Where does one go to learn about DG-algebras?

The theory of differential graded algebras (in char 0) and their modules has numerous applications in rational homotopy theory as well as algebraic geometry.

I'm looking for a reasonably complete reference book/collection of articles that treat the basic algebraic and homotopical aspects of dg-algebras akin to Atiyah-MacDonald's book on commutative algebra.

Hopefully containing the following topics

1. Model structures on dg algebras and simplicial algebras (presenting the "correct" infinity category) and relation between them (concrete description of fibrations and cofibrations).
2. Resolutions and minimal models
3. Localizations and Completions
4. Analogs of the classical types of morphisms (flat, smooth, unramified, etale, open immersion, closed embedding, finite, finite type, etc.).
5. Derived categories of modules (Triangulated / Stable $\infty$) and functors on them (push-pull-shriek functors, perfect complexes vs. bounded vs. unbounded derived categories, dualizing complex, cotangent complex, localization and completion, fourier mukai transforms).
6. Derived descent theorems - descent for dg-modules in the derived categories. (hopefully making it clear what kind of information goes into glueing a dg-module).
7. Hochshild and Cyclic (co-)homology
8. DG lie algebras and Koszul duality

I realize that one can always go to Lurie's books for a comprehensive and much more general treatment but I hope that a more elementary reference exists. I do believe that some aspects of this theory are a lot less technical than they appear.

• A good idea is to look into Keller's ICM paper. – Sasha Jan 24 '17 at 16:30

3 Answers

I am not aware of a single book that has all of these topics, but I can list a couple of very good books that can do the job together. First, Benoit Fresse works in this setting quite a bit. His book Modules over Operads and Functors contains material on (1), (2), (5), (7), and (8). The model categorical work is very explicit, including a description of (co)fibrations as you want. I also think I learned the most about Koszul duality from this source. It does not focus on the triangulated structure for (5). A good reference for that is the paper Stable Model Categories are Categories of Modules. Fresse also does not focus a ton on (2), but Loday and Vallette do in their book Algebraic Operads. This is also a great source for (1), (7), and (8). The model categorical work goes back to Hinich.

For (3), I recommend More Concise Algebraic Topology, by May and Ponto. They also have a discussion of (1).

For (4) and (6), and the more algebraic parts of (5) (e.g. Fourier-Mukai), I recommend Axiomatic, Enriched and Motivic Homotopy Theory, edited by John Greenlees, especially Strickland's article about axiomatic homotopy theory. Related is the monograph Axiomatic Stable Homotopy Theory, by Hovey, Palmieri, Strickland. I confess, this is the part of your question I know the least about. Strickland, however, has written in these sources about the connection between the items you list and homotopy theory. Others may have written more on the subject (e.g. Kaledin), and I hope someone can come along and add a comment if they know of a source.

There is a study of DG algebra (DG rings, DG modules, DG categories, DG functors) in the book below. With the related derived categories, etc.

The Stacks Project at Columbia University hosts a colloborative web-based survey on algebraic stacks, including a survey of Differential Graded Algebras. The whole project is quite an impressive resource.

There is an MSE posting from last year that lists a variety of sources that target more specific aspects, including: Differential Graded Algebras and Applications, for DG Lie algebras and Koszul duality.