Commutative Coherent Rings by Sarah Glaz

By Sarah Glaz

This booklet offers the 1st broad and systematic remedy of the idea of commutative coherent earrings. It blends, and gives a hyperlink, among the 2 occasionally disjoint techniques on hand within the literature, the hoop theoretic process, and the homological algebra technique. The booklet covers so much leads to commutative coherent ring idea identified to this point, in addition to a few effects by no means released ahead of. beginning with ordinary effects, the ebook advances to themes akin to: uniform coherence, commonplace jewelry, jewelry of small homological dimensions, polynomial and tool sequence earrings, staff earrings and symmetric algebra over coherent earrings. the topic of coherence is dropped at the frontiers of analysis, exposing the open difficulties within the box. so much issues are taken care of of their absolutely generality, deriving the implications on coherent earrings as conclusions of the final conception. therefore, the ebook develops the various instruments of recent learn in commutative algebra with numerous examples and counterexamples. even supposing the booklet is largely self-contained, simple wisdom of commutative and homological algebra is usually recommended. It addresses graduate scholars and researchers.

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Coherent rings. and R a where a ~ ~, rings then and, many examples Noetherian To see (For limit x n] provides coherent. for . , polynomial R a module R , of R, let polynomial the flat and . of Ra module R a module of and of J as an R m o d u l e . ) a presentation satisfying Section R is of Xn]]. direct obtained flat J a of R a s a t i s f y i n g R be the a ideal ® RaRe) Xl, , is generated Noetherian~ R[[Xl, R system R~ i s then presentation ring S = I~[R Although is be directed % generated Set the R is Proof: ring that a finite Let over T = R[[Xl,X2~ which see ideal a finite a a ~ ~t every finitely generated w i t h R, we o b t a i n T = lin_~ for for if J is a f i n i t e l y R is R a flat, Assume that e S be ® R alii+m R~ = lim__+ ( I a finitely and~ { R ~a Suppose a is Fix and Let of hence, in such R a are R[x] rings, this, ring example is, nevertheless, without the "flatness" not R be one variable a ring see Ra[x] coherent.

1 0 . every module absolutely pure module N and L. Let R be a r i n g and l e t M be a f i n i t e l y g e n e r a t e d R module s a t i s f y i n g Ext~(M~N) = 0 f o r a l l a b s o l u t e l y pure R modules N~ t h e n M i s a f i n i t e l y p r e s e n t e d R module. Proof: finitely Let O--+K--+F-+M--+ generated the d i a g r a m and K free. ,F 0 be an e x a c t For an R m o d u l e can be c o m p l e t e d . sequence N, with ExtI(M,N) Thus, F = 0 iff it s u f f i c e s to p r o v e . J .

Flat K is a pure submodule of F. 5. presented R module~ (i) modules as projective. " (2) For every d i r e c t e d system of R modules {G j ~ ~ S we have: lim HomR(M,Ga) ~ HomR(_MM,Ii~ G ). (3) For every d i r e c t e d system of R modules { G ~ %m: s S the n a t u r a l map lim Ext~(M,G a) --+Ext~(M,lim G ) is injective for all n ~ i~ and is an isomorphism whenever Gn are submodules of a module G~ for all a and { G ~ a 6 S is ordered by inclusion. Proof: assertions All three finitely generated diagram chasing.

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