Lemma 46.3.20 (06VH)—The Stacks project

Processing math: 100%

Table of contents
Part 3: Topics in Scheme Theory
Chapter 46: Adequate Modules
Section 46.3: Adequate functors
Lemma 46.3.20 (cite)

previous lemma

numberstags

history
statistics
1 tag refers to this tag

Lemma 46.3.20. Let $A \to A'$ be a ring map and let $F$ be a module-valued functor on $\textit{Alg}_ A$ such that

the restriction $F'$ of $F$ to the category of $A'$ -algebras is adequate, and
for any $A$ -algebra $B$ the sequence

$0 \to F(B) \to F(B \otimes _ A A') \to F(B \otimes _ A A' \otimes _ A A')$

is exact.

Then $F$ is adequate.

Proof. The functors $B \to F(B \otimes _ A A')$ and $B \mapsto F(B \otimes _ A A' \otimes _ A A')$ are adequate, see Lemmas 46.3.18 and 46.3.17. Hence $F$ as a kernel of a map of adequate functors is adequate, see Lemma 46.3.11. $\square$

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