to view this structure is the following. One checks easily that for any cyclic $A$-bimodule $M_\#$, the restriction $j^*M_\# \in {\operatorname{Fun}}(\Delta^{opp},k)$ is canonically isomorphic to the simplicial $k$-vector space $M^\Delta_\#$ associated to the underlying $A$-bimodule $M$ as in . By adjunction, we have a natural map $$\tau_\#:j_!M^\Delta_\# \to M_\#.$$ Then $j_!M^\Delta_\#$ in this formula only depends on $M \in
A{\operatorname{\!-\sf bimod}}$, and all the structure maps which turn $M$ into the cyclic bimodule $M_\#$ are collected in the map $\tau_\#$.

We can now define cyclic homology with coefficients. The definition is rather tautological. We note that for any cyclic $A$-bimodule $M_\#$ – or in fact, for any $M_\# \in {\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$ – we can treat $M_\#$ as a cyclic vector space by forgetting the bimodule structure on its components $M_n$.

\[cycl.def\] The [*cyclic homology $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M_\#)$ with coefficients*]{} in a cyclic $A$-bimodule $M$ is equal to $H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Lambda,M_\#)$.

Of course, , being valid for any cyclic $k$-vector space, also applies to $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M_\#)$, so that we automatically get the whole package – the Connes’ exact sequence, the periodicity endomorphism, and the periodic cyclic homology $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M)$. By Lemma \[hoch\], $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M_\#)$ coincides with $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M)$ as defined in .

Gauss-Manin connection. {#getz}
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To illustate the usefulness of the notion of a cyclic bimodule, let us study the behavior of cyclic homology under deformations.

There are two types of deformation theory objects that one can study for an associative algebra $A$. The first is the notion of a [*square-zero extension*]{} of the algebra $A$ by a $A$-bimodule $M$. This is an associative algebra ${\