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The Stacks project

Lemma 38.39.2. Let p be a prime number. Consider an almost blowup square X, X', Z, E in characteristic p as in Example 38.37.10. Then the functor

\mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(X) \longrightarrow \mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(Z) \times _{\mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(E)} \mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(X')

is an equivalence.

Proof. Let A, f, J be as in Example 38.37.10. Since all our schemes are affine and since we have internal Hom in the category of vector bundles, the fully faithfulness of the functor follows if we can show that

\mathop{\mathrm{colim}}\nolimits P \otimes _{A, F^ N} A = \mathop{\mathrm{colim}}\nolimits P \otimes _{A, F^ N} A/J \times _{\mathop{\mathrm{colim}}\nolimits P \otimes _{A, F^ N} A/fA + J} \mathop{\mathrm{colim}}\nolimits P \otimes _{A, F^ N} A/fA

for a finite projective A-module P. After writing P as a summand of a finite free module, this follows from the case where P is finite free. This case immediately reduces to the case P = A. The case P = A follows from Lemma 38.38.2 (in fact we proved this case directly in the proof of this lemma).

Essential surjectivity. Here we obtain the following algebra problem. Suppose P_1 is a finite projective A/J-module, P_2 is a finite projective A/fA-module, and

\varphi : P_1 \otimes _{A/J} A/fA + J \longrightarrow P_2 \otimes _{A/fA} A/fA + J

is an isomorphism. Goal: show that there exists an N, a finite projective A-module P, an isomorphism \varphi _1 : P \otimes _ A A/J \to P_1 \otimes _{A/J, F^ N} A/J, and an isomorphism \varphi _2 : P \otimes _ A A/fA \to P_2 \otimes _{A/fA, F^ N} A/fA compatible with \varphi in an obvious manner. This can be seen as follows. First, observe that

A/(J \cap fA) = A/J \times _{A/fA + J} A/fA

Hence by More on Algebra, Lemma 15.6.9 there is a finite projective module P' over A/(J \cap fA) which comes with isomorphisms \varphi '_1 : P' \otimes _ A A/J \to P_1 and \varphi _2 : P' \otimes _ A A/fA \to P_2 compatible with \varphi . Since J is a finitely generated ideal and f-power torsion we see that J \cap fA is a nilpotent ideal. Hence for some N there is a factorization

A \xrightarrow {\alpha } A/(J \cap fA) \xrightarrow {\beta } A

of F^ N. Setting P = P' \otimes _{A/(J \cap fA), \beta } A we conclude. \square


Comments (2)

Comment #5254 by Janos Kollar on

A small comment. The statement refers to "X,X′,Z,E in characteristic p as in Example 0EVG" but the Example has no X,X', Z, E in it. Same happens in the next Lemma. best János


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