Consider the following
Here $A$ is a flat (commutative, unital) $k$-algebra ($k$ a commutative ring) and $\mu:A\otimes_k A\rightarrow A$ is by $\mu(a\otimes b)=ab$, $\mathcal{M}$ denotes a maximal ideal of $A$. I am asking for a (explicit) description of this "natural map $\theta_n$".
It seems to me that the homomorphism in question is the following (from Bourbaki's algèbre homologique, page 110): given a morphism of rings $A\rightarrow B$ and $A$-modules $E,F$ then one has a natural map $B\otimes_A\mathrm{Tor}^A(E,F)\rightarrow\mathrm{Tor}^B(E\otimes_A B,B\otimes_A F)$. So in my case I take $A=A\otimes_k A$, $B=(A\otimes_k A)_{\mu^{-1}(\mathcal{M})}$, $E=A$, $F=M$. Now we have for any $(A,A)$-bimodule $M$ (i.e. $A\otimes A$-module) an isomorphism $M\otimes_{A\otimes A}(A\otimes A)_{\mu^{-1}(\mathcal{M})}=M_{\mu^{-1}(\mathcal{M})}$ as $(A\otimes_k A)_{\mu^{-1}(\mathcal{M})}$-modules. Is there some known isomorphism which allows me to identify this with $M_{\mathcal{M}}$? Because in that case I have what I want.
By the way, by the "classical homology argument" he means that a functorial isomorphism in degree 0 between two universal delta functors extends to functorial isomorphisms in all degrees, isn't it?