I want to read about graded rings and modules. First, I saw Bruns-Herzog. But it was difficult for a beginner. Then I saw notes of Tom Marley (see Tom Marley's Homepage: Graded rings and modules). It is good. but it has a lot of exercises.
What other references do you know about graded rings? Specially I want to have a intuitive look about it, and know its applications in commutative algebra and algebraic geometry.

Thank you very much.

• What's wrong with lots of exercises? (Also, which Marley?) – darij grinberg Nov 13 '15 at 22:52
• is it "GRADED RINGS AND MODULES" by "Tom Marley"? – user 1 Nov 14 '15 at 6:51

The book *"Graded Syzygies"* by *"Irena Peeva"*.

It starts with a grading on the polynomial ring. So it is good for a beginner. Preface of the book is expressive :

The main goal of the book is to inspire the readers and develop their intuition about syzygies and Hilbert functions. Research on free resolutions and Hilbert functions is a core and beautiful area in Commutative Algebra. Many examples are given in order to illustrate and develop ideas and key concepts. The book contains open problems and conjectures...

It depends a lot on your CA background. However, you can't go wrong with Eisenbud's Comm Algebra with a View to AG. It is a great introduction, graded modules are treated well, and you see a lot of that "intuitive" outlook you are craving. His follow up book Geom. of Syzygies really shows a bunch of common techniques when working with graded modules (keeping track of the grading, that sort of thing). Of course, both have a lot of exercises, but at the end of the day, they're just needed. Hope that helps.

• A glance on OP's "math.stackexchange" profile suggest that (s)he is studying Bruns-Herzog and combinatorics. BTW you can add any other reference that can help other people. +1 – user 1 Nov 16 '15 at 11:08

The following two books treat the basic theory of graded rings and modules and several applications thereof, but rather of algebraic than geometric nature.

C. Nastasescu, F. Van Oystaeyen, Graded Ring Theory, North-Holland, 1982.

C. Nastasescu, F. Van Oystaeyen, Methods of Graded Rings, Springer, 2004.

For geometrical meaning of (algebraic) graded structures, the best I know is the work by Miles Reid : 1) Graded rings and birational geometry (PDF), 2) Graded rings and varieties in weighted projective space (PDF), 3) Graded rings and Fano 3-Folds (an introduction to Tom and Jerry !) : video 68 mn (on line ). May I add that there are two short books , easy to study and to understand, by Miles Reid : Undergraduate Commutative Algebra, and Undergraduate Algebraic Geometry (London Mathematical Society, Student Texts).