p}\cap N^{+}_{q},$

which, assuming that $h^{-}_{p}>h^{-}_{q}$, is a manifold of dimension
$h^{-}_{p}-h^{-}_{q}$. This manifold contains all the flow lines from $p$ to
$q$, that is, paths $\gamma:{\mathbb{R}}\to M$ with
$\lim_{s\to-\infty}-\nabla(f)(\gamma(s))=p$ and
$\lim_{s\to+\infty}-\nabla(f)(\gamma(s))=q$. Hence there is an induced
${\mathbb{R}}$-action on ${\mathcal{A}}(p,q)$ and one obtains

${\mathcal{M}}(p,q)={\mathcal{A}}(p,q)/{\mathbb{R}},$

a manifold of dimension $h^{-}_{p}-h^{-}_{q}-1$. If $h^{-}_{p}-h^{-}_{q}=1$
this is a $0$-dimensional manifold, so a number of points. For all $p\not=q$,
one can orient the manifold ${\mathcal{M}}(p,q)$ as explained in [Hut02]. In
particular, in this way one can count the number of signed points of
${\mathcal{M}}(p,q)$ which is denoted $\\#{\mathcal{M}}(p,q)$. One then sets

$d:C_{i}^{\rm Morse}M\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\quad\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}C_{i-1}^{\rm Morse}M),\quad
d(p)=\sum_{q}\\#{\mathcal{M}}(p,q)\cdot q,\quad h^{-}_{q}=i-1.$

It is easy to show that $d\raise 1.0pt\hbox{{$\scriptstyle\circ$}}d=0$ and the
homology of this complex by definition is the Morse homology $H_{*}^{\rm
Morse}(M)$. It is independent of all choices. Moreover, the map assigning to a
critical point $p$ of Morse index $h^{-}_{p}=i$ its descending manifold
$N^{-}_{p}$ considered as a singular $i$-simplex can be shown to induce an
isomorphism

(5.3) $H_{i}^{\rm Morse}(M)\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\sim\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}H_{i}(M,{\mathbb{Z}}).$

Dualizing the above complex yields Morse cohomology, which therefore is
isomorphic to singular cohomology.

####  5.3.B. Definition of symplectic cohomology for a Liouville domain

While the Bott complex for $M$ is built on the set of critical points of a
Morse function, Floer homology on a symplectic manifold $(N,\omega)$ is built
on the set ${\mathcal{P}}(H)$ of periodic orbits of the Hamiltonian flow of a
function $H:M\to{\mathbb{R}}$, where instead of the gradient of a metric, one
uses the form $\omega$ to define the Hamiltonian vector field $X$ defined by
$\omega(X,-)=-dH$. Since the critical values of $H$ can be considered as
_constant periodic orbits_ , the procedure that will be outlined below indeed
exhibits Floer cohomology as an extension of Morse cohomology. More precisely,
if $H$ and all of its first and second derivatives are small enough, one can
show [Oan04, §1.2] that $H$ has no periodic obits at all and so in that case
Floer cohomology coincides with Morse cohomology.

I shall exclusively deal with symplectic cohomology for Liouville domains, and
so I shift to the usual notation $W$ instead of $N$. The symplectic form is
then exact near the cylindrical end of $W$, say $\omega=d(r\alpha)$ where
$\alpha$ gives $\partial W$ the structure of a compact contact manifold. To
define symplectic (co)homology on $W$ one makes use of the completion
$\widehat{W}$ of $W$. Moreover, in the definition special Hamiltonian
functions on $\widehat{W}$ are used:

######  Definition 5.6.

An _admissible Hamiltonian_ on $\widehat{W}\ $ is a smooth function
$H:\widehat{W}\to{\mathbb{R}}$ with

  1. (i)

$H$ is a general Morse function on $W$ which is small in the $C^{2}$ norm on
the complement in $W$ of the negative cylindrical end;

  2. (ii)

on the positive cylindrical end one has $H=a\cdot r+b$, $a\in{\mathbb{R}}$
positive, $b\in{\mathbb{R}}$.

In addition, one assumes that $\omega$ comes from a Kähler metric $g$ making
$(H,g)$ Morse–Smale. As just was observed, requirement (i) implies that the
Hamilton flow of such a Morse function only has critical points on the
complement in $W$ of the negative cylindrical end so this takes care of the
cohomology of $W$. The symplectic information comes from the Hamiltonian flow
near the cylindrical end of $W$ which coincides with the Reeb flow for
$\alpha$.

The _Floer cochain group_ in degree $k$ is the free group on all periodic
orbits of index $n-k$:

(5.4) $\mathsf{CF}^{k}(H)=\bigoplus_{{{\bf\it x}}}{\mathbb{Z}}\cdot{{\bf\it
x}},\quad{{\bf\it
x}}\in{\mathcal{P}}(H),\,\operatorname{\mu_{\mathsf{C}Z}}({H,{{\bf\it
x}}})=n-k,\quad 2n=\dim W.$

The definition of the boundary operator of the Floer complex resembles that of
the one of the Morse complex, but it is more involved. For details consult
e.g. [Sei08]; in outline this goes as follows.
1. One starts with an almost complex structure $J$ on $\widehat{W}$ compatible with $\omega$ on $W$ and with $d(r\alpha)$ on the cylindrical end. Each $J$ defines a compatible metric on the tangent spaces, that is, a metric $g_{J}$ for which $g_{J}(X,Y)=\omega(X,Y)$, $X,Y\in T_{x}\widehat{W}$. $x\in\widehat{W}$. This metric is used to form the covariant derivative $\nabla_{J}$.
2. The constructions that follow require finite dimensional moduli spaces of ”tamed” smooth maps from the infinite cylinder ${\mathbb{R}}\times S^{1}$ to $W$ and which converge to the given periodic orbit ${{\bf\it x}}$, respectively to the periodic orbit ${{\bf\it y}}$ when one goes to either end of the cylinder. This only turns out to be possible if $J$ is general enough. In particular one cannot expect such a $J$ to be integrable. More precisely, let ${\mathcal{U}}({{\bf\it x}},{{\bf\it y}})$ be the collection of smooth maps $u:{\mathbb{R}}\times S^{1}\to\widehat{W}$ with the following properties:

  1. (i)

$u(s,t)$ converges to ${{\bf\it x}}(t)$, respectively to ${{\bf\it y}}(t)$ if
$s\to-\infty$, respectively $s\to\infty$;

  2. (ii)

$\displaystyle\frac{\partial s}{\partial u}+J\raise
1.0pt\hbox{{$\scriptstyle\circ$}}\frac{\partial t}{\partial u}=\nabla_{J}H$.

The set ${\mathcal{U}}({{\bf\it x}},{{\bf\it y}})$ admits an ${\mathbb{R}}$
action induced by the action which sends $\lambda\in{\mathbb{R}}$ to
$u(s,t)\mapsto u(s,t+\lambda)$.

######  Theorem 5.7.

Assume that $\operatorname{\mu_{\mathsf{C}Z}}({H,{{\bf\it
x}}})>\operatorname{\mu_{\mathsf{C}Z}}({H,{{\bf\it y}}})$ and $J$ is
sufficiently general. Then the quotient ${\mathcal{M}}({{\bf\it x}},{{\bf\it
y}})={\mathcal{U}}({{\bf\it x}},{{\bf\it y}})/{\mathbb{R}}$ is a (non-empty)
finite dimensional topological manifold which can be compactified to an
oriented smooth manifold with corners. Furthermore,
$\dim{\mathcal{M}}({{\bf\it x}},{{\bf\it
y}})=\operatorname{\mu_{\mathsf{C}Z}}({H,{{\bf\it
x}}})-\operatorname{\mu_{\mathsf{C}Z}}({H,{{\bf\it y}}})-1$. In particular, if
$\operatorname{\mu_{\mathsf{C}Z}}({H,{{\bf\it
x}}})-\operatorname{\mu_{\mathsf{C}Z}}({H,{{\bf\it y}}})-1=0$,
${\mathcal{M}}({{\bf\it x}},{{\bf\it y}})$ is a finite set of points. In that
case, the orientation gives each of the points a sign.

3. This clearly suggests to define the operator $\partial:\mathsf{CF}^{k}(H)\to\mathsf{CF}^{k+1}(H)$ by setting

$\partial{{\bf\it y}}:=\sum_{{{\bf\it m}}\in{\mathcal{M}}({{\bf\it
x}},{{\bf\it y}})}\text{sign}({{\bf\it m}})\cdot{{\bf\it
x}},\quad\operatorname{\mu_{\mathsf{C}Z}}({H,{{\bf\it
x}}})=n-k+1,\,\operatorname{\mu_{\mathsf{C}Z}}({H,{{\bf\it y}}})=n-k.$

One can show that $\partial\raise 1.0pt\hbox{{$\scriptstyle\circ$}}\partial=0$
so that $\mathsf{CF}^{*}(H)$ becomes a cochain complex whose cohomology
groups, the _Floer cohomology groups_ are denoted

$\mathsf{HF}^{k}(\widehat{W},H)=\frac{\mathop{\rm
Ker}\nolimits(\partial:\mathsf{CF}^{k}(H)\to\mathsf{CF}^{k+1}(H))}{\text{Im}(\partial:\mathsf{CF}^{k-1}(H)\to\mathsf{CF}^{k}(H))}.$

4. Finally, to arrive at symplectic cohomology, one orders the set ${\mathcal{H}}$ of admissible Hamiltonians as follows. Recall that every $H\in{\mathcal{H}}$ looks on the cylindrical end like $ar+b$ for some $a>0$. Choose $a$ to be a non-period and denote such a Hamiltonian by $H^{a}$. The order on ${\mathcal{H}}$ is induced by the real number $a$. The group

$\mathsf{SH}^{k}(\widehat{W})^{<a}=\mathsf{HF}^{k}(\widehat{W},H^{a}).$

takes care of periodic orbits having periods $<a$ and one can show that it is
essentially independent of the choice of $H$ as long as $a$ is fixed.

If $a\leq b$ there is a chain map from $\mathsf{CF}_{k}(H^{a})$ to
$\mathsf{CF}_{k}(H^{b})$ defined as follows. Pick ${{\bf\it x}}$, ${{\bf\it
y}}$ $1$-periodic with respect to $H^{a}$, respectively $H^{b}$. Choose a non-
decreasing family of admissible Hamiltonians $\\{H_{s}\\}$,
$s\in{\mathbb{R}}$, which joins $H^{a}$ to $H^{b}$ and a family $J_{s}$ of
almost complex structures. Let ${\mathcal{B}}({{\bf\it x}},{{\bf\it y}})$ be
the space of smooth maps $u:{\mathbb{R}}\times S^{1}\to\widehat{W}$ with the
following properties:

  1. (i)

$u(s,t)$ converges to ${{\bf\it x}}(t)$, respectively to ${{\bf\it y}}(t)$ if
$s\to-\infty$, respectively to $\infty$;

  2. (ii)

$\displaystyle\frac{\partial s}{\partial u}+J_{s}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}\frac{\partial t}{\partial u}=\nabla H_{s}$.

This set has an ${\mathbb{R}}$-action (as before) with finite quotient
${\mathcal{B}}({{\bf\it x}},{{\bf\it y}})/{\mathbb{R}}={\mathcal{N}}({{\bf\it
x}},{{\bf\it y}})$ which allows to define a chain map which on ${{\bf\it
x}}\in{\mathcal{P}}(H^{a})$ of index $n-k$ assigns a chain on periodic orbits
for $H^{b}$ of the same index:

$\phi_{ab}({{\bf\it x}}):=\sum_{{{\bf\it
y}}\in{\mathcal{P}}(H^{b})}\\#{\mathcal{N}}({{\bf\it x}},{{\bf\it
y}})\cdot{{\bf\it y}}.$

In cohomology it induces a homomorphism
$H(\phi_{ab}):\mathsf{HF}^{k}(\widehat{W},H^{a})\to\mathsf{HF}^{k}(\widehat{W},H^{b})$,
the _transfer map_. Passing to the direct limit then defines symplectic
cohomology:

$\mathsf{SH}^{k}(W):=\varinjlim_{a}\mathsf{SH}^{k}(\widehat{W})^{<a}.$

It can be shown that $\mathsf{SH}^{k}(W)$ is independent of all choices and
hence it is a symplectic invariant of $W$. If $b_{1}(W)=0$ is turns out to be
also an invariant of $\widehat{W}$. Note however that a priori non-zero groups
$\mathsf{SH}^{k}(W)$ may occur for all integral values of $k$ because this is
true for the Conley–Zehnder index. Finally, as for ordinary cohomology, one
can define a graded product structure on the direct sum of these groups,
resulting in a ${\mathbb{Z}}$-graded ring.

Summarizing the above discussion, one has:

######  Theorem 5.8.

[Sei14, §2.5], [Vit99]

  1. (1)

The ${\mathbb{Z}}$-graded cohomology, $\mathsf{SH}^{*}(W)$ is a symplectic
invariant of $W$ and, if $b_{1}(W)=0$, also of $\widehat{W}$.

  2. (2)

$\mathsf{SH}^{*}(W)$ has an associative graded product giving it a graded ring
structure. This ring structure is a symplectic invariant.

