We (Roger Wiegand and Graham Leuschke) are writing a book about maximal Cohen—Macaulay modules over Cohen—Macaulay local rings. We’ll be putting draft chapters up here as we get them to a presentable state, hoping that friendly passersby will read them and give us feedback.

We appreciate any feedback you’re willing to provide: read a chapter, or a section, or some pages, or some collection of results, or just make sure we’re not omitting an important reference. Contact either one of us:

Of course, we’d also like to hear if you like something you read.

Here is a list of chapters with some downloadable files. We’re working hard and posting updates regularly. See below for some notes on updates as they occur.

1. The Krull-Remak-Schmidt Theorem (9 pages, 2010–06–15)
   1. Krull-Schmidt in an additive category
   2. Krull-Schmidt over Henselian rings
   3. Exercises
2. Dimension Zero (12 pages, 2010–06–30)
   1. Artinian rings with finite Cohen-Macaulay type
   2. Artinian pairs
   3. Exercises
3. Dimension One (24 pages, 2010–06–30)
   1. Necessity of the Drozd-Roiter conditions
   2. Sufficiency of the Drozd-Roiter conditions
   3. Historical Remarks
   4. ADE singularities
   5. Multiplicity two
   6. The analytically ramified case
   7. Ranks of indecomposable MCM modules
   8. Exercises
4. Invariant Theory (23 pages, 2009–12–20)
   1. Group actions on power series rings
   2. The reconstruction algebra
   3. The McKay-Gabriel quiver
   4. Exercises
5. Kleinian Singularities and Finite Representation Type (42 pages, 2010–06–15)
   1. Invariant rings in dimension two
   2. Kleinian singularities
   3. McKay-Gabriel quivers of the Kleinian singularities
   4. Geometric McKay correspondence
   5. Exercises
6. Isolated Singularities and Classification in Dimension Two (21 pages, 2010–08–02)
   1. Criteria for isomorphism
   2. Miyata’s theorem
   3. Auslander’s theorem
   4. Classification in dimension two
   5. Exercises
7. The Double Branched Cover (14 pages, 2010–08–02)
   1. Matrix factorizations
   2. Sharping Things
   3. Characteristic 2
   4. Exercises
8. Hypersurfaces of Finite Cohen-Macaulay Type
   1. Herzog’s theorem
   2. Simple singularities
   3. Exercises
9. Auslander-Buchweitz Theory (43 pages, 2010–06–11)
   1. Canonical modules
   2. MCM approximations and FID hulls
   3. Numerical invariants
   4. The index and applications to finite Cohen-Macaulay type
   5. Exercises
10. Auslander-Reiten Theory (36 pages, 2010–08–02)
    1. AR sequences
    2. AR quivers
    3. Examples
    4. Exercises
11. Ascent and Descent (18 pages, 2010–03–22)
    1. Descent
    2. Ascent to the completion
    3. Ascent along separable field extensions
    4. Equicharacteristic Gorenstein singularities
    5. Exercises
12. Semigroups of Modules (16 pages, 2010–02–07)
    1. Realization theorem in dimension one
    2. Realization theorem in dimension two
    3. Exercises
13. Countable Cohen—Macaulay Type (36 pages, 2010–08–02)
    1. Structure
    2. Burban-Drozd pairs
    3. Hypersurfaces of countable CM type
    4. Other examples
    5. Exercises
14. The Brauer-Thrall Conjectures (28 pages, 2010–06–30)
    1. Efficient systems of parameters
    2. The Harada-Sai lemma
    3. Faithful systems of parameters
    4. Proof of Brauer-Thrall I
    5. Brauer-Thrall II in dimension one
    6. Exercises
15. Bounded Cohen-Macaulay Type (27 pages, 2010–02–08)
    1. Hypersurfaces
    2. Dimension one
    3. Ascent and descent
    4. Exercises
16. Tame and Wild Cohen-Macaulay Type
    1. Artinian algebras and pairs
    2. Curves
    3. Hypersurfaces
    4. The generic determinant
    5. Exercises

Appendices:

1. Basics (7 pages, 2010–03–19)
2. Proof of Necessity of the Drozd-Roiter Conditions (6 pages, 2010–02–07)
3. Ramification Theory (29 pages, 2010–08–02)

Backmatter:

• Bibliography (25 pages, 2010–07–15)
• Index

Changelog

2010–08–02

a busy busy July. New versions of “Isolated singularities…” (rewrote section 4), “Double branched cover” (cosmetic fixes), “Auslander-Reiten” (new section on dimension 2 and other examples), and “Countable” (rewritten completely after first section). Also, first posting of Appendix on Ramification.

2010–07–15

new version of Bibliography

2010–06–30

minor changes to Dim0, restructured Dim1, first posting of Double Branched Cover, first posting of Auslander-Reiten Theory, largely rewritten Brauer-Thrall

2010–06–15

first posting of Bibliography (been forgetting and forgetting), updated Krull-Remak-Schmidt chapter and Auslander-Buchweitz chapter, posted oldish versions of Dimension 0 and Kleinian chapters that hadn’t gotten posted.

2010–04–21

rewritten Auslander-Buchweitz chapter, now featuring complete proof of uniqueness

2010–03–23

new section for Ascent/Descent.

2010–03–22

Ascent/Descent rewritten, new Appendix on Basics, and first posting of Appendix on DR conditions.

2010–03–10

Kleinian chapter got minor repairs and a totally rewritten last section.

2010–02–26

Dimension 1 chapter rewritten, countable chapter got minor changes in “Other examples” section, Brauer-Thrall rewritten, initial posting of bounded type chapter and semigroups chapter

2010–02–17

Krull-Schmidt chapter rewritten

2010–01–26

Initial posting of Brauer-Thrall

2009–12–23

Initial posting of Krull-Schmidt, dimension zero, dimension one, invariant theory, Kleinian singularities, isolated singularities, Auslander-Buchweitz, ascent/descent, and countable.