By Carl Faith
VI of Oregon lectures in 1962, Bass gave simplified proofs of a few "Morita Theorems", incorporating rules of Chase and Schanuel. one of many Morita theorems characterizes whilst there's an equivalence of different types mod-A R::! mod-B for 2 jewelry A and B. Morita's answer organizes rules so successfully that the classical Wedderburn-Artin theorem is a straightforward outcome, and furthermore, a similarity type [AJ within the Brauer staff Br(k) of Azumaya algebras over a commutative ring ok contains all algebras B such that the corresponding different types mod-A and mod-B inclusive of k-linear morphisms are similar through a k-linear functor. (For fields, Br(k) involves similarity periods of straightforward significant algebras, and for arbitrary commutative okay, this can be subsumed less than the Azumaya 1 and Auslander-Goldman [60J Brauer crew. ) a number of different circumstances of a marriage of ring conception and classification (albeit a shot gun wedding!) are inside the textual content. additionally, in. my try and extra simplify proofs, significantly to cast off the necessity for tensor items in Bass's exposition, I exposed a vein of principles and new theorems mendacity wholely inside of ring idea. 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) instructed by means of the Morita context. As a spinoff, this gives starting place for a slightly entire conception of easy Noetherian rings-but extra approximately this within the advent.
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Additional resources for Algebra. Rings, modules and categories
Spec(R) is surjective and the R-algebras C ⌦ B and C n are isomorphic. 2. Let R be a ring, and let A be a degree-n extension of R that is ´etale as an R-algebra. Let G be a subgroup of Sn , and let (B, ') be a G-closure of A over R. Then B is an ´etale degree-|G| extension of R. Proof. 1 to obtain an R-algebra C such that Spec(C) ! Spec(R) is surjective and C ⌦ A ⇠ = Cn as R-algebras. 3, we find that C ⌦B naturally has the structure of being a G-closure of C ⌦ A over C. 3, we must have C ⌦ B ⇠ = C |G| .
Then isomorphism classes of G-closures of A over R correspond to R-algebra homomorphisms R[x1 , . . , xn ]G ! R sending ek (x) 7! sk . Proof. 2, isomorphism classes of G-closures of A over R correspond to R-algebra homomorphisms R[x1 , . . , xn ]G ! R such that ek (x) 7! sk and R ! R[x1 , . . ,xn ]G is injective. 3, since |G| is a non-zerodivisor the latter condition is guaranteed to hold. 4 An-closures In this section we give generators and relations for R[x1 , . . , xn ]An as an R[x1 , . .
R. Then the R-algebra homomorphism (A⌦n )G ! 1, the corresponding G-closure of A is B. 2, we need criteria for proving that R ! R[x1 , . . ,xn ]G is injective. 3. Let R be a ring, let n be a natural number, and let G be a finite group acting on an R-algebra A. Let AG ! R be an R-algebra homomorphism, and N construct the R-algebra homomorphism R ! A AG R. If |G| is a non-zerodivisor in R, then this homomorphism is injective. Proof. Consider the AG -module homomorphism A ! AG given by p 7! p. The composite AG !
Algebra. Rings, modules and categories by Carl Faith