######  Remark 5.9.

Recall that if $\omega$ is the symplectic form on $W$ and $Y$ is a Liouvlille
field, the contact form on the boundary $S=\partial W$ is given by
$\alpha=\iota_{Y}\omega$. By Remark 4.16 two contact forms on the boundary of
$W$ are related by an isotopy and hence are contactomorphic. So, if needed,
one may choose a particular contact form without altering the symplectic
cohomology of $W$. This will be relevant for the discussion in Section 5.3.C .

######  Examples 5.10.

1. I present here a simplified version of the computation in [Oan04, §3.2] which shows that complex unit balls $B_{n}$ in ${\mathbb{C}}^{n}$ for all $n\geq 1$ have zero symplectic cohomology. The argument exhibits the essential role of the transfer maps.

One considers ${\mathbb{C}}^{n}$ as the symplectic completion of $B_{n}$. For
simplicity, consider the symplectization of the unit sphere
$S^{2n-1}\subset{\mathbb{C}}^{n}$ as the complement of the ball $\|z\|^{2}\leq
1-\delta$, $0<\delta<1$, with coordinate $r=\|z\|^{2}$. Then the Hamiltonian
vector field of the function $H_{a,b}=2ar+b$ is $2\pi aX$, $X$ the Reeb vector
field for the contact structure on the sphere of radius $r$. As in Example
5.4.2, the periodic orbits of the linearized flow are given by the diagonal
complex matrix $\text{diag}(e^{2\pi{{\bf\it i}}at},\dots,e^{2\pi{{\bf\it
i}}at})$ for various values of $a$. Assuming that $k<a<k+1$ so that $\lfloor
a\rfloor=k$, Example 5.2 gives Conley–Zehnder index of this orbit $\gamma_{a}$
as $\operatorname{\mu_{\mathsf{C}Z}}({\gamma_{a}})=n(2k+1)$. So for each
choice of $a\in(k,k+1)$ this gives one generator for
$\mathsf{CF}^{-2kn}(H^{a})$ due to the convention of Eqn. (5.4). For this
choice of $a$ the groups $\mathsf{CF}^{j}(H^{a})$, $j\not=-2kn$ all vanish so
that $\mathsf{HF}^{-2kn}(H^{a})$ is $1$-dimensional. If $b-a>1$ the transfer
maps $\mathsf{HF}^{-2kn}(H^{a})\to\mathsf{HF}^{-2kn}(H^{b})$ obviously do not
preserve the cohomological degrees since the latter depend on the choice of
the parameter $a$. Hence in the limit the resulting symplectic cohomology
groups all vanish.
2. The arguments in [Sei08, §6b] show that for this example the symplectic cohomology is determined by the contact structure of the boundary, namely the symplectic cohomology of a any Liouville domain of (real) dimension $\geq 4$ vanishes if its boundary is contactomorphic to the standard contact sphere.
3. In loc. cit. §3.a, one finds a few more examples, including that of an open Riemann surface of genus $g\geq 1$ with one boundary component. This is shown to have symplectic cohomology for infinitely many degrees: it has $2g$ generators in degrees $0$, one generator in degrees $-1,2$ and in degrees $4kg-2k-1$, $k=1,2,\dots$.

####  5.3.C. Symplectic cohomology as a contact invariant

It is not clear that symplectic cohomology for $W$ leads to a contact
invariant of the boundary $S$, but as explained below, a certain convexity
condition does lead to contact invariants.

Observe first that the periodic orbits in the interior of $W$ give rise to a
subcomplex $\mathsf{CF}^{-}_{*}(H)$ of the so-called _negative symplectic
chains_ which reduces to the Morse chain complex for $H$ on $W$. Indeed, these
orbits are constant, and so precisely give the critical points ${{\bf\it x}}$
of the function $H$ and by § 5.3.A the subcomplex $\mathsf{CF}^{-}_{*}(H)$ is
indeed the Morse complex for $H$ where the Morse index of $H$ at ${{\bf\it
x}}$ equals $n-\operatorname{\mu_{\mathsf{C}Z}}({{{\bf\it x}}})$, $2n=\dim W$.
See also [CFO10, Lemma 2.1] and formula (8) in loc. cit.

And so, comparing with the indexing (5.4) for the Floer complex, taking the
limit, and using the (dual of the) isomorphism (5.3), one obtains the usual
$k$-th singular cohomology group of $W$:

(5.5) $\varinjlim_{H\in{\mathcal{H}}}H^{k}(\mathsf{CF}^{*}_{-}(H))=H^{k}(W).$

So there is an induced homomorphism $\mathsf{SH}^{k}(W)\to H^{k}(W)$ which
measures de difference between Morse cohomology and symplectic cohomology. The
quotient complex

$\mathsf{CF}^{+}_{*}(H)=\mathsf{CF}_{*}(H)/\mathsf{CF}^{-}_{*}(H)$

is called the complex of the _positive symplectic chains_.

After taking the limit for admissable Hamiltonians the resulting cohomology,
the _positive symplectic cohomology_

$\mathsf{SH}^{k}_{+}(W)=\varinjlim_{H\in{\mathcal{H}}}H^{k}(\mathsf{CF}^{*}_{+}(H)),$

is independent of the choice of admissible Hamiltonians, but a priori depends
on the contact form for the contact structure on $S$. For the full symplectic
cohomology this is not the case as remarked before (Remark 5.9). K. Cieliebak
and A. Oancea show in [CO18, §9] that this is also true for positive
symplectic cohomology. Surprisingly, if the contact structure on $\partial W$
satisfies the following convexity condition, it does not depend on the
filling.

######  Definition 5.11.

If the Conley–Zehder index for all period orbits ${{\bf\it x}}$ of the Reeb
flow for _some_ contact form $\alpha$ for the contact structure $\xi$ on
$S=\partial W$ satisfies the inequality

(5.6) $\operatorname{\mu_{\mathsf{C}Z}}({{{\bf\it x}}})+n-3>0,\quad
2n=\dim_{\mathbb{R}}W,$

the Reeb flow is called _dynamically convex_ and the contact manifold
$(S,\xi)$, is called _index positive with respect to $\alpha$_.

The criterion then reads:

######  Criterion 5.12 ([CO18, Thm. 9.17]).

If the contact structure on $S$ is index positive, then
$\mathsf{SH}^{*}_{+}(W)$ is independent of the symplectic filling $W$, that
is, it is a contact invariant of $S$.

From this it follows that the full symplectic cohomology groups in negative
degrees give contact invariants for $S$ since
$\mathsf{SH}^{k}_{+}(W)\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\simeq\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}\mathsf{SH}^{k}(W)$ for
$k<0$. This follows directly from the long exact sequence in cohomology for
the pair $(\mathsf{CF}_{*}(H),\mathsf{CF}^{-}_{*}(H))$ which reads

(5.7) $\cdots\to H^{*-1}(W)\to\mathsf{SH}^{*}_{+}(W)\to\mathsf{SH}^{*}(W)\to
H^{*}(W)\to\cdots,$

using that $W$ has no cohomology in degrees $<0$.

By results obtained by McLean on the contribution of the Reeb orbits on the
link of a cDV-singularity discussed in the next section one can draw more
detailed conclusions in that situation. See Proposition 5.25 in the Section
5.4.

###  5.4. McLean’s results

#### 1

The first of McLean’s result is a topological characterization of numerically
Gorenstein singularities which are not necessariily IHS. Recall from § 3.1
that $(X,x)$ is numerically Gorenstein if and only if for some (and hence all)
resolutions $Y\to X$ with smooth normal crossing exceptional divisors $E_{i}$
and for some set of rational numbers $a_{i}$ the ${\mathbb{Q}}$-divisor
$K_{Y}-\sum_{i}a_{i}E_{i}$ is numerically trivial.

As explained in Section 4.3, the link $\mathsf{L}_{X,x}$ has a suitable
symplectic filling which is a Liouville domain and so the Reeb flow exists
near the link, and its periodic orbits can be investigated. Defining their
Conley–Zehnder indices require these orbits to be contractible which in
general is not the case. However, the condition that $(X,x)$ be numerically
Gorenstein allows to weaken this condition in view of the following result of
McLean:

######  Lemma 5.13 ( [McL16, Lemma 3.3]).

An isolated normal singularity $(X,x)$ is numerically Gorenstein if and only
if $c_{1}(TX|_{\mathsf{L}_{X,x}})\in H^{2}(\mathsf{L}_{X,x})$ is torsion. In
particular this is true for canonical singularities such as cDV-singularities.

Using this, one can modifiy the usual Conley–Zehder index as explained in
[McL16, §4.1]. Suppose that for instance $m\cdot
c_{1}(TX|_{\mathsf{L}_{X,x}})=0$ which is the case for index $m$
singularities. This has the effect that all the Conley–Zehder indices belong
to $\displaystyle\frac{1}{m}{\mathbb{Z}}$. In particular, for an index $1$
singularity, one gets integer invariants, as is the case for isolated cDV-
singularities.

#### 2

McLean’s second result implies that the contact structure on the link
determines whether the singularity is canonical or terminal, as will be
explained after having given the relevant definitions.

First, recall the following algebra-geometric notion which generalizes the one
discussed in Remark 3.5:

######  Definition 5.14.

The _minimal discrepancy $\text{md}(X,x)$ of $(X,x)$_ equals the infimum of
$\min(a_{i})$ taken over all non-trivial _divisorial_ resolutions $Y\to X$ of
$(X,x)$ with center $x$111In particular, $\sigma$ is not the identity. for
which $K_{Y}-\sum_{i}a_{i}E_{i}$ is numerically trivial.222Recall also (Remark
3.5) that the minimal discrepancy is attained for any one resolution of the
singularity.

The surface case has been dealt with in Example 3.2: the only canonical
singularities are the A-D-E singularities (with minimal discrepancy equal to
$0$) and the terminal singularities are the smooth points (with minimal
discrepancy equal to $1$). All other surface singularities have (possibly
fractional) minimal discrepancies $<0$.

As shown in Remark 3.5, smooth points have minimal discrepancy equal to $\dim
X-1$. A well-known conjecture states that, as for surfaces, the converse
holds:

######  Conjecture 5.15 (Shokurov’s conjecture [Sho88, Conjecture 2]).

A normal isolated Gorenstein singularity is smooth if and only if
$\text{md}(X,x)=\dim(X)-1$.

In addition to surfaces this conjecture has also been shown in dimension $3$
(see Markushevich ([Mar96]) for index $1$, and Y. Kawamata (appendix to
[Sho92]) for higher index).

For an isolated threefold singularity, by [Kol92, Ch. 17, Prop. 1.8] 333Kollár
considers also blow-ups in centers that strictly contain the singular locus,
while in the present situation only points are blown up. This explains the
different upper bound., two possibilities occur:

  1. (1)

either all $a_{j}\geq-1$, and then $\text{md}(X,x)\in[-1,2]$ and the infimum
is a minimum;

  2. (2)

alternatively, $\text{md}(X,x)=-\infty$.

For canonical singularities $\text{md}(X,x)\geq 0$ and for terminal
singularities of index $1$ one has $\text{md}(X,x)\geq 1$. As a consequence of
the validity of the smoothness criterion in terms of the minimal discrepancy
in dimension $3$, one has:

######  Proposition 5.16.

Let $(X,x)$ be a $3$-dimensional normal (non-smooth) terminal singularity of
index $1$, then $\text{md}(X,x)=1$.

Next, I shall explain the central symplectic notions. Consider the symplectic
filling $W$ of the link $(S,\xi)=(\mathsf{L}_{X,x},\xi)$ of the isolated
normal singularity $(X,x)$ with the contact structure as described in Section
4.3. The invariant of $(S,\xi)$ considered by McLean is constructed from the
collection of all $1$-periodic orbits ${{\bf\it x}}:[0,1]\to S$ of the Reeb
flow. Recall that the Reeb flow is constructed from the contact form $\alpha$
and different contact forms defining the same contact structure may lead to
different Reeb flows and so the Conley–Zehnder index
$\operatorname{\mu_{\mathsf{C}Z}}({{{\bf\it x}}})$ of an orbit of the Reeb
flow, as given in Definition 5.3, also depends on the chosen contact form
$\alpha$. So a contact invariant of the link of $(X,x)$ has to take all of
these into account, which leads to the following invariant introduced by
McLean, the highest minimal index:

######  Definition 5.17.

Set $\text{i}({{\bf\it x}}):=\operatorname{\mu_{\mathsf{C}Z}}({{{\bf\it
x}}})+(n-3)$. Then the _minimal index_ of the Reeb flow for $\alpha$ is
defined as

$\text{mi}(\mathsf{L}_{X,x},\alpha):=\inf_{{\bf\it x}}\text{i}({{\bf\it x}}).$

Then the _highest minimal index_ is defined as

$\displaystyle\text{hmi}(\mathsf{L}_{X,x},\xi)$
$\displaystyle:=\sup_{\alpha}\text{mi}(\mathsf{L}_{X,x},\alpha),$

where the supremum is taken over all contact forms $\alpha$ with $\mathop{\rm
Ker}\nolimits\alpha=\xi$ (but one needs to preserve the orientation of
$TX|_{\xi}$).

