Graded Commutative Algebra: A Comprehensive Reference

In summary, the conversation is about finding a good reference for commutative algebra of graded rings and modules. The individual has only found bits and pieces in other texts and wants to avoid diving fully into the theory of modules over preadditive categories. Another person recommends Eisenbud's Commutative Algebra, but it may not specifically address graded algebras.
  • #1
Hurkyl
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Is there a good reference for commutative algebra of graded rings and modules?

I've only found little bits and pieces in other texts (e.g. Hartshorne's Algebraic Geometry), and I would like to avoid having to dive fully into the theory of modules over preadditive categories! (And I'd prefer not to have to guess at the right way to generalize definitions & theorems from ordinary commutative algebra)
 
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  • #2
Modules over preadditive categories? I don't see that in Hartshorne. Are you talking about sheaves of modules?

Anyway, I recommend Eisenbud's Commutative Algebra. I don't know what you specifically want to know about graded algebras, but I'm sure you can find something useful in that book.
 
  • #3
masnevets said:
Modules over preadditive categories? I don't see that in Hartshorne.
Modules over preadditive categories was a separate thought from the Hartshorne bit. I do have a text on that, but it never specializes any results specifically to graded rings. (It doesn't even do much specialization to ordinary rings!)
 

What is graded commutative algebra?

Graded commutative algebra is a branch of mathematics that studies commutative rings with a grading structure. This means that the elements of the ring are assigned degrees or levels, and the multiplication between elements is required to preserve this grading.

What is the significance of graded commutative algebra?

Graded commutative algebra has many applications in algebraic geometry, topology, and representation theory. It allows for a better understanding of the structure of commutative rings and the behavior of their ideals and modules.

What are some examples of graded commutative algebra?

The polynomial ring in several variables, the coordinate ring of a projective variety, and the symmetric algebra of a vector space are all examples of graded commutative rings. The graded structure in these examples comes from the degrees of the variables or generators.

What are the main properties of graded commutative algebra?

The main properties of graded commutative algebra include the existence of a grading on the ring, the preservation of the grading under multiplication, and the existence of a homogeneous basis for the ring. In addition, graded commutative rings satisfy the graded versions of the commutative and associative laws.

How is graded commutative algebra related to other branches of mathematics?

Graded commutative algebra has connections to many areas of mathematics, such as algebraic geometry, topology, representation theory, and combinatorics. It also has applications in physics, particularly in the study of supersymmetry and quantum field theory.

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