Simplicial modules

There is a chain complex associated to simplicial module

$$M_{\bullet}: \quad\hdots\to M_n\xrightarrow{b_n} M_{n-1} \xrightarrow{b_{n-1}} M_{n-1}\to \hdots$$

where $b=b_n=\sum_{i=0}^n (-1)^i d_i$. We have $b^2=0$ as an immediate consequence of $d_id_j=d_{j-1}d_i$, $i<j$, for example:

$$\hdots M_2\xrightarrow{d_0-d_1+d_2} M_1\xrightarrow{d_0-d_1} M_0$$

$$(d_0-d_1)(d_0-d_1+d_2)=\underbrace{d_0d_0-d_0d_1}_{0}+ \underbrace{d_0d_2-d_1d_0}_{0}+\underbrace{d_1d_1-d_1d_2}_{0}=0$$

We define the homology of a simplicial module as

$$\rH_n(M_{\bullet}):=\ker(b_n)/ \im(b_{n-1})$$

It is well defined for presimplicial module, that is using only face maps.



Define the normalized complex $\Bar{M}_{\bullet}$ as a quotient

$$0\to M'_{\bullet} \to M_{\bullet} \to \Bar{M}_{\bullet}\to 0$$

Let $A$ be the $k$-algebra and $M$ an $A$-module. There is a simplicial module

$$C_{\bullet}(A, M):=M\ox A^{\ox n}$$

$$\begin{align*} d_i(a_0, a_1, \hdots, a_n) &= (a_0, \hdots, a_i a_{i+1}, \hdots, ... ..., a_1, \hdots, a_n) &= (a_0, \hdots, a_j, 1, a_{j+1}, \hdots, a_n ) \end{align*}$$
Define

$$b:=\sum_{i=0}^n (-1)^i d_i$$

Then $(C_{\bullet}(A, M), b)$ is called the Hochschild chain complex, and its homology $\rH_*(A; M)$ the Hochschild homology of $A$ with coefficients in $M$. If $M=A$, then we denote

$$\rH_*(A; A)=: \HH_*(A)$$

Suppose that $A$ is augmented and let $\Bar{A}$ be its augmentation ideal in $A$, that is $A=\Bar{A}\oplus k1$. Define the reduced Hochschild complex as

$$\Bar{C}_n(A, M):= M\ox \Bar{A}^{\ox n}$$

If $M=A=\bar{A}\oplus k1$, then $C_{\bullet}(A, A)$ has extra degeneracy

$$s_{-1}(a_0, \hdots, a_n)=(1, a_0, \hdots, a_n).$$

We have
$$\begin{align*} d_0(1, a_1, \hdots, a_n) &= (a_1, \hdots, a_n) \\ d_n(1, a_1, \hdots, a_n) &= (a_n, \hdots, a_1) \end{align*}$$
Define also two maps on $\Bar{A}^{\ox n}$
$$\begin{align*} t(a_1, \hdots, a_n) &:= (-1)^n(a_n, a_1, \hdots, a_{n-1})\\ b' &:= \sum_{i=0}^{n-1}(-1)^i d_i, \quad (b= b'+ (-1)^n d_n) \end{align*}$$