blob: comparison text/hochschild.tex

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 Thus we can, if we choose, restrict the blob twig labels to things of this form.
 \end{itemize}
 We want to show that $\bc_*(S^1)$ is homotopy equivalent to the
 Hochschild complex of $C$.
-(Note that both complexes are free (and hence projective), so it suffices to show that they
+Note that both complexes are free (and hence projective), so it suffices to show that they
-are quasi-isomorphic.)
+are quasi-isomorphic.
 In order to prove this we will need to extend the blob complex to allow points to also
 be labeled by elements of $C$-$C$-bimodules.
 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$.
 We define a blob-like complex $K_*(S^1, (p_i), (M_i))$.
 $C$-$C$-bimodule, not a category) is quasi-isomorphic to the blob complex
 $\bc_*(S^1; C)$. (Proof later.)
 \end{lem}
 Next, we show that for any $C$-$C$-bimodule $M$,
-\begin{prop}
+\begin{prop} \label{prop:hoch}
 The complex $K_*(M)$ is quasi-isomorphic to $HC_*(M)$, the usual
 Hochschild complex of $M$.
 \end{prop}
 \begin{proof}
 Recall that the usual Hochschild complex of $M$ is uniquely determined,
 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 $M$, $\coinv(M) =
 M/\langle cm-mc \rangle$.
 \item \label{item:hochschild-free}%
-$HC_*(C\otimes C)$ (the free $C$-$C$-bimodule with one generator) is contractible; that is,
+$HC_*(C\otimes C)$ 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 $HC_0 \onto HH_0$.
 \end{enumerate}
 (Together, these just say that Hochschild homology is `the derived functor of coinvariants'.)
 We'll first recall why these properties are characteristic.
 %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
 %hence recursively as coinvariants of some other bimodule.
-The proposition then follows from the following lemmas, establishing that $K_*$ has precisely these required properties.
+Proposition \ref{prop:hoch} then follows from the following lemmas, establishing that $K_*$ has precisely these required properties.
 \begin{lem}
 \label{lem:hochschild-additive}%
 Directly from the definition, $K_*(M_1 \oplus M_2) \cong K_*(M_1) \oplus K_*(M_2)$.
 \end{lem}
 \begin{lem}
 We define a left inverse $s: \bc_*(S^1) \to K_*(C)$ to the inclusion as follows.
 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if
 * is a labeled point in $y$.
 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *.
-Let $x \in \bc_*(S^1)$.
+Extending linearly, we get the desired map $s: \bc_*(S^1) \to K_*(C)$.
-Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in
+%Let $x \in \bc_*(S^1)$.
-$x$ with $s(y)$.
+%Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in
+%$x$ with $s(y)$.
 It is easy to check that $s$ is a chain map and $s \circ i = \id$.
-Let $L_*^\ep \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points
+Let $N_\ep$ denote the ball of radius $\ep$ around *.
-in a neighborhood $B_\ep$ of $*$, except perhaps $*$, and $B_\ep$ is either disjoint from or contained in every blob.
+Let $L_*^\ep \sub \bc_*(S^1)$ be the subcomplex
+spanned by blob diagrams
+where there are no labeled points
+in $N_\ep$, except perhaps $*$, and $N_\ep$ is either disjoint from or contained in
+every blob in the diagram.
 Note that for any chain $x \in \bc_*(S^1)$, $x \in L_*^\ep$ for sufficiently small $\ep$.
 We define a degree $1$ chain map $j_\ep: L_*^\ep \to L_*^\ep$ as follows. Let $x \in L_*^\ep$ be a blob diagram.
-If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding $B_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction
+If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding $N_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction
-of $x$ to $B_\ep$. If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$,
+of $x$ to $N_\ep$. If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$,
-write $y_i$ for the restriction of $z_i$ to $B_\ep$, and let
+write $y_i$ for the restriction of $z_i$ to $N_\ep$, and let
-$x_i$ be equal to $x$ on $S^1 \setmin B$, equal to $z_i$ on $B \setmin B_\ep$,
+$x_i$ be equal to $x$ on $S^1 \setmin B$, equal to $z_i$ on $B \setmin N_\ep$,
-and have an additional blob $B_\ep$ with label $y_i - s(y_i)$.
+and have an additional blob $N_\ep$ with label $y_i - s(y_i)$.
 Define $j_\ep(x) = \sum x_i$.
 \todo{need to check signs coming from blob complex differential}
 \todo{finish this}
 \end{proof}
 \begin{proof}[Proof of Lemma \ref{lem:hochschild-exact}]
 Let $K'_* \sub K_*(C\otimes C)$ be the subcomplex where the label of
 the point $*$ is $1 \otimes 1 \in C\otimes C$.
 We will show that the inclusion $i: K'_* \to K_*(C\otimes C)$ is a quasi-isomorphism.
 Fix a small $\ep > 0$.
-Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$.
+Let $N_\ep$ be the ball of radius $\ep$ around $* \in S^1$.
 Let $K_*^\ep \sub K_*(C\otimes C)$ be the subcomplex
-generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from
+generated by blob diagrams $b$ such that $N_\ep$ is either disjoint from
-or contained in each blob of $b$, and the only labeled point inside $B_\ep$ is $*$.
+or contained in each blob of $b$, and the only labeled point inside $N_\ep$ is $*$.
-%and the two boundary points of $B_\ep$ are not labeled points of $b$.
+%and the two boundary points of $N_\ep$ are not labeled points of $b$.
-For a field $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$
+For a field $y$ on $N_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$
 labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$.
-(See Figure \ref{fig:sy}.) Note that $y - s_\ep(y) \in U(B_\ep)$. We can think of
+(See Figure \ref{fig:sy}.) Note that $y - s_\ep(y) \in U(N_\ep)$. We can think of
-$\sigma_\ep$ as a chain map $K_*^\ep \to K_*^\ep$ given by replacing the restriction $y$ to $B_\ep$ of each field
+$\sigma_\ep$ as a chain map $K_*^\ep \to K_*^\ep$ given by replacing the restriction $y$ to $N_\ep$ of each field
 appearing in an element of  $K_*^\ep$ with $s_\ep(y)$.
 Note that $\sigma_\ep(x) \in K'_*$.
 \begin{figure}[!ht]
 \begin{align*}
 y & = \mathfig{0.2}{hochschild/y} &
 \label{fig:sy}
 \end{figure}
 Define a degree 1 chain map $j_\ep : K_*^\ep \to K_*^\ep$ as follows.
 Let $x \in K_*^\ep$ be a blob diagram.
-If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to
+If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $N_\ep$ to
-$x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$.
+$x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $N_\ep$.
 If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows.
-Let $y_i$ be the restriction of $z_i$ to $B_\ep$.
+Let $y_i$ be the restriction of $z_i$ to $N_\ep$.
-Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$,
+Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin N_\ep$,
-and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$.
+and have an additional blob $N_\ep$ with label $y_i - s_\ep(y_i)$.
 Define $j_\ep(x) = \sum x_i$.
 \nn{need to check signs coming from blob complex differential}
 Note that if $x \in K'_* \cap K_*^\ep$ then $j_\ep(x) \in K'_*$ also.
 The key property of $j_\ep$ is

changeset 66	58707c93f5e7
parent 48	b7ade62bea27
child 68	4f2ea5eabc8f