######  Remark 5.18.

(a) If one chooses the form $\alpha$ such that
$\text{mi}(\alpha)=\text{hmi}(\mathsf{L}_{X,x},\xi)$, the definition of the
minimal index and the highest minimal index implies that for all orbits
${{\bf\it x}}$ of the associated Reeb flow the Conley–Zehder index satisfies
the inequality

$\operatorname{\mu_{\mathsf{C}Z}}({{{\bf\it
x}}})+n-3\geq\text{hmi}(\mathsf{L}_{X,x},\xi).$

Consequently, recalling (5.6), $\text{hmi}(\mathsf{L}_{X,x},\xi)>0$ implies
that the contact manifold is index positive with respect to $\alpha$ in the
sense of Definition 5.11.
(b) Orbits for which linearized return map $D$ has eigenvalue $1$ are
sometimes excluded, but in deformation arguments these come up and in order to
make the highest minimal index well behaved, one subtracts the correction term
$\delta=\frac{1}{2}\dim\mathop{\rm Ker}\nolimits(D-\text{\rm id})$ in the
definition of $\text{i}({{\bf\it x}})$. Since $2\delta$ counts the
multiplicity of the eigenvalue $1$ of the linear symplectic automorphism
$\tilde{D}|_{\xi}$ on a vector space of dimension $2n-2$ which is a symplectic
subpace, it has indeed even dimension so that $0\leq\delta\leq n-1$ with
equality $\delta=n-1$ if and only if $D|_{\xi}=\text{\rm id}$.

McLean’s main result [McL16, Thm. 1.1] reads as follows:

######  Theorem 5.19.

Let $(X,x)$ be a normal isolated numerically ${\mathbb{Q}}$-Gorenstein
singularity with $H^{1}(\mathsf{L}_{X,x},{\mathbb{Q}})=0$. Then:

  * •

if $\text{md}(X,x)\geq 0$, then
$\text{hmi}(\mathsf{L}_{X,x},\xi)=2\text{md}(X,x)$;

  * •

if $\text{md}(X,x)<0$, then $\text{hmi}(\mathsf{L}_{X,x},\xi)<0$.

In particular, the contact structure on the link determines whether the
singularity is canonical ($\text{md}(X,x)\geq 0$) or terminal
($\text{md}(X,x)>0$).

In view of Remark 5.18(a) this implies:

######  Corollary 5.20.

Any a normal isolated numerically ${\mathbb{Q}}$-Gorenstein singularity
$(X,x)$ with $H^{1}(\mathsf{L}_{X,x},{\mathbb{Q}})=0$ (e.g. a cDV-singularity)
has an index positive link if and only if it is terminal.

######  Example 5.21 (The standard sphere $S^{2n-1}$ with contact structure
$\xi$).

Example 5.5 makes it plausible that $\text{hmi}(\xi)=2(n-1)$. Since it is the
link of a smooth point in an $n$-dimensional complex algebraic variety $X$ for
which $\text{md}(X,\mathbf{0})=n-1$, Theorem 5.19 shows that indeed
$\text{hmi}(\xi)=2(n-1)$. This seems quite difficult to show directly, since
the Reeb orbits for $f\alpha$ are difficult to control (here $\alpha$ is the
standard contact form and $f:S^{2n-1}\to{\mathbb{R}}$ is an everywhere
positive smooth function).

Example 5.21 together with Theorem 5.19 suggest a conjectural Mumford-type
result (cf. Theorem 2.11). This is true dimension $3$:

######  Corollary 5.22.

If Shokurov’s conjecture holds, then a normal Gorenstein singularity $(X,x)$
is smooth if and only if its link is contactomorphic to the standard sphere
$S^{2n-1}\subset{\mathbb{C}}^{n}$. So this is in particular true for $n=2$ and
$n=3$.

Coming back to surfaces, this gives a symplectic proof of Mumford’s result
stated here as Theorem 2.11:

######  Corollary 5.23 ([McL16, p. 508]).

A normal surface germ $(X,x)$ is smooth if and only if its link is simply-
connected, i.e., homeomorphic to $S^{3}$.

###### Proof.

Suppose that the link is homeomorphic to $S^{3}$. Then (by the now proven
Poincaré conjecture) it is also diffeomorphic to $S^{3}$. Using a
desingularization of $(X,x)$ Theorem 4.18 shows that the link is strongly
Milnor fillable and so by the classification of contact structures on $S^{3}$
as discussed in Example 1.11.(1) the link $S^{3}$ can be assumed to have its
standard contact structure. Then, since Shokurov’s conjecture holds here,
Corollary 5.22 implies that $(X,x)$ is a smooth germ. ∎

#### 3

Finally I shall explain some further consequences of Theorem 5.19 for the
contact geometry of the link of an isolated 3-dimensional terminal singularity
of index $1$. The link of such singularities is index positive, but since by
Proposition 5.16 such a singularity has minimal discrepancy $1$, McLean’s
theorem together with the validity of Shokurov’s conjecture in this case gives
a stronger inequality for the Conley–Zehnder indices:

######  Corollary 5.24.

For a $3$-dimensional isolated terminal singularity of index $1$ (so in
particular for a cDV-singularity) one has $\text{hmi}(\mathsf{L}_{X,x},\xi)=2$
and so $\operatorname{\mu_{\mathsf{C}Z}}({{{\bf\it x}}})\geq 2$ for all
periodic orbits ${{\bf\it x}}$ of the Reeb flow for the contact form which
realizes the highest minimal index.

This result has consequences for the symplectic cohomology of Milnor fibers of
a cDV-singularity:

######  Proposition 5.25.

For a cDV-singularity $(X,x)$ with Milnor number $\mu$, one has

  * •

$\mathsf{SH}^{k}_{+}(\mathsf{F}_{X,x})$ is a contact invariant;

  * •

$\mathsf{SH}^{k}_{+}(\mathsf{F}_{X,x})=\mathsf{SH}^{k}(\mathsf{F}_{X,x})$ for
$k<0$ and $\mathsf{SH}^{k}_{+}(\mathsf{F}_{X,x})=0$ for $k\geq 2$;

  * •

$\mathsf{SH}^{k}(\mathsf{F}_{X,x})=0\text{ for }k=2,k\geq 4$;

  * •

$\rank(\mathsf{SH}^{3}(\mathsf{F}_{X,x}))=\mu$.

###### Proof.

Since the link of a cDV-singularity is index positive, Criterion 5.12 states
that $\mathsf{SH}^{k}_{+}(\mathsf{F}_{X,x})$ is a contact invariant for the
contact structure on the link. By Remark 5.9 one may then assume that the
contact form on the link is the one realizing $\text{hmi}(\mathsf{L}_{X,x})$.
Note that any Reeb orbit with $\operatorname{\mu_{\mathsf{C}Z}}({{{\bf\it
x}}})=3-\ell$ on the link contributes only to positive symplectic cohomology
in degree $\ell$. Corollary 5.24 states that
$\operatorname{\mu_{\mathsf{C}Z}}({{{\bf\it x}}})\geq 2$ for all periodic Reeb
orbits for the chosen contact form for the link and so
$\mathsf{SH}^{k}_{+}(\mathsf{F}_{X,x})=0$ for $k\geq 2$. Now invoke the long
exact sequence (5.7) to deduce that $\mathsf{SH}^{k}(\mathsf{F}_{X,x})=0$ for
$k=2,k\geq 4$, $\mathsf{SH}^{3}(\mathsf{F}_{X,x})\simeq
H^{3}({\mathsf{F}_{X,x}})$, where we use that the Milnor fiber of a three-
dimensional IHS having the homotopy type of a $3$-sphere, only has cohomology
in ranks $0$ and $3$. ∎

##  Chapter 6 Matrix factorizations and Hochschild cohomology

### Introduction

Since direct calculation of symplectic cohomology is often not possible, in
recent years a roundabout way has been proposed which gives a route to
calculate symplectic cohomology for the Milnor fiber of several classes of
invertible matrix singularities. In outline this goes as follows.

  1. (a)

One first applies the classical theory of matrix factorizations. One can
assign many matrix factorizations to a given IHS, but there is one that plays
a predominant role, its Koszul matrix factorization. All matrix factorizations
come with their cohomology. It should be considered as a first rough
invariant. of the IHS.

  2. (b)

The next step is to reinterpret this invariant via Hochschild cohomology. This
is quite intricate and uses the entire category of matrix factorizations of a
given IHS. Indeed, the supplementary structure of dg-category allows to apply
the machinery of categorical Hochschild cohomology.

  3. (c)

For invertible matrix singularities one refines Koszul cohomology in such a
way that the extra symmetry of these singularities is reflected therein. This
step uses equivariant matrix factorizations. For many classes of invertible
matrix singularities the resulting refined Hochschild cohomology has been
calculated.

  4. (d)

The categorical framework allows comparison with other categories associated
to invertible matrix singularities which are rich enough so that Hochschild
cohomology makes sense for them. These categories are relevant because their
Hochschild cohomology is precisely the symplectic cohomology of the Milnor
fiber of that singularity.111Rather, of the ”dual” singularity associated to
the transpose matrix. The associated homotopy categories are conjecturally
equivalent to the homotopy categories of equivariant matrix factorizations
(for any given invertible matrix singularity, or its ”dual”). Since these
conjectures have been shown to hold under easily verifiable conditions, the
refined Hochschild cohomology in these cases – for which there is an explicit
formula – is thus the same as the symplectic cohomology.

In this, admittedly long chapter only steps (a)–(c) will be dealt with.
Sections 6.1 and 6.2 treat step (a). This basically reports on D. Eisenbud’s
original treatment [Eis80] of matrix factorizations. The categorical
reformulation of step (b) is due to T. Dyckerhoff in [Dyc11]. The main ideas
from loc. cit. will be explained in Section 6.4 after the preliminary Section
6.3.

The chapter ends with a brief summary the calculation of Hochschild cohomology
in the equivariant setting, with the necessary details in the case of
invertible matrix singularities. This will be further detailed in Chapter 8.

Step (d) will be briefly explained in a later chapter; see in particular
Section 8.2.

Up to now IHSs have been defined by polynomials
$w\in{\mathbb{C}}[x_{1},\dots,x_{m}]$ vanishing in the origin, but only the
germ of $w$ at the origin is of importance. The maximal ideal
$(x_{1},\dots,x_{m})$ corresponding to the origin thus plays a special role.
In the algebraic setting of a polynomial ring $R=k[x_{1},\dots,x_{m}]$ over
any algebraically closed field $k$ the maximal ideal
${{\mathfrak{m}}}=(x_{1},\dots,x_{m})$ is called the irrelevant ideal. $S=R/w$
is the ring of functions on the IHS given by $\\{w=0\\}$.

Much of what follows can be rephrased in terms of local rings
$(R,{{\mathfrak{m}}})$, where $w\in{{\mathfrak{m}}}$. For instance, the
${{\mathfrak{m}}}$-adic completion of $R=k[x_{1},\dots,x_{m}]$ is the ring
$\widehat{R}:=k[\\![x_{1},\dots,x_{m}]\\!]$ of formal power series in
$x_{1},\dots,x_{m}$. It is a regular local ring in which the irrelevant ideal
now becomes the maximal ideal. If $k={\mathbb{C}}$, and using the usual
(classical) topology), one can work in the local ring
${\mathcal{O}}_{\mathbf{0}}$, of germs at $\mathbf{0}$ of holomorphic
functions on ${\mathbb{C}}^{m}$ in which the irrelevant ideal is the maximal
ideal and $w=0$ then defines an IHS. The ring ${\mathcal{O}}_{\mathbf{0}}$ is
not only local, it is the subring of ${\mathbb{C}}[\\![x_{1},\dots,x_{m}]\\!]$
consisting of convergent power series.

###  6.1. Basics on matrix factorizations

####  6.1.A. Matrix factorizations are related to singularities

Consider an IHS defined by a polynomial $w\in R=k[x_{1},\dots,x_{m}]$ with an
isolated singularity at the origin. Then $0\to R\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\cdot w\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}R\to S:=R/(f)$ is a minimal
free $R$-resolution of $S$ as an $R$-module. Setting $d_{1}=w\cdot\text{\rm
id}_{R}$ and $d_{0}=\text{\rm id}_{R}$, one has $d_{1}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d_{0}=d_{0}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d_{1}=w\cdot\text{\rm id}$. This is an
example of a matrix factorization of $w$:

######  Definition 6.1.

Let $(R,{{\mathfrak{m}}})$ be as above and let $w\in{{\mathfrak{m}}}$. Then

  1. (i)

A _matrix factorization_ of $w$ is a free ${\mathbb{Z}}/2$-graded $R$-module
$X$ of finite rank equipped with an odd degree $R$-linear map $d:X\to X$ such
that $d\raise 1.0pt\hbox{{$\scriptstyle\circ$}}d=w\cdot\text{\rm id}$. In
other words, $X^{0}\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;d_{0}\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}X^{1}$ and
$X^{1}\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;d_{1}\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}X^{0}$ with $d_{1}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d_{0}=d_{1}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d_{0}=w\cdot\text{\rm id}$ (as above).

  2. (ii)

A morphism $\phi:(X,d)\to(Y,d^{\prime})$ of matrix factorizations is a
${\mathbb{Z}}/2$-graded map $\phi$ such that $\phi\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d^{\prime}=d\raise
1.0pt\hbox{{$\scriptstyle\circ$}}\phi$.

Choosing a basis of the free $R$ modules $X^{0}$ and $X^{1}$ makes clear where
the terminology originates from: a matrix factorization of $w\in R$ is
represented by a matrix of rank $2k$, $k=\rank X^{0}=\rank X^{1}$,

$(A,B):=\begin{pmatrix}0&A\\\ B&0\end{pmatrix}\quad A,B\in R^{k\times
k},\,\,\text{ such that }AB=BA=w\,I_{k}.$

######  Example 6.2.

  1. (i)

The _zero matrix factorization_ : $X^{0}=X_{1}=R$, $d_{0}=d_{1}=0$, is the
matrix factorization of $0\in R$. Note that there is no matrix factorization
of $1$ since $1\not\in{{\mathfrak{m}}}$.

