Functoriality and coefficient long exact sequences for Hochschild homology

Let $f:M\to N$ be a $k$-central $A$-bimodule map. In degree $n\ge 1$ set $C_n(f)=f\otimes 1_{A^{{\otimes }_{k}n}}$, and set $C_0(f)=f$. For the first face, $f(m{a}_1)=f(m)a_1$; for each internal face the map on the coefficient factor does not alter the multiplied algebra entries; for the last face, $f(a_{n}m)=a_{n}f(m)$. Thus $C(f)$ commutes with every face and with every boundary, including $b_0=0$, so it is a chain map. The identity bimodule map gives the identity chain map, and $C(g\circ f)=C(g)\circ C(f)$. By [F6] the induced homology maps obey the same identities. Hence $M\mapsto H{H}_n(A,M)$ is a covariant functor.

The maps just defined agree with the coefficient maps under the preceding Tor comparison. On an elementary bar tensor, $C_n(f)((a_{n+1}m{a}_0)\otimes a_1\otimes \cdots \otimes a_n)=(a_{n+1}f(m)a_0)\otimes a_1\otimes \cdots \otimes a_n,$ which is the image of $(a_0\otimes \cdots \otimes a_{n+1})\otimes f(m)$ under the comparison for $N$. The equality uses that $f$ is a bimodule map. Since elementary tensors span, the comparison square commutes; [F12] therefore identifies the induced Hochschild map with the natural map on Tor. This compatibility uses the completed preceding theorem and adds no projectivity hypothesis on $M$.

For $n\ge 0$ put $V_n=A^{{\otimes }_{k}n}$, with $V_0=k$. By [F3] and [F4], choose bases $B_n$ for this set-indexed family of vector spaces; take $B_0=\{1\}$. Each $V_n$ is then a free, hence flat, $k$-module by [F5]. For any exact sequence of $k$-modules $0\to X^{\mathrm{\prime }}\to X\to X^{\mathrm{\prime }\mathrm{\prime }}\to 0$, tensoring with $V_n$ is exact: using the chosen basis, the tensor sequence identifies with the direct sum over $B_n$ of copies of the original sequence. In particular, for each $n\ge 0$, $0⟶C_n(A,M^{\mathrm{\prime }})⟶C_n(A,M)⟶C_n(A,M^{\mathrm{\prime }\mathrm{\prime }})⟶0$ is exact. At $n=0$, this is the original coefficient sequence under $C_0(A,M)=M$. For $n<0$ all three chain groups are zero.

The inclusions and quotient map in the coefficient sequence are $A$-bimodule maps. By step 1.1, their maps on every chain degree commute with the Hochschild boundaries. By [F8], the degreewise exact sequences in step 1.3, with the chain maps checked in step 1.1, form a short exact sequence of chain complexes.

By [F7] the category of $k$-modules is abelian; apply [F10] to the short exact sequence of complexes from step 2.1, which qualifies by [F8]. This gives, in each degree $n\ge 1$, $\cdots \to H{H}_n(A,M^{\mathrm{\prime }})\to H{H}_n(A,M)\to H{H}_n(A,M^{\mathrm{\prime }\mathrm{\prime }})\stackrel{{\mathrm{\partial }}_n}{\to }H{H}_{n-1}(A,M^{\mathrm{\prime }})\to H{H}_{n-1}(A,M)\to \cdots \mathrm{.}$ At the lower endpoint $C_{-1}=0$, so the sequence ends as $H{H}_0(A,M^{\mathrm{\prime }})\to H{H}_0(A,M)\to H{H}_0(A,M^{\mathrm{\prime }\mathrm{\prime }})\to 0.$ This is the asserted long exact sequence.

A morphism between two short exact sequences of $k$-central $A$-bimodules induces in each degree the corresponding morphism between the short exact sequences of Hochschild chains: the vertical maps are the tensor maps of step 1.1, and commute with the differentials there. By [F9] this is a morphism of short exact sequences of complexes; [F11] makes the square for the homology connecting maps commute. The maps at all other positions are the functorial homology maps of step 1.1, so the entire long exact sequence is natural in the coefficient sequence.

If a coefficient module is zero, all its chain groups and homology groups are zero. A zero bimodule map induces the zero chain and homology maps, while an identity map induces identities; the composition check in step 1.1 covers all composites. As a unit-case check, when $A=k$ the maps in [F13] identify $C_n(k,M)\cong M$ and every face is the identity, so $b_n=0$ for odd $n$ and $b_n=1_M$ for positive even $n$. Thus $H{H}_0(k,M)=M$ and $H{H}_j(k,M)=0$ for $j>0$; the coefficient long exact sequence reduces to the original short exact sequence in degree zero and zeros in positive degrees. No iff claim occurs. AC is used through [F12] for the Tor comparison in step 1.2 and in step 1.3 to supply bases for the tensor powers; the chain-map and connecting-map constructions are choice-free.