Download e-book for kindle: Cohen-Macaulay representations by Graham J. Leuschke

By Graham J. Leuschke

ISBN-10: 0821875817

ISBN-13: 9780821875810

This booklet is a accomplished therapy of the illustration concept of maximal Cohen-Macaulay (MCM) modules over neighborhood earrings. This subject is on the intersection of commutative algebra, singularity thought, and representations of teams and algebras. introductory chapters deal with the Krull-Remak-Schmidt Theorem on area of expertise of direct-sum decompositions and its failure for modules over neighborhood jewelry. Chapters 3-10 examine the primary challenge of classifying the jewelry with in basic terms finitely many indecomposable MCM modules as much as isomorphism, i.e., jewelry of finite CM sort. the basic material--ADE/simple singularities, the double branched disguise, Auslander-Reiten thought, and the Brauer-Thrall conjectures--is lined basically and entirely. a lot of the content material hasn't ever earlier than seemed in publication shape. Examples comprise the illustration thought of Artinian pairs and Burban-Drozd's comparable development in size , an advent to the McKay correspondence from the perspective of maximal Cohen-Macaulay modules, Auslander-Buchweitz's MCM approximation idea, and a cautious therapy of nonzero attribute. the rest seven chapters current effects on bounded and countable CM sort and at the illustration conception of absolutely reflexive modules

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For the converse, choose a Bourbaki sequence 0 −→ G −→ N −→ L −→ 0 over R, so that G is a free R-module and L is an ideal of R. Then cl(L) = cl(N ), and since cl(N ) is in the image of Φ there is a divisorial ideal I of R such that R ⊗R I ∼ = L∗∗ . Set V = L∗∗ /L. 7(i) and the short exact sequence 0 −→ L −→ L∗∗ −→ V −→ 0 that L is extended. Moreover, Rp is a discrete valuation ring for each height-one prime ideal p, so that Ext1R (L, G) has finite length. 7(ii) says that N is extended since G and L are.

Let R be the m-adic completion of R, and write R ⊗R (n ) (n ) M = V1 1 ⊕ · · · ⊕ Vt t , where each Vi is an indecomposable R-module (a ) and each ni > 0. 9. Thus we have a well-defined map j : + (M ) −→ Nt0 , taking [L] to (a1 , . . , at ). 15). If [L] ∈ +(M ) and j([L]) is a minimal non-zero element of j(+(M )), then L is clearly indecomposable. Conversely, if [L] ∈ add(M ) and L is indecomposable, we claim that j([L]) is a minimal non-zero element of j(+(M )). For, suppose that j([X]) < j([L]), where [X] ∈ +(M ) is non-zero.

Since S is a module-finite extension of R contained in the total quotient ring K, the conductor contains a non-zerodivisor (clear denominators), so that the bottom line A = (A →B) is an Artinian pair in the sense of Chapter 3. Suppose that M is a MCM R-module. Then M is torsion-free, so the natural map M −→ K ⊗R M is injective. Let SM be the S-submodule of K ⊗R M generated by the image of M ; equivalently, SM = (S ⊗R M )/torsion. If we furthermore assume that SM is a projective S-module, then the inclusion M/cM →SM/cM gives a module over the Artinian pair A →B.

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Cohen-Macaulay representations by Graham J. Leuschke

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