  2. (ii)

Consider the double point $xy=0$ in ${\mathbb{C}}^{2}$. Setting
$R={\mathbb{C}}[\\![x,y]\\!]$, $S=R/(xy)$, the double point admits a matrix-
factorization

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0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
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The operation of ”adding a double point” to an IHS at $\mathbf{0}$ given by a
polynomial $w\in{\mathbb{C}}[x_{1},\dots,x_{m}]$ consists of replacing $w$ by
$w+xy\in R={\mathbb{C}}[x_{1},\dots,x_{m},x,y]$. If $(A,B)$ gives a matrix
factorization for $w$, then

$\left(\begin{pmatrix}xI_{k}&A\\\
B&-yI_{k}\end{pmatrix},\begin{pmatrix}yI_{k}&A\\\
B&-xI_{k}\end{pmatrix}\right)=\begin{pmatrix}0&0&xI_{k}&A\\\ 0&0&B&-yI_{k}\\\
yI_{k}&A&0&0\\\ B&-xI_{k}&0&0\end{pmatrix}\in R^{2k,2k}$

gives a matrix factorization of $w+xy$.

####  6.1.B. A geometric incarnation of matrix factorizations

Ultimately one wants to apply this to IHS’s, geometric objects, and indeed, a
geometric flavor can be given to the above construction, if instead of free
$R$-modules of finite rank, one considers vector bundles over affine $n$-space
${\mathbb{A}}^{n}$, or, more precisely, locally free
${\mathcal{O}}_{{\mathbb{A}}^{n}}$-modules. So one replaces
$R={\mathbb{C}}[x_{1},\dots,x_{n}]$ with the sheaf of regular functions on the
space $\mathsf{Spec}(R)={\mathbb{A}}^{n}$.

A matrix factorization for a degree $d$ polynomial $w$ is a pair
$({\mathcal{E}}^{\bullet},d)$ consisting of a ${\mathbb{Z}}/2$-graded vector
bundle ${\mathcal{E}}^{\bullet}={\mathcal{E}}^{0}\oplus{\mathcal{E}}^{1}$ and
two vector bundle morphisms222Degree $d$ polynomials are sections of the line
bundle ${\mathcal{O}}_{{\mathbb{A}}^{n}}(d)$, and as usual, for any vector
bundle ${\mathcal{E}}$, one sets
${\mathcal{E}}(d):={\mathcal{E}}\otimes_{{\mathbb{A}}^{n}}(d)$.
$d_{0}:{\mathcal{E}}^{0}\to{\mathcal{E}}^{1}(d)$ and
$d_{1}:{\mathcal{E}}^{1}\to{\mathcal{E}}^{0}$ such that $d_{0}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d_{1}=w\cdot\text{\rm id}$ and $d_{1}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d_{0}=w\cdot\text{\rm id}\otimes 1$. Clearly,
the preceding incarnation is the special case where the vector bundles are
direct sums of line bundles (these are all isomorphic to
${\mathcal{O}}_{{\mathbb{A}}^{n}}(d)$) since morphisms between such vector
bundles are given by matrices of polynomials.

More generally, replacing in he above definition ${\mathbb{A}}^{n}$ with any
variety (or a scheme) $X$ and $w$ by a section $w$ of a line bundle
${\mathcal{L}}$ on $X$, one obtains the definition of a matrix factorization
of $w$.

####  6.1.C. Relation with maximal Cohen–Macaulay modules

First recall two basic concepts for finitely generated modules $M$ over a
regular local ring $(R,{{\mathfrak{m}}})$. A sequence $(x_{1},\dots,x_{k})$,
$x_{i}\in R$, $i=1,\dots,k$, is called $M$-regular if $x_{i}$ is a nonzero
divisor in $M/(x_{1},\dots,x_{i-1})M$ for all $i=1,\dots,k$. The projective
dimension (abbreviated below as ”pd”) and the depth of $M$ are then defined as
follows:

$\displaystyle\text{pd}_{R}(M)$ $\displaystyle=\text{ length of a minimal
$R$-free resolution of }M,$ $\displaystyle\text{depth}(M)$
$\displaystyle=\text{ maximal length of an $M$-regular sequence}.$

These are related as follows:

######  Theorem (Auslander–Buchsbaum [Eis95, Thm. 19.9]).

If $R$ is a regular local ring, then $\text{\rm pd}_{R}(M)=\dim(R)-\text{\rm
depth}(M)$.

Suppose that the $R$-module $M$ is an $S=R/wR$-module for some
$w\in{{\mathfrak{m}}}$ such that $\text{depth}(M)=\dim(S)=\dim(R)-1$, then the
above theorem shows that $\text{\rm pd}_{R}(M)=1$. The $S$-module $M$ is then
called a _maximal Cohen–Macaulay $S$-module_. By assumption there is a minimal
free $R$-resolution $0\to X^{0}\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;d_{0}\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}X^{1}\to M$ and since $w\cdot
M=0$ one can construct $d_{1}:X^{1}\to X^{0}$ such that $d_{0}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d_{1}=w\cdot\text{\rm id}$:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
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29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{X^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
53.54773pt\raise 12.075pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.78612pt\hbox{$\scriptstyle{d_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
71.37013pt\raise 4.02837pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
71.36942pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{X^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
53.54773pt\raise-12.07498pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.78612pt\hbox{$\scriptstyle{d_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
44.56801pt\raise-3.00064pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
96.20845pt\raise-5.18748pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{p}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 113.23883pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
113.23883pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{M}$}}}}}}}\ignorespaces}}}}\ignorespaces,\quad
d_{0}\raise 1.0pt\hbox{{$\scriptstyle\circ$}}d_{1}=w\cdot\text{\rm id}.$

Then from the injectivity of $d_{0}$ one derives that also $d_{1}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d_{0}=w\cdot\text{\rm id}$. It follows that
$X^{0}$ and $X^{1}$ have the same rank and so $d^{0}$ and $d^{1}$ are
represented by square matrices of the same size. So this shows:

######  Lemma 6.3.

Let $(R,{{\mathfrak{m}}})$ be a regular local ring, $w\in{{\mathfrak{m}}}$ and
$S=R/(w)$, A maximal Cohen–Macaulay module $M$ over $S$ gives a canonical
matrix factorization of $w$ over $R$,
$\textstyle{X^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}}$$\textstyle{X^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{1}}$
with $\mathop{\rm Coker}\nolimits d_{0}=M$.

######  Remark 6.4.

Note that $M$, considered as an $S$-module receives a $2$-periodic free
resolution

$\cdots\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;d_{k+1}\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}\overline{X^{k}}\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;d_{k}\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}\overline{X^{k-1}}\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;d_{k+1}\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}\cdots\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;d_{1}\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}\overline{X^{0}}\to M\to 0,$

where $\overline{X^{k}}=X^{0}\otimes_{R}S$ for $k$ even,
$\overline{X^{k}}=X^{1}\otimes_{R}S$ if $k$ is odd, and the $d_{k}$ are
induced by $d_{0}$ and $d_{1}$. Note that this is a complex since $d_{0}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d_{1}=d_{1}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d_{0}=w\cdot\text{\rm id}$ is zero in $S$.

###  6.2. Koszul matrix factorizations

The basic constructions leading to the relevant matrix factorizations can be
performed over an arbitrary commutative ring $R$ with a unit $1$. The point of
departure is a free $R$-module of rank $k$ with given basis ${{\bf\it
e}}_{1},\dots,{{\bf\it e}}_{k}$ and an $R$-module homomorphism $\phi:N\to R$.
This defines a _Koszul sequence_ after identifying $\phi$ with the
corresponding row-vector ${{{\bf\it f}}}=(f_{1},\dots,f_{k})$:

$N^{\bullet}({{\bf\it f}}):=\\{0\to\Lambda^{k}N\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\delta_{{\bf\it f}}\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}\Lambda^{k-1}N\to\cdots\to\Lambda^{2}N\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\delta_{{\bf\it f}}\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}N\\},$

where $\delta_{{\bf\it f}}$ is the $R$-linear map given on $\Lambda^{j}N$ by

(6.1) $\delta_{{\bf\it f}}(e_{i_{1}}\wedge\cdots\wedge{{\bf\it
e}}_{i_{j}})=\sum_{m=1}^{j}f_{i_{j}}(-1)^{m-1}{{\bf\it
e}}_{i_{1}}\wedge\cdots\widehat{e_{i_{m}}}\cdots\wedge{{\bf\it e}}_{i_{j}}.$

The derivations actually do not depend on the choice of a basis for $N$ and
$\delta_{{\bf\it f}}\raise 1.0pt\hbox{{$\scriptstyle\circ$}}\delta_{{\bf\it
f}}=0$. The indexing is chosen such that the complex starts at degree $-k$ and
ends at degree $0$ with $N$. Then the cohomology groups
$H^{m}(N^{\bullet}({{\bf\it f}}))$ vanish for $m>0$ and $m<-k$. If ${{\bf\it
f}}=f_{1},\dots,f_{k})$ is an $R$-regular sequence, only $H^{0}$ survives and
equals the quotient ring $R/(f_{1},\dots,f_{k})$. In other words:

######  Lemma 6.5.

Suppose that ${{\bf\it f}}=(f_{1},\dots,f_{k})$ is an $R$-regular sequence.
Then the Koszul sequence $N^{\bullet}({{\bf\it f}})$ is a resolution of the
$R$-module $R/(f_{1},\dots,f_{k})$, that is, $H^{0}(N^{\bullet}({{\bf\it
f}}))=R/(f_{1},\dots,f_{k})$ and $H^{i}(N^{\bullet}({{\bf\it f}}))=0$ for
$i\not=0$.

There is also a dual version, using a vector in $N$ which one identifies with
a column-vector ${{{\bf\it g}}}^{{\mathsf{T}}}\in\oplus^{k}R$,

$N^{\bullet}({{{\bf\it
g}}}^{{\mathsf{T}}}):=\\{\Lambda^{k}N\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;d_{{\bf\it g}}\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\leftarrowfill\crcr}}\;}\Lambda^{k-1}N\leftarrow\cdots\leftarrow\wedge^{2}N\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;d_{{\bf\it g}}\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\leftarrowfill\crcr}}\;}N\leftarrow 0\\},$

where one reverses the arrows and defines

$d_{{\bf\it g}}(y)={{\bf\it g}}\wedge y,\quad y\in\wedge^{j}N,\quad
j=0,\ldots,k-1.$

Considering this as a cohomological complex starting at degree $-k$, there is
only cohomology in degrees $-k,\dots,0$. If $(g_{1},\dots,g_{n})$ is an
$R$-regular sequence, then only $H^{0}(N^{\bullet}({{\bf\it
g}}))=R/(g_{1},\dots,g_{k})$ survives. Here one uses the isomorphisms
$\Lambda^{k}N\simeq R$, and $\Lambda^{k-1}N\simeq N$.

Observe also that

(6.2) $\displaystyle d_{{\bf\it g}}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}\delta_{{\bf\it f}}+\delta_{{\bf\it f}}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d_{{\bf\it g}}=({{\bf\it f}}\cdot{{{\bf\it
g}}}^{{\mathsf{T}}})\cdot\text{\rm id}_{R}$

which defines a homotopy operator.

######  Example 6.6.

  1. (1)

Let $R={\mathbb{C}}[x_{1},\dots,x_{m}]$ and let $N=R^{m}$. Assume that
$f_{1},\dots,f_{m}$ is an $R$-regular sequence. Then
$H^{k}(N^{\bullet}({{\bf\it f}}))=0$ if $k\not=0$ and
$H^{0}(N^{\bullet}({{\bf\it f}}))=R/(f_{1},\dots,f_{m})$. In particular, if
$f=0$ has an IHS at $\mathbf{0}$, then the partial derivatives of $f$ form an
$R$-regular sequence and the Jacobian ring reappears as the cohomology of the
associated Koszul complex:

$\mathsf{Jac}_{f}=H^{0}(N^{\bullet}({{\bf\it f}})),\,N=Re_{1}\oplus
Re_{2}\oplus\cdots\oplus Re_{m},\,\phi(e_{j})=\frac{\partial f}{\partial
x_{j}}.$

  2. (2)

(See D. Eisenbud [Eis80, Sect. 7]) Let there be given a commutative regular
local ring $R$ with an $R$-regular sequence ${{\bf\it f}}=(f_{1},\dots,f_{m})$
and let $w$ be an element in the ideal $I=(f_{1},\dots,f_{m})$, say
$w={{\bf\it f}}\cdot{{{\bf\it g}}}^{{\mathsf{T}}}$ where ${{\bf\it
g}}=(\cdots,g_{j},\cdots)$ is row vector in $Re_{1}\oplus\cdots\oplus Re_{m}$.
This free $R$-module receives a ${\mathbb{Z}}/2$-grading by declaring
$|e_{j}|=1$, $j=1,\dots,m$, $|r|=0$ for $r\in R$. This induces a degree on
wedge-products by reducing mod $2$. The _Koszul matrix factorization_ of
$w={{\bf\it f}}\cdot{{{\bf\it g}}}^{{\mathsf{T}}}$ is defined as

(6.3) $\\{{{\bf\it f}},{{\bf\it g}}\\}:=\left(\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{(\Lambda^{\bullet}N)^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
66.23662pt\raise 12.25pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.61111pt\hbox{$\scriptstyle{\delta_{{\bf\it f}}+d_{{{\bf\it
g}}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
88.94989pt\raise 5.02165pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
88.9472pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{(\Lambda^{\bullet}N)^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
66.23662pt\raise-12.24998pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.61111pt\hbox{$\scriptstyle{\delta_{{\bf\it f}}+d_{{{\bf\it
g}}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
64.94461pt\raise-5.02165pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
133.01639pt\raise-5.43054pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{{{\bf\it f}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 148.39441pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
148.39441pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{R/wR}$}}}}}}}\ignorespaces}}}}\ignorespaces\right).$

It is indeed a matrix factorization for $w$ with $\mathop{\rm
Coker}\nolimits{{{\bf\it f}}}=R/(f_{1},\dots,f_{m})$, since $(\delta_{{\bf\it
f}}+d_{{{\bf\it g}}})(\delta_{{\bf\it f}}+d_{{{\bf\it g}}})=d_{{\bf\it
g}}\raise 1.0pt\hbox{{$\scriptstyle\circ$}}\delta_{{\bf\it f}}+\delta_{{\bf\it
f}}\raise 1.0pt\hbox{{$\scriptstyle\circ$}}d_{{\bf\it g}}=w\cdot\text{\rm id}$
by (6.2).

