Remark 97.21.3 (Functoriality). Assumptions and notation as in Lemma 97.21.2. Suppose $A \to B$ is a ring map and $y = x|_{\mathop{\mathrm{Spec}}(B)}$. Let $M \in \text{Mod}_ A$, $N \in \text{Mod}_ B$ and let $M \to N$ an $A$-linear map. Then there are canonical maps $\text{Inf}_ x(M) \to \text{Inf}_ y(N)$ and $T_ x(M) \to T_ y(N)$ simply because there is a pullback functor

$\textit{Lift}(x, A[M]) \to \textit{Lift}(y, B[N])$

coming from the ring map $A[M] \to B[N]$. Similarly, given a morphism of deformation situations $(y, B' \to B) \to (x, A' \to A)$ we obtain a pullback functor $\textit{Lift}(x, A') \to \textit{Lift}(y, B')$. Since the construction of the action, the addition, and the scalar multiplication on $\text{Inf}_ x$ and $T_ x$ use only morphisms in the categories of lifts (see proof of Formal Deformation Theory, Lemma 89.11.4) we see that the constructions above are functorial. In other words we obtain $A$-linear maps

$\text{Inf}_ x(M) \to \text{Inf}_ y(N) \quad \text{and}\quad T_ x(M) \to T_ y(N)$

such that the diagrams

$\vcenter { \xymatrix{ \text{Inf}_ y(J) \ar[r] & \text{Inf}(y'/y) \\ \text{Inf}_ x(I) \ar[r] \ar[u] & \text{Inf}(x'/x) \ar[u] } } \quad \text{and}\quad \vcenter { \xymatrix{ T_ y(J) \times \text{Lift}(y, B') \ar[r] & \text{Lift}(y, B') \\ T_ x(I) \times \text{Lift}(x, A') \ar[r] \ar[u] & \text{Lift}(x, A') \ar[u] } }$

commute. Here $I = \mathop{\mathrm{Ker}}(A' \to A)$, $J = \mathop{\mathrm{Ker}}(B' \to B)$, $x'$ is a lift of $x$ to $A'$ (which may not always exist) and $y' = x'|_{\mathop{\mathrm{Spec}}(B')}$.

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