By Carl Faith
VI of Oregon lectures in 1962, Bass gave simplified proofs of a couple of "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 principles so successfully that the classical Wedderburn-Artin theorem is a straightforward end result, and in addition, a similarity classification [AJ within the Brauer workforce Br(k) of Azumaya algebras over a commutative ring okay comprises all algebras B such that the corresponding different types mod-A and mod-B including k-linear morphisms are similar by means of a k-linear functor. (For fields, Br(k) includes similarity periods of straightforward significant algebras, and for arbitrary commutative okay, this is often subsumed below the Azumaya 1 and Auslander-Goldman [60J Brauer crew. ) a variety of different situations of a marriage of ring thought and class (albeit a shot gun wedding!) are inside the textual content. additionally, in. my try and additional simplify proofs, particularly to cast off the necessity for tensor items in Bass's exposition, I exposed a vein of rules and new theorems mendacity wholely inside ring concept. This constitutes a lot of bankruptcy four -the Morita theorem is Theorem four. 29-and the root for it's a corre spondence theorem for projective modules (Theorem four. 7) urged through the Morita context. As a derivative, this gives origin for a slightly entire idea of easy Noetherian rings-but extra approximately this within the advent.
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Extra resources for Algebra. Rings, modules and categories
I let f~S be the g r o u p of the q u a s i - characters of kA/kX'" w h o s e conductor is disjoint f r o m S. Let Z(~o), Z'(~0) be two extended Dirichlet series, both convergent s o m e where; let F, F' be the functions defined on B A by the Fourier series (without constant terms) with the s a m e coefficients as Z'(oj), respectivelY. ~,)(o(a)~(~)Zz~(~ -t) w h e r e notations are as in t h e o r e m , 2. T o say that the condition is n e c e s s a r y is m e r e l y to repeat the statement of t h e o r e m 2.
No a d d i t i o n a l then, 54 according to Chapter VI, §25, T V of Chapter VI, §24, but it is m o r e equation for Co, Co, ~' = ka(~)~'. >.. > p ~ in the f o r m e r case, and u v-1 -I -I -I -I u > P'I ' u > up'1 in the latter case, with P'I = qv Pl ' PZ = qv P2 T o simplify notations, w e will a s s u m e Pl / P2' similar, and the final conclusions are the s a m e , may therefore write uniquely replacing x w e can write by xw ,v oa(d)- 1 [Zn(~0) + in the case Co(XWUv) in the f o r m Pl = PZ" 9.
N o w , divisor VB, of d e g r e e in that lul = i, div(u) m u s t be equal to the d e g r e e u u c h o o s e a n idele m for e a c h positive such that ~ = div(m); in v i e w of w h a t has just b e e n said, w e h a v e I(f, e, t, co) = co(dft)-1 X; w h e r e the s u m c(m)co(m)f~(ed -lf-lmu)~(u)dX u is taken over all positive divisors and the integrals are taken over the g r o u p ~-~r ×. e. unless v occurring in ~ , St~ is disjoint f r o m ~. the T h e n the 51 same results give at once for sition.