Proposition 19.2.12 (Change of algebras). Let \(C\) be a monoidal \(\infty \)-category which admits geometric realizations and whose tensor product preserves geometric realizations separately in both variables. For every morphism \(f\colon A\to B\) in \(\Alg (C)\), the restriction functor from Proposition 19.1.16 \[ f_*\colon \LMod _B(C)\longrightarrow \LMod _A(C) \] admits a left adjoint given by extension of scalars: \[ f^*:=B\otimes _A-\colon \LMod _A(C)\longrightarrow \LMod _B(C). \] Analogously, restriction from right \(B\)-modules to right \(A\)-modules admits the left adjoint \(-\otimes _A B\).
Proof. The morphism \(f\) makes every left \(B\)-module into a left \(A\)-module and makes \(B\) into a \((B,A)\)-bimodule. For a left \(A\)-module \(M\) and a left \(B\)-module \(N\), an \(A\)-linear morphism \(g\colon M\to f_*N\) induces the \(B\)-linear composite \[ B\otimes _A M\xrightarrow {\id _B\otimes _A g}B\otimes _A f_*N\longrightarrow N, \] where the final map is induced by the \(B\)-action on \(N\). Conversely, precomposition with \[ M\iso A\otimes _A M\xrightarrow {f\otimes _A\id _M}B\otimes _A M \] turns every \(B\)-linear morphism \(B\otimes _A M\to N\) into an \(A\)-linear morphism \(M\to f_*N\). These assignments arise from the displayed natural transformations, so they define maps between the corresponding mapping animae. Unitality and associativity of relative tensor products show that these maps are inverse. The argument for right modules is analogous. □
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