Lemma 2

Tag 0E2Q, permanent. Cite it as [Bourbaki, Tag 0E2Q].

Algebra, chapter X, § 9. Complexes de Koszul

If the k-module $L_0$ is free with basis $(e_i)_{i \in I}$ , the k-linear mappings g and h are bijective. This follows from III, p. 79, th. 1.

Let $M$ be a $k$ -module, and let $x = (x_i)_{i \in I}$ be a family of $k$ -endomorphisms of $M$ , pairwise permutable. Consider the polynomial ring $A = k[(X_i)_{i \in I}]$ and endow $M$ with the structure of an $A$ -module such that $P(X_i) m = P(x_i) m$ for $P \in A$ and $m \in M$ . Let moreover $L$ be the free $A$ -module $A^I$ , $(e_i)_{i \in I}$ its canonical basis and $u : L \to A$ the linear form which maps $e_i$ onto $X_i$ for all $i \in I$ . Consider the complexes of $k$ -modules $K_A^*(u, M)$ and $K_A^*(u, M)$ ; one has canonical isomorphisms

K_A^p(u, M) = \mathrm{Hom}_A(\Lambda_A^p(A^I), M) \to \mathrm{Hom}_k(\Lambda_k^p(k^I), M) ,

M \otimes_k \Lambda_k^p(k^I) \to M \otimes_A \Lambda_A^p(A^I) = K_A^p(u, M) ;

whence by composition with the isomorphisms

g

and

h

isomorphisms of k-modules

\theta^p : K_A^p(u, M) \to C_I^p(M) ,

\theta_p : C_I^p(M) \to K_A^p(u, M) .

We denote by

\partial^p : C_I^p(M) \to C_I^{p+1}(M) ,

\partial_p : C_I^p(M) \to C_I^{p-1}(M) ,

the

k

-homomorphisms obtained by transporting the differentials of

K_A^*(u, M)

and

K_A^*(u, M)

by the isomorphisms

\theta

. For example, one has: (12)

(\partial^p m)(\alpha_1, ..., \alpha_{p+1}) = \sum_{j=1}^{p+1} (-1)^{j+1} x_{\alpha_j} m(\alpha_1, ..., \alpha_{j-1}, \alpha_{j+1}, ..., \alpha_{p+1}) .

The complex formed by the

C_I^p(M)

and the

\partial^p

(resp. the

\partial_p

) is denoted by

K^*(x, M)

(resp.

K.(x, M)

) and is called the ascending (resp. descending) Koszul complex associated with the module M and the sequence of endomorphisms

(x_1, ..., x_n)

. One therefore has isomorphisms of complexes of

k

-modules

\theta^*: K_A^*(u, M) \to K^*(x, M),

\theta_*: K_*(x, M) \to K_A^*(u, M).

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

Cited by

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Cites

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