The Stacks project

Remark 97.21.9 (Canonical automorphism). Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $\mathcal{X}$ satisfies condition (RS*). Let $A$ be an $S$-algebra such that $\mathop{\mathrm{Spec}}(A) \to S$ maps into an affine open and let $x, y$ be objects of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(A)$. Further, let $A \to B$ be a ring map and let $\alpha : x|_{\mathop{\mathrm{Spec}}(B)} \to y|_{\mathop{\mathrm{Spec}}(B)}$ be a morphism of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(B)$. Consider the ring map

\[ B \longrightarrow B[\Omega _{B/A}], \quad b \longmapsto (b, \text{d}_{B/A}(b)) \]

Pulling back $\alpha $ along the corresponding morphism $\mathop{\mathrm{Spec}}(B[\Omega _{B/A}]) \to \mathop{\mathrm{Spec}}(B)$ we obtain a morphism $\alpha _{can}$ between the pullbacks of $x$ and $y$ over $B[\Omega _{B/A}]$. On the other hand, we can pullback $\alpha $ by the morphism $\mathop{\mathrm{Spec}}(B[\Omega _{B/A}]) \to \mathop{\mathrm{Spec}}(B)$ corresponding to the injection of $B$ into the first summand of $B[\Omega _{B/A}]$. By the discussion of Remark 97.21.4 we can take the difference

\[ \varphi (x, y, \alpha ) = \alpha _{can} - \alpha |_{\mathop{\mathrm{Spec}}(B[\Omega _{B/A}])} \in \text{Inf}_{x|_{\mathop{\mathrm{Spec}}(B)}}(\Omega _{B/A}). \]

We will call this the canonical automorphism. It depends on all the ingredients $A$, $x$, $y$, $A \to B$ and $\alpha $.


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