###  6.3. Matrix factorizations form a dg-category

####  6.3.A. Some basic constructions

Fix a commutative ring $R$ with unit $1$. Recall that a complex
$(X^{\bullet},d^{\bullet})$ of $R$-modules is a ${\mathbb{Z}}$-graded
$R$-module equipped with a degree $1$ derivation $d^{\bullet}:X^{\bullet}\to
X^{\bullet+1}$, A homogeneous element $x\in X^{\bullet}$ by definition belongs
to some graded piece $X^{p}$, and $p=\deg x$ is called the degree of $x$.

######  Definition 6.7.

The category $\underline{C}(R)$ of $R$-complexes has for its objects complexes
of $R$-modules. A morphism $f:X^{\bullet}\to Y^{\bullet}$ in
$\underline{C}(R)$ is a degree-preserving $R$-linear map respecting the
differentials.

Note that there are also maps $X^{\bullet}\to Y^{\bullet+d}$ between complexes
of any degree $d\in{\mathbb{Z}}$. These give $\underline{C}(R)$ an enriched
structure, that of a dg-category to be defined below in § 6.3.B. Let me
observe here only that $\underline{C}(R)$ is a so-called tensor category, that
is, one has a tensor product of two complexes defined by

$(X^{\bullet}\otimes Y^{\bullet})^{k}=\bigoplus_{a+b=k}X^{a}\otimes
Y^{b},\quad d_{X^{\bullet}\otimes Yv}(x\otimes y)=d_{X}x\otimes
y+(-1)^{a}x\otimes d_{Y}y,\,x\in X^{a},y\in Y^{b}.$

The tensor product $f\otimes g:X^{\bullet}\otimes
Y^{\bullet}\to(X^{\prime})^{\bullet}\otimes(Y^{\prime})^{\bullet}$ is defined
as

(6.4) $f\otimes g(x\otimes y)=(-1)^{pq}f(x)\otimes g(y),\quad\deg g=p,\,\deg
y=q.$

This sign-rule is known as the _Koszul sign convention_.

####  6.3.B. Introducing dg-categories

A category $\underline{A}$ is a _dg-category (over $R$)_ if

  1. (1)

For all objects $X,Y$ of $\underline{A}$ the set $\underline{A}(X,Y)$ of
morphisms from $X$ to $Y$ is a complex of $R$-modules, i.e.,
$\underline{A}(X,Y)$ is an object in $\underline{C}(R)$. Moreover,
$d(\text{\rm id}_{X})=0$, where $\text{\rm id}_{X}\in\underline{A}^{0}(X,X)$.

  2. (2)

The composition
$\underline{A}(Z,Y)\otimes_{R}\underline{A}(X,Y)\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\mu_{X,Y,Z}\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}\underline{A}(X,Z)$ (tensor
product as graded $R$-modules) is a morphism of complexes of $R$-modules, that
is, $d(f\otimes g)$ maps to $d(f\raise 1.0pt\hbox{{$\scriptstyle\circ$}}g)$.
Moreover, the composition is associative, i.e.,

$\mu_{W,Y,Z}\,\raise 1.0pt\hbox{{$\scriptstyle\circ$}}\,(\text{\rm
id}_{A(Y,Z)}\otimes\mu_{W,X,Y})=\mu_{W,X,Z}\,\raise
1.0pt\hbox{{$\scriptstyle\circ$}}\,(\mu_{X,Y,Z}\otimes\text{\rm
id}_{A(W,Z)}).$

The associated _homotopy category_ $[\underline{A}]$ has the same objects as
$\underline{A}$ but the morphisms are the homotopy classes of morphisms in
$\underline{A}(X,Y)$.

######  Example 6.8.

  1. 1.

A differential graded $R$-algebra $A$ is a ${\mathbb{Z}}$-graded associative
$R$-algebra $A=\oplus_{j\in{\mathbb{Z}}}A^{j}$ with _algebra_ derivations $d$
of degree $1$, that is, $d:A^{j}\to A^{j+1}$, $d\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d=0$ and $d(ab)=da\cdot b+(-1)^{j}a\cdot db$
for all $a\in A^{j},b\in A$ (the _graded Leibniz rule_). This can be made into
a $dg$-category ${\underline{A}}$ with one object, $A$, viewed as an
$R$-module, whose morphism are the elements of the algebra $A$. Its structure
as a differential graded algebra makes ${\underline{A}}$ a dg-category. Note
that for this to be a dg-category, the Leibniz rule is not required.

  2. 2.

Another basic example is obtained by transforming $\underline{C}(R)$ into a
dg-category: The category $\underline{C}_{dg}(R)$ has the same objects and
morphisms as $\underline{C}(R)$. The extra structure comes from observing that
the direct sum $\bigoplus_{d\in{\mathbb{Z}}}\mathop{\rm
Hom}\nolimits^{d}_{R}(X^{\bullet},Y^{\bullet})$ of $R$-homomorphisms
$X^{\bullet}\to Y^{\bullet+d}$ of all degrees receives a derivation from the
derivations on $X^{\bullet}$ and on $Y^{\bullet}$:

$d(f)=d_{Y}\raise 1.0pt\hbox{{$\scriptstyle\circ$}}f-(-1)^{n}f\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d_{X}\in\mathop{\rm
Hom}\nolimits_{R}^{d+1}(X^{\bullet},Y^{\bullet}),\quad\forall f\in\mathop{\rm
Hom}\nolimits^{d}(X^{\bullet},Y^{\bullet}).$

In this way, $\mathop{\rm Hom}\nolimits^{\bullet}(X^{\bullet},Y^{\bullet})$
becomes a complex, the _hom-complex_ associated to
$(X^{\bullet},Y^{\bullet})$. Usually, in a $dg$-category one just focusses
solely on their hom-complexes.

####  6.3.C. The dg-category of matrix factorizations of $w$

Recall that a matrix factorization of $w$ is a free ${\mathbb{Z}}/2$-graded
$R$-module $X$ of finite rank equipped with an odd degree $R$-linear map
$d:X\to X$ such that $d\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d=w\cdot\text{\rm id}$. These are the objects
of $\underline{Mat\\!f}_{R,w}$. If one allows arbitrary free $R$-modules,
these are the objects of $\underline{Mat\\!f}_{R,w}^{\infty}$. As in Example
6.8.(2) there are associated hom-complexes $\mathop{\rm
Hom}\nolimits_{R}^{\bullet}(X,Y)$. Taking into account the
${\mathbb{Z}}/2$-grading, such complexes have essentially two components,
$\mathop{\rm Hom}\nolimits_{R}^{0}(X,Y)$ and $\mathop{\rm
Hom}\nolimits_{R}^{1}(X,Y)$ and two differentials

$\textstyle{\mathop{\rm
Hom}\nolimits_{R}^{0}(X,Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{0}}$$\textstyle{\mathop{\rm
Hom}\nolimits_{R}^{1}(X,Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{1}}$

given by $(d^{0}f)^{m}=d^{Y}\,f^{m}-f^{m+1}\,d^{X}$ and
$(d^{1}g)^{m}=d^{Y}\,g^{m}+g^{m+1}\,d^{X}$. Now $X$ and $Y$ are not complexes,
but note that $d^{1}\raise 1.0pt\hbox{{$\scriptstyle\circ$}}d^{0}f=0$, for
instance,

$\displaystyle(d^{1}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d^{0}f)^{0}=(d^{Y}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d^{Y}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}f^{0}-d^{Y}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}f^{1}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d^{X})+(d^{Y}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}f^{1}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d^{X}-f^{0}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d^{X}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d^{X})=0,$

since $d^{X}\raise 1.0pt\hbox{{$\scriptstyle\circ$}}d^{X}=d^{Y}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d^{Y}=w\cdot\text{\rm id}$. The resulting
hom-complex being ${\mathbb{Z}}/2$-graded is denoted $\mathop{\rm
Hom}\nolimits_{R}^{{\mathbb{Z}}/2}(X,Y)$. In the remainder of these notes,
also the categorical notation $\underline{Mat\\!f}_{R,w}(X,Y)$ is employed,
instead of $\mathop{\rm Hom}\nolimits_{R}^{{\mathbb{Z}}/2}(X,Y)$. Note that in
the homotopy category $[\underline{Mat\\!f}_{R,w}]$ the complex $\mathop{\rm
Hom}\nolimits_{R}^{{\mathbb{Z}}/2}(X,Y)$ gets replaced by
$[X,Y]=H^{0}\mathop{\rm Hom}\nolimits_{R}^{{\mathbb{Z}}/2}(X,Y)$.

######  Example 6.9 (The Koszul matrix factorization revisited).

Consider the Koszul matrix factorization $\\{{{\bf\it f}},{{\bf\it g}}\\}$ of
$w$ given by (6.3) which is associated to an $R$-regular sequence ${{\bf\it
f}}$ of length $m$ and for which $w={{\bf\it f}}\cdot{{{\bf\it
g}}}^{{\mathsf{T}}}$. Place it in the dg-category $\underline{Mat\\!f}_{R,w}$
which consists of one object $\Lambda^{\bullet}N$, where
$N=\oplus_{j=1}^{m}Re_{j}$, and has
$\underline{Mat\\!f}_{R,w}(\Lambda^{\bullet}N,\Lambda^{\bullet}N)=\mathop{\rm
End}\nolimits_{R}(\\{{{\bf\it f}},{{\bf\it g}}\\})$ as its hom-complex . Note
that the latter $R$-algebra itself can also be considered as a
(${\mathbb{Z}}/2$-graded) dg-category with one object and with endomorphism
algebra $\mathop{\rm End}\nolimits_{R}(\\{{{\bf\it f}},{{\bf\it g}}\\})$. As
such it is full subcategory of $\underline{Mat\\!f}_{R,w}$.

####  6.3.D. Tensors and matrix factorizations

The tensor product of two matrix factorizations is defined using signs as for
complexes:

######  Definition 6.10 (_Tensor products of matrix factorizations_).

Let $(X,d^{X})$ be a matrix factorization of $w$ and $(Y,d^{Y})$ a matrix
factorization of $w^{\prime}$ over $R$. Put

$\displaystyle\begin{matrix}Z^{0}=X^{0}\otimes_{R}Y^{0}\oplus
X^{1}\otimes_{R}Y^{1},&\qquad\quad Z^{1}=X^{0}\otimes_{R}Y^{1}\oplus
X^{1}\otimes_{R}Y^{0}\end{matrix}\hskip 70.0001pt$
$\displaystyle\begin{matrix}d_{0}^{Z}(x_{0}\otimes
y_{0})=d_{0}^{X}(x_{0})\otimes y_{0}+x_{0}\otimes
d^{Y}_{0}y_{0},&d_{0}^{Z}(x_{1}\otimes y_{1})=d_{1}^{X}(x_{1})\otimes
y_{1}-x_{1}\otimes d^{Y}_{1}y_{1}\\\ d_{1}^{Z}(x_{0}\otimes
y_{1})=d_{0}^{X}(x_{0})\otimes y_{1}+x_{0}\otimes
d^{Y}_{1}y_{1},&d_{1}^{Z}(x_{1}\otimes y_{0})=d_{1}^{X}(x_{1})\otimes
y_{0}-x_{1}\otimes d^{Y}_{0}y_{0}.\end{matrix}$

Then $(Z,d^{Z})=(X,d^{X})\otimes(Y,d^{Y})$ is a matrix factorization of
$w_{Z}=-w\otimes 1+1\otimes w^{\prime}$. Indeed, the signs are such that
$d_{0}^{Z}d_{1}^{Z}=d_{1}^{Z}d_{0}^{Z}=-w\otimes 1+1\otimes w^{\prime}$.

In particular, the tensor product does not preserve
$\underline{Mat\\!f}_{R,w}$.

