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% The Continuous Hochschild Cochain Complex of a Scheme (Survey)
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% ** top matter **
\title[Hochschild Cochain Complex]
{The Continuous Hochschild Cochain Complex of a Scheme (Survey)}
\author{Amnon Yekutieli}
\address{Department of Mathematics,
Ben Gurion University, Be'er Sheva 84105, ISRAEL}
\email{amyekut@math.bgu.ac.il \newline \indent
http://www.math.bgu.ac.il/$\sim$amyekut}
\date{19.3.02}
\subjclass{Primary 16E40; Secondary 14F10, 18G10, 13H10}
\thanks{\textit{Keywords}: Hochschild cohomology; schemes;
derived categories.}

\begin{document}
\maketitle

This is a revision of the notes to my lecture at the conference 
``R\'esolution des singularit\'es et g\'eom\'etrie non commutative,
\`a la M\'emoire de Ruth Michler'' held at CIRM, Luminy in
July  2001. The title of the lecture was ``Decomposition of the 
Hochschild complex of a scheme.''

The work discussed in my lecture began when I visited 
Monique Lejeune-Jalabert in Grenoble in February 1997. We had been 
trying to understand a certain construction of Chern classes, and 
this led to a discussion of Hochschild complexes and formal 
schemes.

The main result stated in my lecture 
(decomposition in arbitrary characteristic, using minimal
injective resolutions) was obtained a few 
months later, in September 1997. This was during a conference in 
Obergurgle. It was there, in Obergurgle 1997, that I met Ruth 
Michler.  

Two months after the Luminy conference, in September 2001, Michel Van 
den Bergh discovered a serious gap in my proof of the decomposition. 
I could not repair the gap, and subsequently the paper had to be 
withdrawn for revision. The new version of the paper \cite{Ye3}
concentrates on the use of 
formal completions to study Hochschild cohomology, which I believe 
is interesting in itself. This revision is reflected in the 
present notes.


\section{Hochschild Complex of a $\mbb{K}$-Algebra}

Let $\mbb{K}$ be a commutative ring and $A$ a commutative 
$\mbb{K}$-algebra. The (unnormalized) bar resolution of $A$ is the 
exact sequence
\[ \cdots \to \mcal{B}_{2}(A) \xrightarrow{\partial}
\mcal{B}_{1}(A) \xrightarrow{\partial} \mcal{B}_{0}(A) 
\to A \to 0 , \]
where 
\[ \mcal{B}_{q}(A) := A^{\otimes (q + 2)} = 
A \otimes_{\mbb{K}} \cdots \otimes_{\mbb{K}} A. \]
The formula for $\partial$ is
\[ \partial(a_{0} \otimes \cdots \otimes a_{q + 1}) :=  
\sum_{i = 0}^{q} (-1)^{i} a_{0} \otimes \cdots \otimes 
a_{i} a_{i + 1} \otimes \cdots \otimes a_{q + 1} . \]

Write 
\[ A^{\mrm{e}} := A \otimes_{\mbb{K}} A 
= \mcal{B}_{0}(A) . \] 
Then $\mcal{B}_{q}(A)$ is an $A^{\mrm{e}}$-module via 
\[ a_{1} \otimes a_{2} \mapsto 
a_{1} \otimes 1 \otimes \cdots \otimes 1 \otimes a_{2} , \]
and $\partial$ is $A^{\mrm{e}}$-linear. 

Since we will work with derived categories (with upper indices and 
differentials of degree $+1$) it will be convenient to write
\[ \mcal{B}^{-q}(A) := \mcal{B}_{q}(A) . \]

The Hochschild cochain complex of $A$ is 
$\mcal{C}^{\bdot}_{\mrm{dual}}(A)$ where
\[ \begin{aligned}
\mcal{C}^{q}_{\mrm{dual}}(A) & :=  
\opn{Hom}_{A^{\mrm{e}}} (\mcal{B}_{q}(A), A)  \\
& \cong \opn{Hom}_{\mbb{K}}(A^{\otimes q}, A) . 
\end{aligned} \]
In the literature usually one finds the notation 
$\mcal{C}^{\bdot}(A)$ or $\mcal{C}^{\bdot}(A, A)$ for this 
complex.
The Hochschild cohomology of $A$ is 
\[ \mrm{HH}^{i}(A) := \mrm{H}^{i} \mcal{C}^{\bdot}_{\mrm{dual}}(A)
. \]
It is well known that Hochschild cohomology controls
deformations of $A$ as associative algebra. 
More on that in the Appendix.

