Lemma 56.3.9 (0GP1)—The Stacks project

Loading web-font TeX/Math/Italic

Table of contents
Part 3: Topics in Scheme Theory
Chapter 56: Functors and Morphisms
Section 56.3: Functors between categories of modules
Lemma 56.3.9 (cite)

previous lemma
next remark

numberstags

history
statistics

Lemma 56.3.9. Let $R$ be a ring. Let $A$ and $B$ be $R$ -algebras. If

$F : \text{Mod}_ A \longrightarrow \text{Mod}_ B$

is an $R$ -linear equivalence of categories, then there exists an isomorphism $A \to B$ of $R$ -algebras and an invertible $B$ -module $L$ such that $F$ is isomorphic to the functor $M \mapsto (M \otimes _ A B) \otimes _ B L$ .

Proof. We get $A \to B$ and $L$ from Lemma 56.3.8. To finish the proof, we need to show that the $R$ -linearity of $F$ forces $A \to B$ to be an $R$ -algebra map. We omit the details. $\square$

Comments (0)

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment