64 $\bc_*(S^1; C)$. (Proof later.) |
64 $\bc_*(S^1; C)$. (Proof later.) |
65 \end{lem} |
65 \end{lem} |
66 |
66 |
67 Next, we show that for any $C$-$C$-bimodule $M$, |
67 Next, we show that for any $C$-$C$-bimodule $M$, |
68 \begin{prop} \label{prop:hoch} |
68 \begin{prop} \label{prop:hoch} |
69 The complex $K_*(M)$ is quasi-isomorphic to $HC_*(M)$, the usual |
69 The complex $K_*(M)$ is quasi-isomorphic to $\HC_*(M)$, the usual |
70 Hochschild complex of $M$. |
70 Hochschild complex of $M$. |
71 \end{prop} |
71 \end{prop} |
72 \begin{proof} |
72 \begin{proof} |
73 Recall that the usual Hochschild complex of $M$ is uniquely determined, |
73 Recall that the usual Hochschild complex of $M$ is uniquely determined, |
74 up to quasi-isomorphism, by the following properties: |
74 up to quasi-isomorphism, by the following properties: |
75 \begin{enumerate} |
75 \begin{enumerate} |
76 \item \label{item:hochschild-additive}% |
76 \item \label{item:hochschild-additive}% |
77 $HC_*(M_1 \oplus M_2) \cong HC_*(M_1) \oplus HC_*(M_2)$. |
77 $\HC_*(M_1 \oplus M_2) \cong \HC_*(M_1) \oplus \HC_*(M_2)$. |
78 \item \label{item:hochschild-exact}% |
78 \item \label{item:hochschild-exact}% |
79 An exact sequence $0 \to M_1 \into M_2 \onto M_3 \to 0$ gives rise to an |
79 An exact sequence $0 \to M_1 \into M_2 \onto M_3 \to 0$ gives rise to an |
80 exact sequence $0 \to HC_*(M_1) \into HC_*(M_2) \onto HC_*(M_3) \to 0$. |
80 exact sequence $0 \to \HC_*(M_1) \into \HC_*(M_2) \onto \HC_*(M_3) \to 0$. |
81 \item \label{item:hochschild-coinvariants}% |
81 \item \label{item:hochschild-coinvariants}% |
82 $HH_0(M)$ is isomorphic to the coinvariants of $M$, $\coinv(M) = |
82 $\HH_0(M)$ is isomorphic to the coinvariants of $M$, $\coinv(M) = |
83 M/\langle cm-mc \rangle$. |
83 M/\langle cm-mc \rangle$. |
84 \item \label{item:hochschild-free}% |
84 \item \label{item:hochschild-free}% |
85 $HC_*(C\otimes C)$ is contractible. |
85 $\HC_*(C\otimes C)$ is contractible. |
86 (Here $C\otimes C$ denotes |
86 (Here $C\otimes C$ denotes |
87 the free $C$-$C$-bimodule with one generator.) |
87 the free $C$-$C$-bimodule with one generator.) |
88 That is, $HC_*(C\otimes C)$ is |
88 That is, $\HC_*(C\otimes C)$ is |
89 quasi-isomorphic to its $0$-th homology (which in turn, by \ref{item:hochschild-coinvariants}, is just $C$) via the quotient map $HC_0 \onto HH_0$. |
89 quasi-isomorphic to its $0$-th homology (which in turn, by \ref{item:hochschild-coinvariants}, is just $C$) via the quotient map $\HC_0 \onto \HH_0$. |
90 \end{enumerate} |
90 \end{enumerate} |
91 (Together, these just say that Hochschild homology is `the derived functor of coinvariants'.) |
91 (Together, these just say that Hochschild homology is `the derived functor of coinvariants'.) |
92 We'll first recall why these properties are characteristic. |
92 We'll first recall why these properties are characteristic. |
93 |
93 |
94 Take some $C$-$C$ bimodule $M$, and choose a free resolution |
94 Take some $C$-$C$ bimodule $M$, and choose a free resolution |
108 \begin{align*} |
108 \begin{align*} |
109 \cP_i(F_*) & \xrightarrow{\cP_i(\pi)} \cP_i(M) \\ |
109 \cP_i(F_*) & \xrightarrow{\cP_i(\pi)} \cP_i(M) \\ |
110 \intertext{and} |
110 \intertext{and} |
111 \cP_*(F_j) & \xrightarrow{\cP_0(F_j) \onto H_0(\cP_*(F_j))} \coinv(F_j). |
111 \cP_*(F_j) & \xrightarrow{\cP_0(F_j) \onto H_0(\cP_*(F_j))} \coinv(F_j). |
112 \end{align*} |
112 \end{align*} |
113 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. |
113 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. |
114 In the second case, this is because the `columns' indexed by $j$ are acyclic, since $F_j$ is free. |
114 In the second case, this is because the `columns' indexed by $j$ are acyclic, since $F_j$ is free. |
115 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 |
115 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 |
116 $$\cP_*(M) \quismto \coinv(F_*).$$ |
116 $$\cP_*(M) \quismto \coinv(F_*).$$ |
117 |
117 |
118 %If $M$ is free, that is, a direct sum of copies of |
118 %If $M$ is free, that is, a direct sum of copies of |
119 %$C \tensor C$, then properties \ref{item:hochschild-additive} and |
119 %$C \tensor C$, then properties \ref{item:hochschild-additive} and |
120 %\ref{item:hochschild-free} determine $HC_*(M)$. Otherwise, choose some |