When $\mbb{K}$ is a field, $\mcal{B}^{\bdot}(A)$ is a free 
resolution of $A$ as $A^{\mrm{e}}$-module. Hence
\[ \begin{aligned}
\mcal{C}^{\bdot}_{\mrm{dual}}(A) & = 
\opn{Hom}_{A^{\mrm{e}}} (\mcal{B}^{\bdot}(A), A) \\
& \cong \opn{RHom}_{A^{\mrm{e}}}(A, A)
\end{aligned}  \]
in derived category $\msf{D}(\cat{Mod} A^{\mrm{e}})$ of 
$A^{\mrm{e}}$-modules. In cohomology we get
\begin{equation} \label{eqn4}
\begin{aligned}
\mrm{HH}^{i}(A) & = 
\mrm{H}^{i} \opn{Hom}_{A^{\mrm{e}}}(\mcal{B}^{\bdot}(A), A) \\
& \cong \opn{Ext}^{i}_{A^{\mrm{e}}}(A, A) .
\end{aligned} 
\end{equation}

Now suppose $\mbb{K}$ is a noetherian ring (e.g.\ a finitely 
generated algebra over a field or over $\mbb{Z}$). 
Even though the Hochschild cohomology 
$\mrm{H}^{i} \mcal{C}^{\bdot}_{\mrm{dual}}(A)$
still makes sense and has the same deformation  meaning, 
we choose to concentrate on 
$\opn{Ext}^{i}_{A^{\mrm{e}}}(A, A)$
as the definition of Hochschild cohomology.
This definition will extend well to schemes. 
Also, as we shall see, the Ext definition coincides with 
continuous Hochschild cohomology, when $A$ is smooth.

Let $A$ be a smooth $\mbb{K}$-algebra. 
By smooth I mean that $A$ is finitely generated and formally 
smooth -- i.e.\ it has the lifting property.
(E.g.\ $A = \mbb{K}[t_{1}, \cdots, t_{n}]$, the polynomial algebra.)
This implies $A$ is flat, and the module of differentials 
$\Omega^{1}_{A} = \Omega^{1}_{A / \mbb{K}}$ is projective.
Write 
\[ \mcal{T}_{A /\mbb{K}} := \opn{Der}_{\mbb{K}}(A) = \opn{Hom}_{A}
(\Omega^{1}_{A}, A) \]
and
\[ \bwedge^{q} \mcal{T}_{A} := \bwedge^{q}_{A} \mcal{T}_{A / 
\mbb{K}} . \]

\begin{thm}[Hochschild-Kostant-Rosenberg]
Suppose $\mbb{K}$ is noetherian and $A$ is a smooth 
$\mbb{K}$-algebra. Then there is a canonical isomorphism
\[ \opn{Ext}^{i}_{A^{\mrm{e}}}(A, A)
\cong \bwedge^{i} \mcal{T}_{A} . \] 
\end{thm}

In the derived category $\msf{D}(\cat{Mod} A^{\mrm{e}})$
we have the complex of poly-tangents
\[ \boplus_{q} \bigl( \bwedge^{q} \mcal{T}_{A} \bigr)[-q] = 
\bigl( \cdots \to 0 \to A \xrightarrow{0} \mcal{T}_{A} 
\xrightarrow{0} \bwedge^{2} \mcal{T}_{A} \to \cdots \bigr) \]
with $A$ in degree $0$. 

Because each $A$-module 
$\bwedge^{q} \mcal{T}_{A}$ is projective, an easy calculation 
(cf.\ \cite{Ye3}) shows that the HKR Theorem implies:

\begin{prop} \label{prop1}
There is an isomorphism
 \[ \opn{RHom}_{A^{\mrm{e}}}(A, A) \cong 
\boplus_{q} \bigl( \bwedge^{q} \mcal{T}_{A} \bigr)[-q] \]
in $\msf{D}(\cat{Mod} A^{\mrm{e}})$.
\end{prop}



\section{The Hochschild Complex of a Scheme}

$\mbb{K}$ is still any noetherian ring. 
Let $X$ be a separated finite type scheme over $\mbb{K}$. For 
example, $X \subset \mbf{P}^{n}_{\mbb{K}}$, a subvariety of 
projective $n$-space over $\mbb{K}$. Then $\mcal{O}_{X}$ is a 
coherent $\mcal{O}_{X^{2}}$-module via the diagonal embedding.

