Reduction results for Grothendieck groups

Abstract

15.1 Let k be an algebraically closed field of characteristic p and O the unramified complete discrete valuation ring of characteristic zero with residue field k; denote by O-mod the category of finitely generated O-modules. Let Ĝ be a finite k^* -group — we already have seen the interest of replacing finite groups by k^* -groups with a finite k^* -quotient G (cf. 1.23) — and b be a primitive idempotent of Z(k_*Ĝ) — called a block of Ĝ (cf. 1.25). Recall that, according to Proposition 5.15 in [42], b is also a block of a finite subgroup G′ of Ĝ fulfilling kG′b = k_*Ĝb and therefore all the terminology developed for G′ can be easily translated to Ĝ. Thus, let (P, e) be a maximal Brauer (b, Ĝ)-pair (cf. 1.16 and 1.28) and F=F_{(b, Ĝ)} the corresponding Frobenius P-category (cf. Theorem 3.7); recall that we denote by F^nc the full subcategory of F over the set of F-nilcentralized subgroups of P (cf. 11.1).