# 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