The naive attempt to define a Hochschild complex on $X$ does not 
work (even if $\mbb{K}$ is a field). 
The modules $\mcal{C}^{q}_{\mrm{dual}}(A)$ are not 
covariant in $A$. On the other hand the bar sheaves $\mcal{B}^{-q}(X)$
on $X$, associated to the presheaves $U \mapsto \mcal{B}^{-q}(A)$,
are ill-behaved. 

There is however a way to get around this problem, 
due to Swan \cite{Sw}, which gives a good definition of Hochschild 
cohomology of $X$. 

Another option is to simply define the Hochschild complex of $X$ to 
be
\[ \mrm{R} \mcal{H}om_{\mcal{O}_{X^{2}}}(\mcal{O}_{X}, 
\mcal{O}_{X}) . \]
This is an object in  the derived category 
$\msf{D}(\cat{Mod} \mcal{O}_{X^{2}})$ of 
$\mcal{O}_{X^{2}}$-modules.
It has the benefit that on any affine open set
\[ U = \opn{Spec} A \subset X  \]
one has
\[ \mrm{H}^{i}(U, \mrm{R} \mcal{H}om_{\mcal{O}_{X^{2}}}(\mcal{O}_{X}, 
\mcal{O}_{X})) \cong \opn{Ext}^{i}_{A^{\mrm{e}}}(A, A) . \]
Furthermore:

\begin{thm}[Y]
Let $\mbb{K}$ be a noetherian ring.
If $X$ is flat finite type separated scheme over 
$\mbb{K}$ then Swan's definition of the Hochschild cochain complex
of $X$ coincides with 
$\mrm{R} \mcal{H}om_{\mcal{O}_{X^{2}}}(\mcal{O}_{X}, 
\mcal{O}_{X})$.
\end{thm}

Here is a less obvious way to proceed. The bad behavior of the bar 
resolution $\mcal{B}^{\bdot}(X)$ can be fixed by passing to the 
completion along the diagonal. 

For $q \geq 0$ let 
$\mfrak{X}^{q + 2}$ be the completion of $X^{q + 2}$ along the 
diagonal embedding of $X$. It is a noetherian formal scheme (see 
\cite{EGA-I}). Define 
\[ \what{\mcal{B}}^{-q}(X) := \mcal{O}_{\mfrak{X}^{q + 2}} ; \]
it is a sheaf of $\mcal{O}_{\mfrak{X}^{2}}$-modules with an adic 
topology. 

\begin{exa}
Take 
\[ X := \mbf{A}^{1}_{\mbb{K}} = \opn{Spec} \mbb{K}[t] . \]
Then 
\[ X^{2} = \opn{Spec} \mbb{K}[t \otimes 1, 1 \otimes t] . \]
Let 
\[ \tilde{\mrm{d}} t := t \otimes 1 - 1 \otimes t \]
be the function defining the diagonal.
The formal scheme $\mfrak{X}^{2}$ is then 
\[ \mfrak{X}^{2} \cong \opn{Spf} \mbb{K}[t] 
[[ \tilde{\mrm{d}} t ]]  . \]
For any affine open set $U = \opn{Spec} A \subset X$ one has
\[ \Gamma(U, \what{\mcal{B}}^{0}(X)) = 
\Gamma(U, \mcal{O}_{\mfrak{X}^{2}}) \cong
A[[ \tilde{\mrm{d}} t ]] . \] 
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\bigskip
The sheaves of continuous Hochschild cochains of $X$ are
\[ \mcal{C}_{\mrm{cd}}^{q}(X) := 
\mcal{H}om^{\mrm{cont}}_{\mcal{O}_{\mfrak{X}^{2}}}
(\what{\mcal{B}}^{-q}(X), \mcal{O}_{X}) . \] 
Here ``cd'' stands for continuous dual.
The sheaves $\mcal{C}_{\mrm{cd}}^{q}(X)$ are quasi-coherent 
$\mcal{O}_{X}$-modules. Furthermore on an affine open set 
$U = \opn{Spec} A \subset X$ one has
\[ \begin{aligned}
& \Gamma(U, \mcal{C}^{q}_{\mrm{cd}}(X)) \cong \\
& \quad \quad \{ f \in \opn{Hom}_{\mbb{K}}(A^{\otimes q}, A) \mid
f \text{ is a differential operator in each factor} \} .
\end{aligned} \]
Thus $\mcal{C}_{\mrm{cd}}^{\bdot}(X)$ is also called the complex of 
poly-differential operators. 

