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105 the free $C$-$C$-bimodule with one generator.) |
105 the free $C$-$C$-bimodule with one generator.) |
106 That is, $\HC_*(C\otimes C)$ is |
106 That is, $\HC_*(C\otimes C)$ is |
107 quasi-isomorphic to its $0$-th homology (which in turn, by \ref{item:hochschild-coinvariants} |
107 quasi-isomorphic to its $0$-th homology (which in turn, by \ref{item:hochschild-coinvariants} |
108 above, is just $C$) via the quotient map $\HC_0 \onto \HH_0$. |
108 above, is just $C$) via the quotient map $\HC_0 \onto \HH_0$. |
109 \end{enumerate} |
109 \end{enumerate} |
110 (Together, these just say that Hochschild homology is `the derived functor of coinvariants'.) |
110 (Together, these just say that Hochschild homology is ``the derived functor of coinvariants".) |
111 We'll first recall why these properties are characteristic. |
111 We'll first recall why these properties are characteristic. |
112 |
112 |
113 Take some $C$-$C$ bimodule $M$, and choose a free resolution |
113 Take some $C$-$C$ bimodule $M$, and choose a free resolution |
114 \begin{equation*} |
114 \begin{equation*} |
115 \cdots \to F_2 \xrightarrow{f_2} F_1 \xrightarrow{f_1} F_0. |
115 \cdots \to F_2 \xrightarrow{f_2} F_1 \xrightarrow{f_1} F_0. |
128 \cP_i(F_*) & \xrightarrow{\cP_i(\pi)} \cP_i(M) \\ |
128 \cP_i(F_*) & \xrightarrow{\cP_i(\pi)} \cP_i(M) \\ |
129 \intertext{and} |
129 \intertext{and} |
130 \cP_*(F_j) & \xrightarrow{\cP_0(F_j) \onto H_0(\cP_*(F_j))} \coinv(F_j). |
130 \cP_*(F_j) & \xrightarrow{\cP_0(F_j) \onto H_0(\cP_*(F_j))} \coinv(F_j). |
131 \end{align*} |
131 \end{align*} |
132 The cone of each chain map is acyclic. |
132 The cone of each chain map is acyclic. |
133 In the first case, this is because the `rows' indexed by $i$ are acyclic since $\cP_i$ is exact. |
133 In the first case, this is because the ``rows" indexed by $i$ are acyclic since $\cP_i$ is exact. |
134 In the second case, this is because the `columns' indexed by $j$ are acyclic, since $F_j$ is free. |
134 In the second case, this is because the ``columns" indexed by $j$ are acyclic, since $F_j$ is free. |
135 Because the cones are acyclic, the chain maps are quasi-isomorphisms. |
135 Because the cones are acyclic, the chain maps are quasi-isomorphisms. |
136 Composing one with the inverse of the other, we obtain the desired quasi-isomorphism |
136 Composing one with the inverse of the other, we obtain the desired quasi-isomorphism |
137 $$\cP_*(M) \quismto \coinv(F_*).$$ |
137 $$\cP_*(M) \quismto \coinv(F_*).$$ |
138 |
138 |
139 %If $M$ is free, that is, a direct sum of copies of |
139 %If $M$ is free, that is, a direct sum of copies of |