Example 2

Tag 0DTV, permanent. Cite it as [Bourbaki, Tag 0DTV].

Suppose A commutative; let E be an A-module and u an endomorphism of E. Denote by $E_u$ the $A[X]$ -module obtained by endowing E with the structure defined by

(p, x) \mapsto p(u)(x) \quad \text{for} \quad p \in A[X] \quad \text{and} \quad x \in E.

By III, p. 106, one has an exact sequence:

0 \to A[X] \otimes_A E \xrightarrow{\psi} A[X] \otimes_A E \xrightarrow{\varphi} E_u \to 0

where $\varphi(p \otimes x) = p.x$ and $\psi(p \otimes x) = Xp \otimes x - p \otimes u(x)$ for $p \in A[X]$ and $x \in E$ . This exact sequence is a resolution of length 1 of $E_u$ , free (resp. projective, resp. of finite type) if E is a free (resp. projective, resp. of finite type) A-module.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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