Remark 3

Tag 0E2P, permanent. Cite it as [Bourbaki, Tag 0E2P].

Algebra, chapter X, § 9. Complexes de Koszul

Let G be a group and $\rho : G \to \mathbf{GL}(L)$ a linear representation of G in a flat A-module L. Then the $(\mathscr{E}_n)$ are exact sequences of linear representations. Suppose L finitely generated projective, and denote by $R_A(G)$ the ring of representations of G in finitely generated projective A-modules. It follows from prop. 3 that one has in R_A(G) the relations

\sum_{i=0}^{n} (-1)^i [\mathbf{S}^i(L)] [\Lambda^{n-i}(L)] = 0 , \quad n > 0 .

If one considers the formal series

s(T) = \sum_{i=0}^{\infty} [\mathbf{S}^i(L)] T^i \in R_A(G)[[T]] ,

\lambda(T) = \sum_{i=0}^{\infty} [\Lambda^i(L)] T^i \in R_A(G)[[T]] ,

the relations (10) are written

s(T) \lambda(-T) = 1_* .

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

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