By Carl Faith
VI of Oregon lectures in 1962, Bass gave simplified proofs of a few "Morita Theorems", incorporating principles of Chase and Schanuel. one of many Morita theorems characterizes while there's an equivalence of different types mod-A R::! mod-B for 2 earrings A and B. Morita's resolution organizes principles so successfully that the classical Wedderburn-Artin theorem is an easy end result, and additionally, a similarity category [AJ within the Brauer crew Br(k) of Azumaya algebras over a commutative ring ok includes all algebras B such that the corresponding different types mod-A and mod-B including k-linear morphisms are an identical by means of a k-linear functor. (For fields, Br(k) includes similarity sessions of easy critical algebras, and for arbitrary commutative okay, this is often subsumed lower than the Azumaya 1 and Auslander-Goldman [60J Brauer team. ) various different cases of a marriage of ring concept and class (albeit a shot gun wedding!) are inside the textual content. additionally, in. my try and additional simplify proofs, significantly to dispose of the necessity for tensor items in Bass's exposition, I exposed a vein of rules and new theorems mendacity wholely inside of ring conception. This constitutes a lot of bankruptcy four -the Morita theorem is Theorem four. 29-and the foundation for it's a corre spondence theorem for projective modules (Theorem four. 7) urged through the Morita context. As a spinoff, this gives origin for a slightly entire idea of easy Noetherian rings-but extra approximately this within the creation.
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Extra resources for Algebra. Rings, modules and categories
Then the uniqueness of the decomposition follows from the existence of a progenerator which we will prove in this subsection. However, the case when R is a field is an easy case, and we record here the following basic results. The proof is by standard arguments. 17 (Dunkl–Opdam). Let R be a field of characteristic 0. E/ admits the eigenspace decomposition with respect to Eu. Further, the set of eigenvalues that appear in the eigenspace decomposition coincides with cE C Z 0 . Ä; h/-submodule and the eigenvalue cE does not appear in its eigenspace decomposition with respect to Eu.
E/ is a Proof. Let M ! N ! E/; N /. Fix 0 ¤ v 2 E. E/; M / ! v/, (ii) RW m ' E ˝C R, (iii) V m D 0. 14(2), we may choose m 2 M cE such that m satisfies (i) and (ii). Suppose that I r m ¤ 0 and I rC1 m D 0. V / ˝C E. Then we have cE 0 Ä a by M 2 O Äa and cE 0 D cE r cE . Thus, r D 0 by the assumption and (iii) is also satisfied. A similar argument shows the following. 20. Suppose that R is a local ring such that the residue field F contains C. Let E; E 0 2 Irr W be such that E 6< E 0 . E 0 // D 0.
P/ ! R=p/- mod. M / is a finitely generated projective R=pmodule. We also consider the following conditions. R=m/- mod ! R=m/- mod is a 0-faithful cover, for the maximal ideal m 2 Spec R. (V) Let K be the quotient field of R. Then A ˝R K is a split semisimple K-algebra. v; q/ and the above conditions are satisfied. 42]. 47. Let R be a regular local ring whose residue field contains C, A a module-finite projective R-algebra, P a finitely generated projective A-module such that the conditions (I)–(V) are satisfied.
Algebra. Rings, modules and categories by Carl Faith