Observe that the hom-construction $\mathop{\rm
Hom}\nolimits_{R}^{{\mathbb{Z}}/2}(X,Y)$ applies also in the situation where
$X$ gives a matrix factorization of $w\in R$ and $Y$ gives a matrix
factorization of $w^{\prime}\in R$ and then it is the hom-complex of a matrix
factorization of $-w+w^{\prime}$. In particular, $(Y^{0},Y^{1})=(R,0)$ is a
matrix factorization of $0$ (with zero differentials) and then
$X^{\bullet}:=\mathop{\rm Hom}\nolimits_{R}^{{\mathbb{Z}}/2}(X,Y)$ is a matrix
factorization of $-w$. Explicitly: for $g\in(X^{\bullet})^{1}$ one has
$dg(x)=g(dx)$ and for $f\in(X^{\bullet})^{0}$ one has $df=0$ so that in view
of the signs $d\raise 1.0pt\hbox{{$\scriptstyle\circ$}}dg(x)=-g(d\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d(x))=-wg(x)$. In particular, if $w\not=0$,
the category $\underline{Mat\\!f}_{R,w}$ is not stable under duality.

###  6.4. Matrix factorizations as stabilizions

####  6.4.A. Koszul matrix factorization as a stabilization

As demonstrated in Lemma 6.3, for a regular local ring $(R,{{\mathfrak{m}}})$
and $w\in{{\mathfrak{m}}}$, a maximal Cohen–Macaulay module $M$ over $S=R/w$
gives a matrix factorization of $w$ whose cokernel equals $M$. The category of
such $S$-modules can be “stabilized” if homomorphisms $g,g^{\prime}:M\to
M^{\prime}$ are declared to be identical if $g^{\prime}=t\raise
1.0pt\hbox{{$\scriptstyle\circ$}}g\raise
1.0pt\hbox{{$\scriptstyle\circ$}}t^{\prime}$, where $t\in\mathop{\rm
Hom}\nolimits(M,M)$ and $t^{\prime}\in\mathop{\rm
Hom}\nolimits(M^{\prime},M^{\prime})$ factor over some free $S$-module. This
procedure gives the _stable category_ $\underline{CM}^{\rm stab}(S)$ of
Cohen–Macaulay modules over $S$. It turms out that the above cokernel
assignment functor (on homotopy level)

$\mathop{\rm Coker}\nolimits:[\underline{Mat\\!f}_{R,w}]\to\underline{CM}^{\rm
stab}(R/w),$

is an equivalence of categories. By definition, given a maximal Cohen–Macaulay
module $M$ over $S$, the homotopy class of a corresponding matrix
factorization is called _the stabilization $M^{\rm stab}$ of $M$_.

This generalizes to the situation of Example 6.6.(2):

######  Proposition 6.11 ([Dyc11, Cor.2.7]).

Let $(R,{{\mathfrak{m}}})$ be a regular local ring, $I\subset{{\mathfrak{m}}}$
an ideal generated by a regular sequence ${{\bf\it f}}=(f_{1},\dots,f_{m})$.
Suppose $w\in I$, set $L:=R/I$ and write $w={{\bf\it f}}\cdot{{{\bf\it
g}}}^{{\mathsf{T}}}$ as before.

Put $S=R/w$ (so that $L$ is an $S$-module). Then the stabilization
$L^{\rm{stab}}$ of $L$ (as an $S$-module) is the Koszul matrix factorization
given in Example 6.6.(2), i.e. $L^{\rm{stab}}=\\{{{\bf\it f}},{{\bf\it
g}}\\}\in\underline{Mat\\!f}_{R,w}$

######  Example 6.12.

If the IHS is given by a _weighted homogeneous hypersurface_
$w(x_{1},\dots,x_{m})=0$ of degree $d$, where $x_{i}$ has weight $d_{i}$,
$i=1,\dots,m$, the Euler formula gives $\sum_{j}x_{j}d_{j}w_{x_{j}}=d\cdot w$.
Replacing $x_{j}$ with $x^{\prime}_{j}=d_{j}/d$, this shows that the Koszul
matrix factorization $\\{{{\bf\it x}},\nabla w\\}$ of $w$ represents $k^{\rm
stab}$ as an $R/w$-module while the factorization $\\{\nabla w,{{\bf\it
x}}\\}$ represents the stabilization of the Jacobian algebra
$\mathsf{Jac}_{w}$ as an $(R/w)$-algebra.

####  6.4.B. Technical interlude

The constructions in this subsection, which use the concept of stabilization,
will be used in an essential way in § 6.4.C and § 6.4.D. The main goal is to
replace Hochschild cohomology of the category $\underline{Mat\\!f}_{R,w}$ by
the (classical) Hochschild cohomology of the algebra $\widehat{M_{R,w}}$ which
will be introduced in Corollary 6.18. It requires passing to
$\underline{Mat\\!f}_{R,w}^{\infty}$ where Toën’s results [Toë06] in homotopy
theory can be used. I won’t detail these techniques but only quote the results
that Dyckerhoff obtains using these.

######  Lemma 6.13 ([Dyc11, Lemma 4.2]).

As above, let $(R,{{\mathfrak{m}}})$ be a local ring, $I\subset R$ an ideal
generated by a regular sequence and $w\in I$.

Set $S=R/w$ and let $L$ be an $S$-module whose stabilization $L^{\rm stab}$
belongs to $\underline{Mat\\!f}_{R,w}^{\infty}$. Let $X$ be an object of
$\underline{Mat\\!f}_{R,w}$, and let $A$ be the set of morphisms from $X$ to
itself in the category $\underline{Mat\\!f}_{R,w}^{\infty}$ considered as a
ring under composition and let $A^{o}$ be the opposite ring.

Then composition with $f\in A$ gives
$\underline{Mat\\!f}_{R,w}^{\infty}(X,L^{\rm stab})$ as well as $\mathop{\rm
Hom}\nolimits^{{\mathbb{Z}}/2}(X,L)$ the structure of an $A^{o}$-module. One
has an isomorphism

$[\underline{Mat\\!f}_{R,w}^{\infty}(X,L^{\rm stab})]\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\sim\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}[\mathop{\rm
Hom}\nolimits^{{\mathbb{Z}}/2}(X,L)]$

in the derived category of $A^{o}$-modules.

Now return to the situation of Proposition 6.17. So $L=R/I$, and $I$ is
generated by a regular sequence ${{\bf\it f}}=\\{f_{1},\dots,f_{m}\\}$, and
$L$ is to be considered as an $S=R/(w)$-module. Recall that the Koszul
resolution of $L$ associated to ${{\bf\it f}}$ reads as follows:

$0\to\Lambda^{k}N\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\delta_{{\bf\it f}}\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}\Lambda^{k-1}N\to\cdots\to\Lambda^{2}N\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\delta_{{\bf\it f}}\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}N\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\phi\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}L=R/I\to 0,$

where $\delta_{{\bf\it f}}$ is defined by Eqn. 6.1 and $\phi$ is the natural
map onto $\mathop{\rm Coker}\nolimits\delta_{{\bf\it f}}$. Applying the
preceding lemma, one finds:

######  Corollary 6.14 ([Dyc11, Prop. 4.3]).

For a free $R$-module $N$ of rank $m$, let

$h:\Lambda^{\bullet}N\to\Lambda^{0}N=R\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\phi\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}R/I=L$

the composition of the projection and the map $\phi$ resulting from the Koszul
resolution of $L$. Then for all objects $X$ in $\underline{Mat\\!f}_{R,w}$ and
$f\in\mathop{\rm Hom}\nolimits(X,L^{\rm stab})$ the map $f\mapsto h\raise
1.0pt\hbox{{$\scriptstyle\circ$}}f$ establishes a quasi-isomorphism

$\underline{Mat\\!f}_{R,w}^{\infty}(X,L^{\rm
stab})\xrightarrow[\simeq]{\rm{qiso}}\mathop{\rm
Hom}\nolimits^{{\mathbb{Z}}/2}_{R}(X,L),$

where

$L^{\rm stab}=(\Lambda^{\bullet}N,\delta_{{\bf\it f}}+d_{{{\bf\it g}}})$

is the Koszul matrix factorisation of $w\in I$ (cf. Eqn. (6.3)).

####  6.4.C. Compact generators

In this section $R=k[x_{1},\dots,x_{m}]$ but later, for technical reasons, it
will be replaced by its completion
$\widetilde{R}=k[\\![x_{1},\dots,x_{m}]\\!]$. Since matrix-factorizations take
place in $R$, this is always possible. The goal of this section is to find
$R$-algebras which as dg-algebra are homotopically the same as the two
homotopy categories $[\underline{Mat\\!f}_{R,w}]$ and
$[\underline{Mat\\!f}_{R,w}^{\infty}]$ and that $w\in R$ has an IHS at the
origin. Assuming that $w={{\bf\it g}}\cdot{{\bf\it x}}$, ${{\bf\it
g}}=(g_{1},\dots,g_{m})$, ${{\bf\it x}}=(x_{1},\dots,x_{m})$, the two main
players are

(6.5) $\displaystyle E$ $\displaystyle=$ $\displaystyle k^{\rm
stab}=\\{{{\bf\it g}},{{\bf\it x}}\\}$ (6.6) $\displaystyle M_{R,w}$
$\displaystyle=$ $\displaystyle\underline{Mat\\!f}_{R,w}(E,E).$

The $R$-algebra $M_{R,w}$ will serve as the building block for constructing
the desired $R$-algebra.

The entire construction depends on a crucial feature of these categories,
namely that they are _triangulated_. See Appendix A.3.14 of [Eis95] for more
details on this concept. Subcategories of a triangulated category stable under
shifts, triangles, isomorphisms and direct sums (coproducts) are called _thick
subcategories_. Certain objects in such categories play the role of generators
and in the present setting yield the desired $R$-algebras. The required
technical definitions are as follows.

######  Definition 6.15.

An object $X$ of a category $\underline{C}$ admitting arbitrary direct sums is
said to be _compact_ if $\mathop{\rm Hom}\nolimits(X,-)$ commutes with
coproducts, i.e.,

$\mathop{\rm Hom}\nolimits(X,\coprod_{j\in J}Y_{j})\simeq\coprod_{j\in
J}\mathop{\rm Hom}\nolimits(X,Y_{j})$

for all objects $Y_{j}$ of $\underline{C}$, $j\in J$, and $X$ is said to be a
_compact generator of $\underline{C}$_ if $X$ is compact and if the smallest
thick subcategory of $\underline{C}$ containing $X$ is the entire category
$\underline{C}$.

Compact generators in this sense can only exist within the category
$[\underline{Mat\\!f}_{R,w}^{\infty}]$ and not in
$[\underline{Mat\\!f}_{R,w}]$ since the latter does not admit infinite direct
sums. The following principle describes the functorial role of a compact
generator.

######  Theorem 6.16 ([Dyc11, Thm. 5.1]).

Let $\underline{C}$ be a triangulated 2-periodic dg-category admitting
arbitrary direct sums and admitting a compact generator $X$. Let
$[X^{o}\text{-mod}]$ be the localization (as dg-modules) of the category of
$X^{o}$-modules (in the set of equivalences). Then the functor

(6.7)
$\displaystyle\underline{F}_{X}:\underline{C}\to\widehat{\underline{C}}:=[X^{o}\text{\rm-
mod}],\quad Y\mapsto\mathop{\rm Hom}\nolimits_{\underline{C}}(X,Y)$

induces an isomorphism in the homotopy category of 2-periodic dg-categories.

Consider $B:=(X,\mathop{\rm End}\nolimits X)$ as a dg-category. Then
$\underline{F}_{X}$ sends the homotopy class of $B$ to $\widehat{B}$, that is,
$B$ considered as a $B^{o}$-module.

The concept of $X^{o}$-module requires some explanation. Normally the action
of $X$ on and $X$-module is from the left. To indicate that the action is from
the right, one speaks of $X^{o}$-modules, indicating that the action of $X$ is
”reversed”.

Compact generators do exits for the category
$[\underline{Mat\\!f}_{R,w}^{\infty}]$ due to Dyckerhoff:

######  Proposition 6.17 ([Dyc11, Thm. 4.1 and Corollary 4.12]).

In the present situation $E$ (cf. (6.5)) is a compact generator of the
homotopy category $[\underline{Mat\\!f}_{R,w}^{\infty}]$.

It follows that taking in Theorem 6.16 for $\underline{C}$ the homotopy
category of $\underline{Mat\\!f}_{R,w}^{\infty}$, the object $E=k^{\rm stab}$
is a compact generator. So one can form the category
$\widehat{\underline{Mat\\!f}_{R,w}^{\infty}}$ as in that theorem. Note also
that the objects of $\underline{Mat\\!f}_{R,w}$, being bounded complexes, are
compact. Recalling the formula (6.6), one then deduces from the theorem:

######  Corollary 6.18.

The functor $\underline{F}_{E}$ induces an equivalence of categories

$[\underline{Mat\\!f}_{R,w}^{\infty}]\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\,\,\sim\,\,\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}\widehat{M_{R,w}}.$

In other words, $\widehat{M_{R,w}}$ is a model for the derived category of
$\underline{Mat\\!f}_{R,w}^{\infty}$.

