By Graham J. Leuschke

ISBN-10: 0821875817

ISBN-13: 9780821875810

This publication is a accomplished therapy of the illustration thought of maximal Cohen-Macaulay (MCM) modules over neighborhood earrings. This subject is on the intersection of commutative algebra, singularity idea, and representations of teams and algebras. introductory chapters deal with the Krull-Remak-Schmidt Theorem on strong point of direct-sum decompositions and its failure for modules over neighborhood jewelry. Chapters 3-10 examine the significant challenge of classifying the earrings with purely finitely many indecomposable MCM modules as much as isomorphism, i.e., jewelry of finite CM kind. the elemental material--ADE/simple singularities, the double branched hide, Auslander-Reiten thought, and the Brauer-Thrall conjectures--is coated essentially and fully. a lot of the content material hasn't ever sooner than seemed in ebook shape. Examples contain the illustration thought of Artinian pairs and Burban-Drozd's similar development in size , an advent to the McKay correspondence from the viewpoint of maximal Cohen-Macaulay modules, Auslander-Buchweitz's MCM approximation thought, and a cautious remedy 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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**Extra info for Cohen-Macaulay representations**

**Example text**

4) B = 0, A = Q, A((t − u)I + H) = HA . §2. 2) that A has to be invertible. 4) gives a contradiction, since the left side is invertible and the right side is not. Thus (Vt → W) ∼ = (Vu → W ) if t = u. To see that (Vt → W ) is indecomposable, we take u = t and suppose that ϕ, as above, is idempotent. 2), and comparing “1” and “A” terms, we see that A2 = A and P = AP + P A. 4) says that AH = HA, and it follows that A is in k[H], which is a local ring. Therefore A = 0 or I, and either possibility forces P = 0.

Then A is a two-dimensional normal domain, by [Har77, Chap. 4(b)]. By [Har77, Appendix B, Sect. 5], Pic0 (V ) ∼ = D := 0 1 2g (R/Z) , where g = 2 (d−1)(d−2), the genus of V . Here Pic (V ) is the kernel of the degree map Pic(V ) −→ Z, so Cl(V ) = Pic(V ) = D ⊕ Zσ, where σ is the class of a divisor of degree 1. There is a short exact sequence 0 −→ Z −→ Cl(V ) −→ Cl(A) −→ 0 , in which 1 ∈ Z maps to the divisor class τ := [H · V ], where H is a line in P2C . (Cf. [Har77, Chap. ) Thus Cl(A) ∼ = Cl(V )/Zτ .

Cf. 4). 14. The following lemma, whose proof is left as an exercise, is useful in treating the remaining case, when every element of D satisﬁes a monic quadratic equation over k: 36 3. 17. Lemma. Let be a ﬁeld, and let A be a ﬁnite-dimensional algebra with dim (A) 3. Assume that {1, α, α2 } is linearly dependent over for every α ∈ A. Write A = A1 × · · · × As , where each Ai is local, with maximal ideal mi . Let N = m1 × · · · × ms , the nilradical of A. (i) (ii) (iii) (iv) If x ∈ N, then x2 = 0.

### Cohen-Macaulay representations by Graham J. Leuschke

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