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Let $R$ be a commutative ring with 1 and $A, B$ be $R$-algebras. If $N$ is a $B$-module and $\phi:A\to B$ is an $R$-algebra homomorphism, then $N$ admits as $A$-module structure via $\phi$. Now we can easily check that there exists an exact sequence $$ 0\to \mathrm{Der}_{A}(B, N)\to \mathrm{Der}_{R}(B, N) \xrightarrow{\phi^{*}} \mathrm{Der}_{R}(A, N) $$ where $\mathrm{Der}_{R}(B, N)$ is a set of $R$-linear dervations $D:B\to N$ and others are defined in the similar way. $\phi^{*}$ is a pullback map $D\mapsto D\circ \phi$.

My question is: is it possible to complete the sequence in the following way: $$ 0\to \mathrm{Der}_{A}(B, N)\to \mathrm{Der}_{R}(B, N) \xrightarrow{\phi^{*}} \mathrm{Der}_{R}(A, N) \\ \,\,\to \mathrm{Der}_{A}^{1}(B, N) \to \mathrm{Der}_{R}^{1}(B, N) \xrightarrow{\phi^{*}} \mathrm{Der}^{1}_{R}(A, N) \\ \,\,\to \mathrm{Der}_{A}^{2}(B, N) \to \mathrm{Der}_{R}^{2}(B, N) \xrightarrow{\phi^{*}} \mathrm{Der}^{2}_{R}(A, N) \\ \cdots $$ For some nice groups $\mathrm{Der}^{1}, \mathrm{Der}^{2}, ...$? Here nice means that it satiesfies some functoriality.

Seewoo Lee
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Yes, this is the theory of André-Quillen cohomology.

JHF
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  • And in the non-commutative setting, it is just Hochschild cohomology. More generally, deriving the functor of derivations (which exists for algebras over operads) gives operadic cohomology shifted by one degree, as proved by Reszk in his thesis (under certain reasonable hypotheses). – Pedro Sep 03 '18 at 10:19