120 %\ref{item:hochschild-free} determine $\HC_*(M)$. Otherwise, choose some |
121 %free cover $F \onto M$, and define $K$ to be this map's kernel. Thus we |
121 %free cover $F \onto M$, and define $K$ to be this map's kernel. Thus we |
122 %have a short exact sequence $0 \to K \into F \onto M \to 0$, and hence a |
122 %have a short exact sequence $0 \to K \into F \onto M \to 0$, and hence a |
123 %short exact sequence of complexes $0 \to HC_*(K) \into HC_*(F) \onto HC_*(M) |
123 %short exact sequence of complexes $0 \to \HC_*(K) \into \HC_*(F) \onto \HC_*(M) |
124 %\to 0$. Such a sequence gives a long exact sequence on homology |
124 %\to 0$. Such a sequence gives a long exact sequence on homology |
125 %\begin{equation*} |
125 %\begin{equation*} |
126 %%\begin{split} |
126 %%\begin{split} |
127 %\cdots \to HH_{i+1}(F) \to HH_{i+1}(M) \to HH_i(K) \to HH_i(F) \to \cdots % \\ |
127 %\cdots \to \HH_{i+1}(F) \to \HH_{i+1}(M) \to \HH_i(K) \to \HH_i(F) \to \cdots % \\ |
128 %%\cdots \to HH_1(F) \to HH_1(M) \to HH_0(K) \to HH_0(F) \to HH_0(M). |
128 %%\cdots \to \HH_1(F) \to \HH_1(M) \to \HH_0(K) \to \HH_0(F) \to \HH_0(M). |
129 %%\end{split} |
129 %%\end{split} |
130 %\end{equation*} |
130 %\end{equation*} |
131 %For any $i \geq 1$, $HH_{i+1}(F) = HH_i(F) = 0$, by properties |
131 %For any $i \geq 1$, $\HH_{i+1}(F) = \HH_i(F) = 0$, by properties |
132 %\ref{item:hochschild-additive} and \ref{item:hochschild-free}, and so |
132 %\ref{item:hochschild-additive} and \ref{item:hochschild-free}, and so |
133 %$HH_{i+1}(M) \iso HH_i(F)$. For $i=0$, \todo{}. |
133 %$\HH_{i+1}(M) \iso \HH_i(F)$. For $i=0$, \todo{}. |
134 % |
134 % |
135 %This tells us how to |
135 %This tells us how to |
136 %compute every homology group of $HC_*(M)$; we already know $HH_0(M)$ |
136 %compute every homology group of $\HC_*(M)$; we already know $\HH_0(M)$ |
137 %(it's just coinvariants, by property \ref{item:hochschild-coinvariants}), |
137 %(it's just coinvariants, by property \ref{item:hochschild-coinvariants}), |
138 %and higher homology groups are determined by lower ones in $HC_*(K)$, and |
138 %and higher homology groups are determined by lower ones in $\HC_*(K)$, and |
139 %hence recursively as coinvariants of some other bimodule. |
139 %hence recursively as coinvariants of some other bimodule. |
140 |
140 |
141 Proposition \ref{prop:hoch} then follows from the following lemmas, establishing that $K_*$ has precisely these required properties. |
141 Proposition \ref{prop:hoch} then follows from the following lemmas, establishing that $K_*$ has precisely these required properties. |
142 \begin{lem} |
142 \begin{lem} |
143 \label{lem:hochschild-additive}% |
143 \label{lem:hochschild-additive}% |
388 \end{proof} |
388 \end{proof} |
389 |
389 |
390 \medskip |
390 \medskip |
391 |
391 |
392 For purposes of illustration, we describe an explicit chain map |
392 For purposes of illustration, we describe an explicit chain map |
393 $HC_*(M) \to K_*(M)$ |
393 $\HC_*(M) \to K_*(M)$ |
394 between the Hochschild complex and the blob complex (with bimodule point) |
394 between the Hochschild complex and the blob complex (with bimodule point) |
395 for degree $\le 2$. |
395 for degree $\le 2$. |
396 This map can be completed to a homotopy equivalence, though we will not prove that here. |
396 This map can be completed to a homotopy equivalence, though we will not prove that here. |
397 There are of course many such maps; what we describe here is one of the simpler possibilities. |
397 There are of course many such maps; what we describe here is one of the simpler possibilities. |
398 Describing the extension to higher degrees is straightforward but tedious. |
398 Describing the extension to higher degrees is straightforward but tedious. |
399 \nn{but probably we should include the general case in a future version of this paper} |
399 \nn{but probably we should include the general case in a future version of this paper} |
400 |
400 |
401 Recall that in low degrees $HC_*(M)$ is |
401 Recall that in low degrees $\HC_*(M)$ is |
402 \[ |
402 \[ |
403 \cdots \stackrel{\bd}{\to} M \otimes C\otimes C \stackrel{\bd}{\to} |
403 \cdots \stackrel{\bd}{\to} M \otimes C\otimes C \stackrel{\bd}{\to} |
404 M \otimes C \stackrel{\bd}{\to} M |
404 M \otimes C \stackrel{\bd}{\to} M |
405 \] |
405 \] |
406 with |
406 with |