The next theorem has a ``folklore'' status; it was certainly known 
to Konstevich in some version. A proof can be found in \cite{Ye3}. 

\begin{thm}
Let $\mbb{K}$ be a noetherian ring.
Assume $X$ is a separated smooth scheme over $\mbb{K}$. 
Then there is an isomorphism
\[ \mcal{C}_{\mrm{cd}}^{\bdot}(X) \cong 
\mrm{R} \mcal{H}om_{\mcal{O}_{X^{2}}}(\mcal{O}_{X}, 
\mcal{O}_{X}) \]
in $\msf{D}(\cat{Mod} \mcal{O}_{X^{2}})$.
\end{thm}


\section{Decomposition in Characteristic $0$}

There is a standard homomorphism
\[ \pi_{\mrm{cd}}: \bwedge^{q} \mcal{T}_{X} \to
\mcal{C}^{q}_{\mrm{cd}}(X) . \]
Locally it is given by the formula
\[ \begin{aligned}
\pi_{\mrm{cd}}(v_{1} \wedge \cdots \wedge v_{q})
(1 \otimes a_{1} \otimes \cdots \otimes a_{q} \otimes 1) & \\
= \sum_{\sigma \in \Sigma_{q}} \opn{sgn}(\sigma)
v_{\sigma(1)}(a_{1}) \cdots v_{\sigma(q)}(a_{q}) 
\end{aligned} \]
for $v_{i} \in \mcal{T}_{A} = \opn{Der}_{\mbb{K}}(A)$ and 
$a_{i} \in A$, where $\opn{sgn}(\sigma)$ denotes the sign 
of the permutation $\sigma$. 

\begin{thm} \label{thm2}
Let $\mbb{K}$ be a noetherian ring and let
$X$ be a separated smooth scheme over $\mbb{K}$ of relative 
dimension $n$. If $n!$ is invertible in $\mbb{K}$ then 
\[ \pi_{\mrm{cd}}: 
\boplus_{q} \bigl( \bwedge^{q} \mcal{T}_{X} \bigr)[-q] \to
\mcal{C}^{\bdot}_{\mrm{cd}}(X) \]
is a quasi-isomorphism.
\end{thm}

This result (in characteristic $0$)
seems to have been known for some time. The 
quasi-isomorphism $\pi_{\mrm{cd}}$, up to rational constants, is 
the one underlying the Formality Theorem of Kontsevich (see  
Appendix). In fact Kontsevich \cite{Ko} has a proof of this theorem 
when $X$ a $\mrm{C}^{\infty}$ real manifold. 
The proof of Theorem \ref{thm2} can be found in \cite{Ye3}.

A consequence is a Hodge-type decomposition globally:

\begin{cor}
With $\mbb{K}$ and $X$ as above there is a canonical decomposition 
\[ \opn{Ext}^{i}_{\mcal{O}_{X^{2}}}(\mcal{O}_{X}, \mcal{O}_{X})
\cong \boplus_{q} \mrm{H}^{i - q} 
\bigl(X, \bwedge^{q} \mcal{T}_{X} \bigr) . \]
\end{cor}

The assumption on the characteristic is important. 

\begin{thm}[Y]
Assume $\mbb{K}$ is a Gorenstein noetherian ring of finite 
Krull dimension and $X$ is smooth over $\mbb{K}$ of relative 
dimension $n$. Then $\pi_{\mrm{cd}}$ is a quasi-isomorphism iff
$n!$ is invertible in $\mcal{O}_{X}$. 
\end{thm}

Here are a couple of questions.

\begin{que}
Suppose $\mbb{K}$ has mixed characteristics and
$n!$ is invertible in $\mbb{K}$. 
Over $\mbb{K} \otimes_{\mbb{Z}} \mbb{Q}$
there is a Hodge-type 
decomposition of the Hochschild complex due to 
Gerstenhaber-Schack \cite{GS} and Loday. 
What is the connection to the decomposition we get over
$\mbb{K}$ in Theorem \ref{thm2}?
\end{que}

\begin{que}
Suppose $\mbb{K}$ is a field of characteristic $p$, $A$ is a 
smooth $\mbb{K}$-algebra of dimension $n$ and 
$n! \geq p > 0$. Find an explicit formula for a quasi-isomorphism
\[ \phi: \boplus_{q} \bigl( \bwedge^{q} \mcal{T}_{A} \bigr)[-q] \to
\mcal{C}^{\bdot}_{\mrm{cd}}(A)  \]
representing the decomposition from Proposition \ref{prop1}.
\end{que}


\appendix
\section{Deformations}

Here is an overview of the role of the Hochschild complex in 
deformation theory. This will help put the decomposition results 
in context. 