One can describe $\underline{Mat\\!f}_{R,w}$ in a similar fashion in case the
ring $R$ is a complete local ring. In the situation of a polynomial IHS $w$,
one may assume this and then [Dyc11, Thm. 5.7] implies:

######  Corollary 6.19.

If $w=0$ determines an IHS (with singular point at $\mathbf{0}$) the functor
$\underline{F}_{E}$ (defined by (6.7)) induces an equivalence of categories

$[\underline{Mat\\!f}_{R,w}]\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\,\,\sim\,\,\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}\widehat{M_{R,w}},$

i.e., in the derived category one can replace $\underline{Mat\\!f}_{R,w}$ by
$\widehat{M_{R,w}}$.

Concluding, the algebra $\widehat{M_{R,w}}$ represents the derived category of
the category $\underline{Mat\\!f}_{R,w}$, by which the goal set at the
beginning of this subsection now has been achieved.

####  6.4.D. The diagonal construction

A further crucial ingredient for calculating Hochschild cohomology comes from
the diagonal construction explained in this subsection: see the proof of
Theorem 6.25 In this construction tensor products of matrix factorizations for
$\underline{Mat\\!f}_{R,w}$ and $\underline{Mat\\!f}_{R^{\prime},w^{\prime}}$
play a role for the special case where
$R^{\prime}=R=k[\\![x_{1},\dots,x_{m}]\\!]$, specifically, one uses

$\displaystyle\widetilde{R}$
$\displaystyle=R\otimes_{k}R=k[\\![y_{1},\dots,y_{m},z_{1},\dots,z_{m}]\\!],\quad
y_{j}=x_{j}\otimes 1,z_{j}=1\otimes x_{j}$ $\displaystyle\widetilde{w}$
$\displaystyle=-w\otimes 1+1\otimes w$ $\displaystyle\Delta$
$\displaystyle=\widetilde{R}/I_{\Delta},\quad
I_{\Delta}=(y_{1}-z_{1},\dots,y_{m}-z_{m})\quad\text{the "diagonal" of $R$ in
$\widetilde{R}$.}$ Since $\displaystyle
w\in{{\mathfrak{m}}}=(x_{1},\dots,x_{m})\text{ it follows that
}\widetilde{w}\in I_{\Delta}.$

If $X$ is an object of $\underline{Mat\\!f}_{R,w}$, then $X^{*}\otimes X$ is
an object of $\underline{Mat\\!f}_{\widetilde{R},\widetilde{w}}$. Recall
further that to be able to speak of ”stabilization” in the category
$\underline{Mat\\!f}_{R,w}$, one works over $S=R/(w)$ and over
$\widetilde{S}=\widetilde{R}/\widetilde{w}$ in the category
$\underline{Mat\\!f}_{\widetilde{R},\widetilde{w}}$. The preceding
considerations applies to this situation. Indeed, set

$F:=\Delta^{\rm stab}\text{ as an $\widetilde{S}$-module}\text{ (an object of
$\underline{Mat\\!f}_{\widetilde{R},\widetilde{w}}$)}.$

Then using the notation of (6.5) and (6.6), Corollary 6.19 in the setting of
matrix factorizations over $\widetilde{R}$ gives an equivalence of derived
categories

$\underline{F}_{E^{*}\otimes_{k}E}:[\underline{Mat\\!f}_{\widetilde{R},\widetilde{w}}]\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\sim\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}[M_{R,w}\otimes
M_{R,w}^{o}\text{\rm-mod}],\quad Y\mapsto\mathop{\rm
Hom}\nolimits(E^{*}\otimes_{k}E,Y)$

where one uses that $E^{*}\otimes_{k}E$ is a compact generator of
$\underline{Mat\\!f}_{\widetilde{R},\widetilde{w}}$.

######  Proposition 6.20.

The functor $\underline{F}_{E^{*}\otimes_{k}E}$ sends the stabilized diagonal
$F=\Delta^{\rm stab}$ to $M_{R,w}=\mathop{\rm
Hom}\nolimits^{{\mathbb{Z}}/2}_{R}(E,E)$ considered as an $M_{R,w}\otimes
M_{R,w}^{o}$-module.

###### Proof.

The aim is to show that $\underline{F}_{E^{*}\otimes_{k}E}$ sends $F$ to
$M_{R,w}$ considered as an $M_{R,w}^{o}\otimes M_{R,w}$-module. Apply Lemma
6.13 to $X=E^{*}\otimes_{k}E$ and $L=F$ which represents $\Delta$. One finds
quasi-isomorphisms (in the category
$\underline{Mat\\!f}_{\widetilde{R},\widetilde{w}}$ of matrix factorizations)

$\displaystyle\underline{Mat\\!f}_{\widetilde{R},\widetilde{w}}(E^{*}\otimes_{k}E,F)$
$\displaystyle\simeq\mathop{\rm
Hom}\nolimits^{{\mathbb{Z}}/2}_{R\otimes_{k}R}(E^{*}\otimes_{k}E,R)$
$\displaystyle\simeq\mathop{\rm
Hom}\nolimits^{{\mathbb{Z}}/2}_{R}(E,E)=M_{R,w}.$

Note that $M_{R,w}$ under left and right composition is an $M_{R,w}\otimes
M_{R,w}^{o}$-module: $(f,g)h=f\raise 1.0pt\hbox{{$\scriptstyle\circ$}}h\raise
1.0pt\hbox{{$\scriptstyle\circ$}}g$ for all $f,g,h\in M_{R,w}$. Also
$\underline{Mat\\!f}_{\widetilde{R},\widetilde{w}}(E^{*}\otimes_{k}E,F)$ and
$\mathop{\rm
Hom}\nolimits^{{\mathbb{Z}}/2}_{R\otimes_{k}R}(E^{*}\otimes_{k}E,R)$ are
$M_{R,w}\otimes M_{R,w}^{o}$-modules and one can check that the isomorphisms
preserve this structure. ∎

Recall that
$\widetilde{R}=R\otimes_{k}R=k[\\![y_{1},\dots,y_{m},z_{1},\dots,z_{m}]\\!]$
and that the polynomial $w({{\bf\it x}})\in{{\mathfrak{m}}}\subset
k[x_{1},\dots,x_{m}]$ yields $w({{\bf\it y}})\in k[y_{1},\dots,y_{m}]$ and
hence $\widetilde{w}=-w\otimes 1+1\otimes w\in I_{\Delta}\subset\widetilde{R}$
can be written as
$\widetilde{w}=\sum\widetilde{w}_{j}(y_{j}-z_{j})=\widetilde{{\bf\it
w}}\cdot({{\bf\it y}}-{{\bf\it z}})$. Using this, the central result which
will be used for calculating Hochschild cohomology is as follows:

######  Proposition 6.21.

  1. 1)

The stabilized diagonal $\Delta^{\rm stab}$, an object in
$\underline{Mat\\!f}_{\widetilde{R},\widetilde{w}}$, is represented by the
Koszul matrix factorization $\\{\widetilde{{\bf\it w}},{{\bf\it y}}-{{\bf\it
z}}\\}$.

  2. 2)

One has $\widetilde{w}_{j}=w_{x_{j}}\mod I_{\Delta}$.

  3. 3)

$\mathop{\rm End}\nolimits(\Delta^{\rm stab})$ – considered as a complex – is
isomorphic to the Koszul complex for the sequence
$\widetilde{w}_{1},\dots,\widetilde{w}_{m}$ modulo the diagonal ideal
$I_{\Delta}$, considered as a ${\mathbb{Z}}/2$-graded complex, that is, if
${{\bf\it w}}=(w_{x_{1}},\dots,w_{x_{m}})$, then $\mathop{\rm
End}\nolimits(\Delta^{\rm stab})\simeq N^{\bullet}({{\bf\it w}})$ (see Example
6.6(1)).

  4. 4)

$H^{k}(\mathop{\rm End}\nolimits(\Delta^{\rm stab}))=0$ for $k$ odd and equal
to $\mathsf{Jac}_{w}$ if $k$ is even.

###### Proof.

1) is clear and 2) is left as an exercise.
3) Apply Corollary 6.14 with $X=\widetilde{R}$, $L=R$ and remark that $R$ is
an $\widetilde{R}/\widetilde{w}$-module whose stabilization is $\Delta^{\rm
stab}$. Hence $\mathop{\rm End}\nolimits{\Delta^{\rm stab}}=\mathop{\rm
Hom}\nolimits^{{\mathbb{Z}}/2}_{\widetilde{R}}(\Delta^{\rm stab},R)$. Note
that $R$ as ${\mathbb{Z}}/2$-graded complex has $R$ in even degrees and $0$ in
odd degrees and so all derivatives are $0$. Moreover, $R$ is considered as an
$\widetilde{R}$-module and so ${{\bf\it y}}-{{\bf\it z}}$ maps to $0$ under
any morphism $\Lambda^{2j}N\to R$, $N$ a free $R$-module of rank $m$. So in
the complex $\mathop{\rm
Hom}\nolimits^{{\mathbb{Z}}/2}_{\widetilde{R}}(\Delta^{\rm stab},R)$ only the
derivatives from $\widetilde{{\bf\it w}}=\nabla w\mod I_{\Delta}$ survive
which gives the Koszul complex for $\widetilde{w}_{1},\dots,\widetilde{w}_{m}$
modulo the diagonal ideal $I_{\Delta}$, i.e. for the ideal generated by the
partial derivatives of $w$.
4) Follows from the above by applying Lemma 6.5. ∎

######  Remark 6.22.

Using the diagonal construction one can define a product structure on
$\underline{Mat\\!f}_{R,w}$ which under the equivalence of categories
$M_{R,w}=\underline{Mat\\!f}_{R,w}(E,E)\simeq[\underline{Mat\\!f}_{R,w}]$
corresponds to the $R$-algebra structure on $M_{R,w}$. So one might consider
the homotopy category $[\underline{Mat\\!f}_{R,w}]$ as a categorical
incarnation of the $R$-algebra $M_{R,w}$.

###  6.5. Hochschild cohomology

In this section I shall introduce Hochschild cohomology, first for algebras,
and then for categories. The aim is to understand Hochschild cohomology for
the category of matrix factorizations over a commutative ring $R$ with a unit.
As before, $R$ will be a polynomial algebra over a field $k$, or its
completion.

####  6.5.A. Hochschild cohomology for algebras

Let $A$ be any associative algebra over $R$. So $A$ is an $R$-module equipped
with an associative product. Now pass to the $R$-module

$C^{d}(A)=\mathop{\rm Hom}\nolimits_{R}(A\otimes\cdots\otimes A\to
A),\quad\text{($d$ factors)}.$

By convention, $C^{0}(A)=A$. The modules $C^{d}(A)$ can be made into a
cohomological complex with derivations $\delta^{d}:C^{d}(A)\to C^{d+1}(A)$
given by

$\displaystyle\delta^{d}f(a_{0},\dots,a_{d})$ $\displaystyle=a_{0}\cdot
f(a_{1},\dots,a_{d})-$ (6.8)
$\displaystyle\sum_{i=0}^{d}(-1)^{i}f(a_{0},\dots,a_{i}a_{i+1},\dots,a_{d})+(-1)^{d+1}f(a_{0},\dots,a_{d})\cdot
a_{d}.$

Its cohomology is the _Hochschild cohomology_ of the algebra $A$:

$\mathsf{HH}^{d}(A)=H^{d}(C^{\bullet}(A),\delta^{\bullet}),$

named after Hochschild’s article [Hoc45]. As for ordinary cohomology, this
group carries a graded cup-product structure coming from the product on co-
cycles $\gamma\in C^{n}(A$), $\gamma^{\prime}\in C^{m}(A)$ given by

$\gamma\cup\gamma^{\prime}(a_{1},\dots,a_{n+m})=(-1)^{nm}\gamma(a_{1},\dots,a_{n})\gamma^{\prime}(a_{n+1},\dots,a_{n+m}),\quad\forall\,a_{1},\dots,a_{n+m}\in
A.$

The Hochschild cohomology can be also be described in terms of the _enveloping
algebra_

$A^{e}:=A\otimes_{A}A^{o},\quad A^{o}=\text{opposite algebra of }A$

as will be explained next. This is based on the observation that the action of
$A$ on $A$ by left multiplication makes $A$ into an $A$-module while
multiplication on the right gives $A$ the structure of an $A^{o}$-module.
There is indeed a complex of free $A^{e}$-modules that computes Hochschild
cohomology, the so-called _bar-complex_ $C^{\rm
bar}_{\bullet}(A)=A^{\otimes{\bullet+2}}$, a homological complex starting in
degree $0$ with $A\otimes A$ and with derivations given by

$d_{n}(a_{0}\otimes\cdots\otimes
a_{n+1})=\sum_{i=0}^{n}(-1)^{i}a_{0}\otimes\cdots\otimes
a_{i}a_{i+1}\otimes\cdots\otimes a_{n+1}.$

The modules $C^{\rm bar}_{n}(A)$ are free $A^{e}$-modules under the operation
$(a\otimes b)\cdot(a_{0}\otimes\cdots a_{n+1})=a\cdot a_{0}\otimes\cdots
a_{n+1}\cdot b$ since $C^{\rm bar}_{n}(A)\simeq A^{e}\otimes A^{\otimes
n}\simeq\oplus_{j}(A^{e}\otimes 1)\otimes e_{j}$, where the $e_{j}$ form a
$k$-basis of $A^{\otimes n}$. By definition, the associated cohomological
complex is

$C^{\bullet}(A,A)=\mathop{\rm Hom}\nolimits_{A^{e}}(C^{\rm
bar}_{\bullet}A,A).$

There is an isomorphism of $k$-vector spaces $C^{n}(A)\to C^{n}(A,A)$ given by
$f\mapsto\\{a_{0}\otimes\cdots\otimes a_{m}\to
a_{0}f(a_{1}\otimes\cdots\otimes a_{n})a_{m}\\}$ whose inverse is
$g\mapsto\\{a_{1}\otimes\cdots\otimes a_{n}\mapsto g(1\otimes
a_{1}\otimes\cdots\otimes a_{n}\otimes 1)\\}$, as one easily verifies. Hence

(6.9)
$\displaystyle\mathsf{HH}^{d}(A)=H^{d}(C^{\bullet}A)=H^{d}(C^{\bullet}(A,A)).$

The bar-resolution is a free resolution of $A$ as an $A^{e}$-module by
extending it to the right by the multiplication map $A\otimes_{k}A\to A$. The
Ext-groups are thus the cohomology groups of the complex $\mathop{\rm
Hom}\nolimits_{A^{e}}(C^{\rm bar}_{\bullet}A,A)$. Next, pass to the derived
category333See for example Appendix A.3.14 in [Eis95].
${{\mathsf{D}}}({\underline{A}}^{e})$ which is built on complexes of
$A$-bimodules such as the bar-complex, or its dual $C^{\bullet}(A,A)$. In the
derived language this gives $H^{d}(R\mathop{\rm
Hom}\nolimits_{{{\mathsf{D}}}(A^{e})}(A,A))\simeq\mathop{\rm
Hom}\nolimits_{{{\mathsf{D}}}(A^{e})}(A,A[d])$. Summarizing, one has:

######  Proposition 6.23.

$\mathsf{HH}^{d}(A)\simeq\mathop{\rm Ext}\nolimits^{d}_{A^{e}}(A,A)$ which in
turn is isomorphic to $H^{d}(R\mathop{\rm
Hom}\nolimits_{{{\mathsf{D}}}(A^{e})}(A,A))\simeq\mathop{\rm
Hom}\nolimits_{{{\mathsf{D}}}(A^{e})}(A,A[d])$.

