Example 1

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

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

Let K be a commutative field. Let G be a group, and let K[G] be the algebra of the group G. We identify G with its canonical image in K[G] (III, §2, No. 6, p. 446).

We endow K[G] with the structure of a bigebra with coproduct $c$ and counit $\gamma$ given by

c(g) =g\otimes g ,\gamma (g) = 1(g\in G) \tag{15}

Let E and F be K[G]-modules; by formula (12), the K[G]-module structure on $E\otimes F$ is given by

g(x\otimes y) =gx\otimes gy(g\in G, x\in E, y\in F) \tag{16}

The K[G]-module $K_{\gamma}$ is the vector space K endowed with the action of G defined by $g\lambda =\lambda$ for $g\in G$ and $\lambda \in K$ .

The ring $R_K(K[G])$ is also denoted by $R_K(G)$ . It is commutative; its multiplication is given by $[E] [F] = [E\otimes_KF]$ , and the unit element of $R_K(G)$ is $[K_{\gamma}]$ .

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.

Cited by

Cites

III, §2, No. 6, p. 446