The following approach to deformation problems is attributed to 
Deligne (cf.\ \cite{GM}). Here $\mbb{K}$ is a field of 
characteristic $0$. 

Let $\mfrak{g}$ be a differential graded (DG) Lie algebra over 
$\mbb{K}$. Thus $\mfrak{g} = \boplus \mfrak{g}^{i}$ is a complex 
equipped with a graded Lie bracket compatible with the differential.
For $\lambda \in \mfrak{g}^{0}$, the map 
$\rho(\lambda): \mfrak{g}^{1} \to \mfrak{g}^{1}$,
\[ \rho(\lambda)(\alpha) = [\lambda, \alpha] - \mrm{d} \lambda ,\]
is an affine transformation of the vector space $\mfrak{g}^{1}$. 
For any vector space $V$ let us write
$\opn{Aff}(V) = \opn{Gl}(V) \ltimes V$
for the group of affine transformations, and 
$\mfrak{aff}(V) = \mfrak{gl}(V) \oplus V$
for the Lie algebra. Then 
$\rho: \mfrak{g}^{0} \to \mfrak{aff}(\mfrak{g}^{1})$
is a Lie algebra homomorphism. 

If $\mfrak{g}^{0}$ is nilpotent there is a 
nilpotent algebraic group $G^{0}$ (defined using the 
Campbell-Baker-Hausdorff formula), for which the exponential map
$\opn{exp}: \mfrak{g}^{0} \to G^{0}$ is a bijection. Hence the 
Lie algebra homomorphism $\rho$ induces a group homomorphism
$G^{0} \to \opn{Aff}(\mfrak{g}^{1}).$
In other words, $G^{0}$ acts on $\mfrak{g}^{1}$ by affine 
transformations. Define
\[ \opn{MC}(\mfrak{g}) := \{ \alpha \in \mfrak{g}^{1} \mid
\mrm{d} \alpha + \tfrac{1}{2} [\alpha, \alpha] = 0 \} 
/ G^{0} , \]
the set of solutions of the Maurer-Cartan equation modulo gauge 
equivalence.

Suppose now that $\mfrak{g}$ is any DG Lie algebra. Let $R$ be an
artinian local $\mbb{K}$-algebra with maximal ideal $\mfrak{m}$ 
and $R / \mfrak{m} = \mbb{K}$. Then $\mfrak{m} \otimes \mfrak{g}$ is a 
nilpotent DG Lie algebra. Define
\[ \opn{Def}_{\mfrak{g}}(R) := 
\opn{MC}(\mfrak{m} \otimes \mfrak{g}) . \]

Many infinitesimal deformation problems are described by this 
construction. 

\begin{exa}
Suppose $A$ is a commutative $\mbb{K}$-algebra. Define a DG Lie 
algebra $\mfrak{g}$ as follows. As complex,
\[ \mfrak{g} := \mcal{C}^{\bdot}_{\mrm{dual}}(A)[1] , \]
i.e.\
\[ \mfrak{g}^{i} = \mcal{C}^{i + 1}_{\mrm{dual}}(A) \cong
\opn{Hom}_{\mbb{K}}(A^{\otimes (i + 1)}, A) . \]
The Lie bracket is the Gerstenhaber bracket. Note that 
$\mfrak{g}^{1} = \opn{Hom}_{\mbb{K}}(A^{\otimes 2}, A)$,
which is where $\mbb{K}$-algebra structures on the vector space 
$A$ will live. 
And indeed, for any artinian $\mbb{K}$-algebra $R$ as above, the 
set $\opn{Def}_{\mfrak{g}}(R)$ coincides with the set of flat 
associative $R$-algebras $\tilde{A}$, such that 
$\tilde{A} \otimes_{R} \mbb{K} \cong A$, up to isomorphism. 

