--- a/text/hochschild.tex Mon Oct 26 17:14:35 2009 +0000
+++ b/text/hochschild.tex Tue Oct 27 02:11:36 2009 +0000
@@ -66,7 +66,7 @@
Next, we show that for any $C$-$C$-bimodule $M$,
\begin{prop} \label{prop:hoch}
-The complex $K_*(M)$ is quasi-isomorphic to $HC_*(M)$, the usual
+The complex $K_*(M)$ is quasi-isomorphic to $\HC_*(M)$, the usual
Hochschild complex of $M$.
\end{prop}
\begin{proof}
@@ -74,19 +74,19 @@
up to quasi-isomorphism, by the following properties:
\begin{enumerate}
\item \label{item:hochschild-additive}%
-$HC_*(M_1 \oplus M_2) \cong HC_*(M_1) \oplus HC_*(M_2)$.
+$\HC_*(M_1 \oplus M_2) \cong \HC_*(M_1) \oplus \HC_*(M_2)$.
\item \label{item:hochschild-exact}%
An exact sequence $0 \to M_1 \into M_2 \onto M_3 \to 0$ gives rise to an
-exact sequence $0 \to HC_*(M_1) \into HC_*(M_2) \onto HC_*(M_3) \to 0$.
+exact sequence $0 \to \HC_*(M_1) \into \HC_*(M_2) \onto \HC_*(M_3) \to 0$.
\item \label{item:hochschild-coinvariants}%
-$HH_0(M)$ is isomorphic to the coinvariants of $M$, $\coinv(M) =
+$\HH_0(M)$ is isomorphic to the coinvariants of $M$, $\coinv(M) =
M/\langle cm-mc \rangle$.
\item \label{item:hochschild-free}%
-$HC_*(C\otimes C)$ is contractible.
+$\HC_*(C\otimes C)$ is contractible.
(Here $C\otimes C$ denotes
the free $C$-$C$-bimodule with one generator.)
-That is, $HC_*(C\otimes C)$ is
-quasi-isomorphic to its $0$-th homology (which in turn, by \ref{item:hochschild-coinvariants}, is just $C$) via the quotient map $HC_0 \onto HH_0$.
+That is, $\HC_*(C\otimes C)$ is
+quasi-isomorphic to its $0$-th homology (which in turn, by \ref{item:hochschild-coinvariants}, is just $C$) via the quotient map $\HC_0 \onto \HH_0$.
\end{enumerate}
(Together, these just say that Hochschild homology is `the derived functor of coinvariants'.)
We'll first recall why these properties are characteristic.
@@ -110,32 +110,32 @@
\intertext{and}
\cP_*(F_j) & \xrightarrow{\cP_0(F_j) \onto H_0(\cP_*(F_j))} \coinv(F_j).
\end{align*}
-The cone of each chain map is acyclic. In the first case, this is because the `rows' indexed by $i$ are acyclic since $HC_i$ is exact.
+The cone of each chain map is acyclic. In the first case, this is because the `rows' indexed by $i$ are acyclic since $\HC_i$ is exact.
In the second case, this is because the `columns' indexed by $j$ are acyclic, since $F_j$ is free.
Because the cones are acyclic, the chain maps are quasi-isomorphisms. Composing one with the inverse of the other, we obtain the desired quasi-isomorphism
$$\cP_*(M) \quismto \coinv(F_*).$$
%If $M$ is free, that is, a direct sum of copies of
%$C \tensor C$, then properties \ref{item:hochschild-additive} and
-%\ref{item:hochschild-free} determine $HC_*(M)$. Otherwise, choose some
+%\ref{item:hochschild-free} determine $\HC_*(M)$. Otherwise, choose some
%free cover $F \onto M$, and define $K$ to be this map's kernel. Thus we
%have a short exact sequence $0 \to K \into F \onto M \to 0$, and hence a
-%short exact sequence of complexes $0 \to HC_*(K) \into HC_*(F) \onto HC_*(M)
+%short exact sequence of complexes $0 \to \HC_*(K) \into \HC_*(F) \onto \HC_*(M)
%\to 0$. Such a sequence gives a long exact sequence on homology
%\begin{equation*}
%%\begin{split}
-%\cdots \to HH_{i+1}(F) \to HH_{i+1}(M) \to HH_i(K) \to HH_i(F) \to \cdots % \\
-%%\cdots \to HH_1(F) \to HH_1(M) \to HH_0(K) \to HH_0(F) \to HH_0(M).
+%\cdots \to \HH_{i+1}(F) \to \HH_{i+1}(M) \to \HH_i(K) \to \HH_i(F) \to \cdots % \\
+%%\cdots \to \HH_1(F) \to \HH_1(M) \to \HH_0(K) \to \HH_0(F) \to \HH_0(M).
%%\end{split}
%\end{equation*}
-%For any $i \geq 1$, $HH_{i+1}(F) = HH_i(F) = 0$, by properties
+%For any $i \geq 1$, $\HH_{i+1}(F) = \HH_i(F) = 0$, by properties
%\ref{item:hochschild-additive} and \ref{item:hochschild-free}, and so
-%$HH_{i+1}(M) \iso HH_i(F)$. For $i=0$, \todo{}.
+%$\HH_{i+1}(M) \iso \HH_i(F)$. For $i=0$, \todo{}.
%
%This tells us how to
-%compute every homology group of $HC_*(M)$; we already know $HH_0(M)$
+%compute every homology group of $\HC_*(M)$; we already know $\HH_0(M)$
%(it's just coinvariants, by property \ref{item:hochschild-coinvariants}),
-%and higher homology groups are determined by lower ones in $HC_*(K)$, and
+%and higher homology groups are determined by lower ones in $\HC_*(K)$, and
%hence recursively as coinvariants of some other bimodule.
Proposition \ref{prop:hoch} then follows from the following lemmas, establishing that $K_*$ has precisely these required properties.
@@ -390,7 +390,7 @@
\medskip
For purposes of illustration, we describe an explicit chain map
-$HC_*(M) \to K_*(M)$
+$\HC_*(M) \to K_*(M)$
between the Hochschild complex and the blob complex (with bimodule point)
for degree $\le 2$.
This map can be completed to a homotopy equivalence, though we will not prove that here.
@@ -398,7 +398,7 @@
Describing the extension to higher degrees is straightforward but tedious.
\nn{but probably we should include the general case in a future version of this paper}
-Recall that in low degrees $HC_*(M)$ is
+Recall that in low degrees $\HC_*(M)$ is
\[
\cdots \stackrel{\bd}{\to} M \otimes C\otimes C \stackrel{\bd}{\to}
M \otimes C \stackrel{\bd}{\to} M