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