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Cohen-Macaulay representations. (English) Zbl 1252.13001
Mathematical Surveys and Monographs 181. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-7581-0/hbk). xvii, 367 p. (2012).
The book under review is mainly concerned with the problem of the classification of Cohen-Macaulay local rings \((R,\, \mathfrak{m},\, k)\) having only a finite number of indecomposable maximal Cohen-Macaulay module (one says that such a ring has finite CM type). Compared with the canonical reference for the subject, namely the book of Y. Yoshino [Cohen-Macaulay modules over Cohen-Macaulay rings. London Mathematical Society Lecture Note Series. 146. Cambridge (UK): Cambridge University Press (1990; Zbl 0745.13003)], it contains a number of new results and topics. Among them, the elementary proof of C. Huneke and G. J. Leuschke [Math. Ann. 324, No. 2, 391–404 (2002; Zbl 1007.13005)] of the theorem of M. Auslander [Lect. Notes Math. 1178, 194–242 (1986; Zbl 0633.13007)] asserting that if \(R\) has finite CM type then \(R_{\mathfrak{p}}\) is a regular local ring, \(\forall \, \mathfrak{p} \in \text{Spec}\, R \setminus \{\mathfrak{m}\}\), and a proof, due to R. Wiegand [J. Algebra 203, No. 1, 156–168 (1998; Zbl 0921.13015)] and G. Leuschke and R. Wiegand [J. Algebra 228, No. 2, 674–681 (2000; Zbl 0963.13020)] of a conjecture of F.-O. Schreyer [Lect. Notes Math. 1273, 9–34 (1987; Zbl 0719.14024)] asserting that \(R\) has finite CM type if and only if its \(\mathfrak{m}\)-adic completion \(\widehat R\) has finite CM type. Actually, for the proof of the “only if” part of the later result, the authors have to assume that \(R\) is excellent (it is unknown whether this condition is necessary or not). Besides, the proof of the “only if” part depends on a difficult result of R. Elkik [Ann. Sci. Éc. Norm. Supér. (4) 6, 553–603 (1973; Zbl 0327.14001)] for which the authors refer to Elkik’s paper.
For the majority of the theorems contained in the book under review the authors provide complete and (essentially) self-contained proofs, preferring explicit constructions to categorical techniques. There are, however, a number of important results whose proofs depend on difficult facts for which the authors refer to the original papers (we saw an example above).
Here is a brief description of the content of the book.
The first two chapters are concerned with the proof of the Krull-Remak-Schmidt uniqueness theorem for \(R\) complete and with the analysis of how badly can it fail for a general \(R\).
Chapters 3 and 4 treat the case \(\text{dim}\, R \leq 1\) where, essentially, everything is known: if \(R\) has finite CM type and \(\text{dim}\, R = 0\) then \(R\) is a principal ideal ring, and if \(\text{dim}\, R = 1\) then it is characterized by the conditions of Y. A. Drozd and A. V. Roĭter [Izv. Akad. Nauk SSSR, Ser. Mat. 31, 783–798 (1967; Zbl 0164.04103)].
Chapters 5 to 7 are devoted to the classification of complete, local, 2-dimensional \(\mathbb C\)-algebras of finite CM type and of their relations with Invariant Theory, Kleinian singularities, ADE hypersurface singularities, McKay correspondence etc.
Chapters 8 and 9 contain the classification (due to H. Knörrer [Invent. Math. 88, No. 1, 153–164 (1987; Zbl 0617.14033)] and R.-O. Buchweitz, G.-M. Greuel and F.-O. Schreyer [Invent. Math. 88, No. 1, 165–182 (1987; Zbl 0617.14034)]) of equicharacteristic complete hypersurface rings having finite CM type. The main ingredients of the proof of this result are matrix factorizations, the double branched cover construction, and Knörrer’s periodicity theorem.
In Chapter 10 the authors prove F.-O. Schreyer’s conjecture mentioned at the beginning of the review. Chapter 11 contain a complete and explicit treatment of the theory of MCM approximations of M. Auslander and R.-O. Buchweitz [Mém. Soc. Math. Fr., Nouv. Sér. 38, 5–37 (1989; Zbl 0697.13005)].
Chapter 12 is concerned with totally reflexive modules, formerly known as modules of Gorenstein dimension 0, a notion introduced by M. Auslander and M. Bridger [Mem. Am. Math. Soc. 94, 146 p. (1969; Zbl 0204.36402)]. Over a Gorenstein local ring a finitely generated module is totally reflexive if and only if it is maximal Cohen-Macaulay. The main result of this chapter is due to L. W. Christensen, G. Piepmeyer, J. Striuli and R. Takahashi [Adv. Math. 218, No. 4, 1012–1026 (2008; Zbl 1148.14004)] and asserts that if \(R\) has at least one non-free totally reflexive module but only finitely many indecomposable ones then it must be a hypersurface ring of finite CM type.
Chapter 13 presents the Auslander-Reiten theory of almost split sequences and of the Auslander-Reiten quiver, cf. M. Auslander and I. Reiten [Adv. Math. 73, No. 1, 1–23 (1989; Zbl 0744.13003)]. The Auslander-Reiten quiver gives, in some sense, a picture of the whole category of MCM modules.
The last chapters, 14 to 17, consider other CM representation types, namely countable (it is obvious what this means) and, respectively, bounded (which is equivalent to the fact that there is a bound on the minimal number of generators of indecomposable MCM modules). The authors use recent results of I. Burban and Y. Drozd [in: Proceedings of the 12th international conference on representations of algebras and workshop (ICRA XII), Toruń, Poland, August 15–24, 2007. Zürich: European Mathematical Society (EMS). EMS Series of Congress Reports, 101–166 (2008; Zbl 1200.14011)] to prove the Buchweitz-Greuel-Schreyer classification of hypersurface rings with countable CM type. Chapter 15 contains a proof of the first Brauer-Thrall conjecture asserting that an excellent isolated singularity with bounded CM type has finite CM type. This proof follows the original proofs of E. Dieterich [Comment. Math. Helv. 62, No. 4, 654–676 (1987; Zbl 0654.14002)] and Y. Yoshino [J. Math. Soc. Japan 39, No. 4, 719–739 (1987; Zbl 0615.13008)].
The authors use, in Chapter 16, the Brauer-Thrall theorem to present the only two non-Gorenstein examples of dimension \(> 2\) of CM rings \(R\) with finite CM type. These examples appear in the paper of Auslander and Reiten quoted above.
Finally, in Chapter 17, the authors classify the complete CM local rings \(R\) of bounded CM type if either \(R\) is 1-dimensional or an equicharacteristic hypersurface ring.

13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
14B05 Singularities in algebraic geometry
14H20 Singularities of curves, local rings
14J17 Singularities of surfaces or higher-dimensional varieties
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
13C14 Cohen-Macaulay modules