Roger Wiegand and I are writing a book about maximal Cohen—Macaulay modules. 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: pick 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. At this early date, many of them are not available yet, and the ones that are available are a bit rough. We’re working hard, though, and hope to post updates regularly. See below for some notes on updates as they occur.

1. The Krull-Remak-Schmidt Theorem (9 pages, 2010–02–17)
   1. Krull-Schmidt in an additive category
   2. Krull-Schmidt over Henselian rings
   3. Exercises
2. Dimension Zero (11 pages)
   1. Artinian rings with finite Cohen-Macaulay type
   2. Artinian pairs
   3. Exercises
3. Dimension One (21 pages, 2010–02–26)
   1. Necessity of the Drozd-Roiter conditions
   2. The analytically unramified case
   3. Historical Remarks
   4. Ranks of indecomposable MCM modules
   5. ADE singularities
   6. The analytically ramified case
   7. Multiplicity two
   8. Exercises
4. Invariant Theory (23 pages)
   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 Δ (42 pages, 2010–03–10)
   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)
   1. Criteria for isomorphism
   2. Miyata’s theorem
   3. Auslander’s theorem
   4. Classification in dimension two
   5. Exercises
7. The Double Branched Cover
   1. Matrix factorizations
   2. Knörrer’s periodicity
   3. Exercises
8. Hypersurfaces of Finite Cohen-Macaulay Type
   1. Herzog’s theorem
   2. Simple singularities
   3. Exercises
9. Auslander-Buchweitz Theory (44 pages)
   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. Ascent and Descent of Finite Cohen-Macaulay Type (18 pages)
    1. Descent
    2. Ascent to the completion
    3. Ascent along separable field extensions
    4. Exercises
11. Semigroups of Modules (16 pages, 2010–02–07)
    1. Realization theorem in dimension one
    2. Realization theorem in dimension two
    3. Exercises
12. Auslander-Reiten Sequences
    1. Basic properties
    2. Existence of AR sequences
    3. Lifting property of AR sequences
    4. Duality and AR sequences
    5. AR quivers
    6. Exercises
13. Countable Cohen—Macaulay Type (30 pages, 2010–02–26)
    1. Structure
    2. Burban-Drozd pairs
    3. Hypersurfaces of countable CM type
    4. Other examples
    5. Exercises
14. Brauer-Thrall Theorems (31 pages, 2010–02–26)
    1. Efficient systems of parameters
    2. The Harada-Sai lemma
    3. Proof of Brauer-Thrall I
    4. Brauer-Thrall II in dimension one
    5. 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. Proof of Necessity of the Drozd-Roiter Conditions
2. Ramification Theory

Backmatter:

• References
• Index

Changelog

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.