####  6.5.B. Hochschild cohomology for dg-categories

B. Keller [Kel94] has introduced an analog of the bar-resolution for any dg-
category $\underline{A}$ which serves as a means to define Hochschild
cohomology of $\underline{A}$. In order to carry this out, the first task is
to define tensor products of dg-categories in order to define the analog of
$A^{e}$.

  * •

The tensor product $\underline{A}\otimes\underline{B}$ of dg-categories: its
objects are pairs $(x,y)$ of objects $x$ of $\underline{A}$ and $y$ of
$\underline{B}$ and its morphisms are given by
$\underline{A}\otimes\underline{B}((x,y),(x^{\prime},y^{\prime}))=\underline{A}(x,x^{\prime})\otimes\underline{B}(y,y^{\prime})$
as dg-modules;

  * •

the dg-category $\underline{A}^{o}$, the one opposite to $\underline{A}$, has
the same objects as $\underline{A}$ but
$\underline{A}^{o}(x,y)=\underline{A}(y,x)$.

  * •

the enveloping dg-category is
$\underline{A}^{e}:=\underline{A}\otimes\underline{A}^{o}$.

One can attempt to define Hochschild cohomology by imitating what has been
done for algebras:

$\displaystyle\mathsf{HH}^{d}(\underline{A})=H^{d}(R\mathop{\rm
Hom}\nolimits_{{{\mathsf{D}}}(\underline{A}^{e})}(\underline{A},\underline{A}))\simeq\mathop{\rm
Hom}\nolimits_{{{\mathsf{D}}}(\underline{A}^{e})}(\underline{A},\underline{A}[d]).$

The problem is then to find a substitute for the bar-complex which should
represent $R\mathop{\rm
Hom}\nolimits_{{{\mathsf{D}}}(\underline{A}^{e})}(\underline{A},\underline{A})$.
Any such complex is called a _Hochschild complex_. There is indeed a
categorical version of the bar-complex as explained in [Kel94] but this
complex usually is unsuitable for concrete calculations. There exists a more
suitable Hochschild complex via the diagonal construction, as now will be
explained. But first, some more categorical constructions are needed.

  1. (i)

A dg-functor $\underline{F}:\underline{A}\to\underline{B}$ between two dg-
categories $\underline{A},\underline{B}$ consists of a map
$x\mapsto\underline{F}(x)$ from objects in $\underline{A}$ to objects in
$\underline{B}$, and for any two objects $x,y$, a $k$-linear morphism
$\underline{A}(x,y)\to\underline{B}(\underline{F}(x),\underline{F}(y))$
preserving the identity and satisfying the usual associativity condition.

  2. (ii)

Given a dg-category $\underline{A}$, a left $\underline{A}$-module consists of
a functor $\underline{M}:\underline{A}\to\underline{C}_{\rm dg}(k)$. So for
objects $x$ of $\underline{A}$ the image $\underline{M}(x)$ is a $k$-complex
and for any two objects $x,y$, there is a $k$-linear morphism
$\underline{A}(x,y)\to\underline{C}_{\rm
dg}(\underline{M}(x),\underline{M}(y))=\mathop{\rm
Hom}\nolimits(\underline{M}(x),\underline{M}(y))$. In other words, this gives
morphisms of complexes
$\underline{A}(x,y)\otimes\underline{M}(x)\to\underline{M}(y)$ which describes
the action of $\underline{A}$ on $\underline{M}$.

  3. (iii)

An _$\underline{A}$ -bimodule_ $M$ is a dg-functor
$M:\underline{A}^{e}\to\underline{C}_{\rm dg}(k)$.

######  Examples 6.24.

1. The _identity $\underline{A}$-bimodule_ or the _diagonal bimodule_ $\Delta_{\underline{A}}:\underline{A}^{e}\to\underline{C}_{\rm dg}(k)$ of $\underline{A}$ is defined by $\Delta_{\underline{A}}(x,y)=\underline{A}(x,y)$ on objects $(x,y)$ of $\underline{A}\otimes\underline{A}^{o}$ and

$\Delta_{\underline{A}}(\underline{A}(x,y)\otimes\underline{A}(y^{\prime},x^{\prime}))=\mathop{\rm
Hom}\nolimits(\underline{A}(x,y),\underline{A}(y^{\prime},x^{\prime}))$

on morphisms of $\underline{A}^{e}$. This is a left $\underline{A}^{e}$-module
with action
$(\underline{A}^{e}((x,y),(x^{\prime},y^{\prime}))\otimes\underline{A}(x,y)\to\underline{A}(y^{\prime},x^{\prime})$.
In the case of an $R$-algebra considered as category, $\Delta(A)$ is the
algebra $A$ considered as an $A^{e}$-bimodule.
2. A special case of _the identity functor._ Let $X$ be a variety over the field $k$ and let $\delta:X\hookrightarrow X\times X$ be the diagonal embedding with image $\Delta$. Let $p,q:X\times X\to X$ be the two projections. Note that $\delta_{*}{\mathcal{O}}_{X}={\mathcal{O}}_{\Delta}$ and so for a locally free sheaf ${\mathcal{E}}$ on $X$ one has $p^{*}{\mathcal{E}}\otimes\delta_{*}{\mathcal{O}}_{X}=q^{*}{\mathcal{E}}\otimes\delta_{*}{\mathcal{O}}_{X}$. It follows that there are canonical isomorphisms

$q_{*}(p^{*}{\mathcal{E}}\otimes_{O_{X\times
X}}\delta_{*}{\mathcal{O}}_{X})\simeq
q_{*}(q^{*}{\mathcal{E}}\otimes_{O_{X\times
X}}\delta_{*}{\mathcal{O}}_{X})={\mathcal{E}}\otimes_{O_{X}}(q_{*}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}\delta_{*}{\mathcal{O}}_{X})\simeq{\mathcal{E}}.$

So the functor on the category of locally free ${\mathcal{O}}_{X}$-sheaves
given by

${\mathcal{E}}\mapsto q_{*}(p^{*}{\mathcal{E}}\otimes_{O_{X\times
X}}{\mathcal{O}}_{\Delta})$

represents the identity functor. In this sense $\Delta$ ”is” the identity
functor on the category of locally free sheaves on $X$. This functor can be
extended to the dg category of complexes of locally free sheaves on $X$, or to
the category of matrix factorizations in the sense of § 6.1.B.

The same construction for pairs $(X,Y)$ of $k$-varieties, ${\mathcal{E}}$ a
coherent ${\mathcal{O}}_{X}$-module, and with the structure sheaf of the
diagonal replaced with any coherent ${\mathcal{O}}_{X\times Y}$-sheaf
${\mathcal{K}}$ defines the _Fourier-Mukai transform of ${\mathcal{E}}$ with
kernel_ ${\mathcal{K}}$, a functor on the category of coherent
${\mathcal{O}}_{X}$-modules to the category of coherent
${\mathcal{O}}_{Y}$-modules.

The diagonal also allows to define Hochschild cohomology for the scheme $X$ as

(6.10) $\mathsf{HH}^{d}(X):=H^{d}(X\times
X,R{\mathcal{H}}om_{{\mathcal{O}}_{X\times
X}}({\mathcal{O}}_{\Delta},{\mathcal{O}}_{\Delta})).$

Equation (6.10) can be seen as an example of a general result due to B. Toën
[Toë06, Cor. 8.1] stating that the usual bar complex is homotopic to the
endomorphism complex of the identity bimodule

(6.11) $\displaystyle\mathop{\rm
End}\nolimits_{\underline{A}^{e}}(\Delta_{\underline{A}})=\mathop{\rm
Hom}\nolimits_{\underline{A}^{e}}(\Delta_{\underline{A}},\Delta_{\underline{A}}),$

and hence serves as a Hochschild complex. In particular, equation (6.10) shows
that any complex representing $R{\mathcal{H}}om_{{\mathcal{O}}_{X\times
X}}({\mathcal{O}}_{\Delta},{\mathcal{O}}_{\Delta})$ is a Hochschild complex
for ${\mathcal{O}}_{X}$ (identified with ${\mathcal{O}}_{\Delta}$).

By making use of the diagonal construction in § 6.4.D the preceding
observations can be applied to the category $\underline{Mat\\!f}_{R,w}$ of
matrix factorizations of an IHS at ${{\bf\it 0}}$ given by a polynomial $w\in
R={\mathbb{C}}[x_{1},\dots,x_{m}]$. Indeed, by Proposition 6.20 the identity
functor is represented by the stabilized diagonal. Its endomorphism algebra as
well as its cohomology has been calculated in Proposition 6.21. The results
thus reads

######  Theorem 6.25.

The Koszul complex on the derivatives $\displaystyle\\{w_{x_{1}},\dots\
w_{x_{m}}\\}$ of $w\in R={\mathbb{C}}[x_{1},\dots,x_{m}]$, viewed as a
${\mathbb{Z}}/2$-graded complex serves as a Hochschild complex for
$\underline{Mat\\!f}_{R,w}$ and hence

$\mathsf{HH}^{d}(\underline{Mat\\!f}_{R,w})=\begin{cases}\mathsf{Jac}_{w}={\mathbb{C}}[x_{1},\dots,x_{m}]/(w_{x_{1}},\dots\
w_{x_{m}})&\text{ if $d$ is even}\\\ 0&\text{ if $d$ is odd.}\end{cases}$

######  Remark 6.26.

As explained in Remark 6.22, one can equip the homotopy category
$[\underline{Mat\\!f}_{R,w}]$ with an $R$-algebra structure through the quasi-
isomorphism $\underline{Mat\\!f}_{R,w}(E,E)\simeq\underline{Mat\\!f}_{R,w}$.
The Hochschild complex respects the algebra structure and so this structure
survives on the level of its cohomology. Clearly, this is reflected in the
above theorem: the jacobian ring $\mathsf{Jac}_{w}$ is an
$R={\mathbb{C}}[x_{1},\dots,x_{m}]$-algebra.

###  6.6. The equivariant case

####  6.6.A. Equivariant matrix factorizations

Assume that the IHS given by $\\{\\{w=0\\},{{\bf\it 0}}\\}$ admits symmetries
in the weak sense that $w\in{\mathbb{C}}[x_{1},\dots,x_{m}]$ is a semi-
invariant with respect to a group $\Gamma$ of linear transformations of
${\mathbb{C}}^{m}$, that is, there is a character
$\chi:\Gamma\to{\mathbb{C}}^{\times}$ with

$\gamma(w)=\chi(\gamma)w,\quad\forall\gamma\in\Gamma.$

A $\Gamma$-character $\chi$ can be viewed as a rank $1$ free $R$ module with
obvious $\Gamma$-action, and so, if $V$ admits a $\Gamma$-action , also
$V(\chi):=V\otimes_{R}\chi$ admits a canonical $\Gamma$-action.

######  Definition 6.27.

Let $(R,{{\mathfrak{m}}})$, $\Gamma$, $\chi$ and $w\in{{\mathfrak{m}}}$ as
above.

  1. (i)

A _$\Gamma$ -equivariant matrix factorization_ of $w$ consists of a pair
$X^{0},X^{1}$ of free $R$-modules of finite rank equipped with an action of
$\Gamma$, together with equivariant $R$-linear morphisms
$X^{0}\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;d_{0}\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}X^{1}(\chi)$ and
$X^{1}\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;d_{1}\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}X^{0}$ such that $d_{1}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d_{0}=w\cdot\text{\rm id}$ (as above).

  2. (ii)