Corollary

Tag 00DK, permanent. Cite it as [Bourbaki, Tag 00DK].

Algebra, VIII, p. 198, § 11. Grothendieck Groups

Let A be a bigebra with counit $\gamma$ over a commutative field K. Then there exists a unique ring structure on the additive group $R_K(A)$ whose multiplication satisfies

[E]_{\mathscr{C}}[F]_{\mathscr{C}}= [E\otimes_KF]_{\mathscr{C}}

for all A-modules E and F that are finite-dimensional over K. The unit element of $R_K(A)$ is $[K_{\gamma}]_{\mathscr{C}}$ . If the bigebra A is cocommutative, then the ring $R_K(A)$ is commutative.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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