If instead we take for $\mfrak{g}$ the subalgebra 
\[ \mcal{D}^{\bdot}_{A, \mrm{poly}} := 
\mcal{C}^{\bdot}_{\mrm{cd}}(A)[1] \]
of poly-differential operators, then $\opn{Def}_{\mfrak{g}}(R)$ 
describes deformations of $A$ which depend differentially on $A$. 
Such a deformation will localize on $\opn{Spec} A$.
\end{exa}


\begin{exa}
Suppose $A$ is a smooth $\mbb{K}$-algebra. A Poisson 
structure is a bi-derivation
$\{-,-\}: A \otimes A \to A$ satisfying certain axioms. 
Let $R$ be a local artinian $\mbb{K}$-algebra. The set of 
equivalence classes of Poisson structures on $A \otimes R$
that vanish modulo $\mfrak{m}$, up to gauge 
transformation, is isomorphic to $\opn{Def}_{\mfrak{g}}(R)$, 
where $\mfrak{g}$ is the DG Lie algebra
\[ \mfrak{g} = \mcal{T}^{\bdot}_{A, \mrm{poly}} := 
\boplus_{i} \bigl( \bwedge^{i} \mcal{T}_{A} \bigr) [i + 1] \]
equipped with $0$ differential and the Schouten-Nijenhuis bracket. 
\end{exa}

For other examples cf.\ \cite{HS}.

Define
\[ U_{1}: \mfrak{g}_{1} = \mcal{T}^{\bdot}_{A, \mrm{poly}} 
\to \mfrak{g}_{2} = \mcal{D}^{\bdot}_{A, \mrm{poly}} \]
by
\[ U_{1}(\alpha) = \tfrac{1}{q!} \pi_{\mrm{cd}}(\alpha), \quad
\alpha \in \bwedge^{q} \mcal{T}_{A} . \]
The quasi-isomorphism $U_{1}$ does not respect the Lie algebra
structures, but in cohomology
\[ \mrm{H}(U_{1}):  \mrm{H} \mfrak{g}_{1} 
\to \mrm{H} \mfrak{g}_{2} \]
is an isomorphism of graded Lie algebras.

The Kontsevich Formality Theorem \cite{Ko} says that $U_{1}$
extends to an $L_{\infty}$ morphism 
$\mfrak{g}_{1} \to \mfrak{g}_{2}$. This means there is a sequence 
of morphisms $U_{1}, U_{2}, U_{3} \ldots$ that constitute a 
``higher homotopy'' map. It follows that for any artinian 
$\mbb{K}$-algebra $R$, 
$\opn{Def}_{\mfrak{g}_{1}}(R) \cong \opn{Def}_{\mfrak{g}_{2}}(R)$.
Taking $R = \mbb{K}[\hslash] / (\hslash^{i})$ and passing to the 
limit one gets
\[ \opn{Def}_{\mfrak{g}_{1}}(\mbb{K}[[\hslash]]) \cong 
\opn{Def}_{\mfrak{g}_{2}}(\mbb{K}[[\hslash]]) , \]
so that formal deformations of $A$ over the power series ring
$\mbb{K}[[\hslash]]$ correspond bijectively (modulo gauge
equivalence) to formal Poisson structures
$\alpha(\hslash) = \sum_{i=1}^{\infty} \alpha_{i} \hslash^{i}$,
$\alpha_{i} \in \mfrak{g}_{2}^{1} = \bwedge^{2} \mcal{T}_{A}$. 
In particular, given any
Poisson structure $\alpha_{1}$ there exists a formal deformation 
of $A$, corresponding to $\alpha(\hslash) = \alpha_{1} \hslash$. 

Kontsevich's result was proved when $A$ is the ring of 
$\mrm{C}^{\infty}$ 
functions on a real manifold $X$. But it seems to hold also for a 
smooth algebraic variety $X$ over a field $\mbb{K}$ of 
characteristic $0$. A similar line of reasoning is claimed in \cite{Ko}
to imply that the isomorphism
\[ \boplus_{i} \opn{Ext}^{i}_{\mcal{O}_{X^{2}}}
(\mcal{O}_{X}, \mcal{O}_{X})
\cong \boplus_{i, q} \mrm{H}^{i - q} 
\bigl(X, \bwedge^{q} \mcal{T}_{X} \bigr) \]
sends the Yoneda product on the left side 
to the cup product on the right.


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