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For my bachelor thesis my goal is to understand the reasoning behind "Hilbert series" and how they connect to the idea of "dimension".

https://en.wikipedia.org/wiki/Hilbert_series_and_Hilbert_polynomial

Concepts like modules and graded algebras are rather unfamiliar to me. I have not encountered these before.

I am familiar with:

  • Rings, groups and fields, both from linear algebra and a standard course in abstract algebra.

  • Gröbner basis, monomials ideals. From a course on computational algebraic geometry.

I am looking for some books/notes that could help me understand the concepts that surround Hilbert Series, preferably an introductory book along with a recommendation for further reading. It seems that I am requesting an "introductory commutative algebra" book.

If anybody also knows a good place for video lectures, this would be greatly appreciated.

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    $\begingroup$ Chapter 1 of Eisenbud's "Commutative Algebra with a View Toward Algebraic Geometry" does a really great job of motivating and situating the "four fundamental theorems of Hilbert," in my opinion. I would read that, if you have not already. $\endgroup$
    – Sam Hopkins
    Commented Oct 23, 2019 at 23:08

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In addition to Eisenbud's book which has already been pointed out, I would like to suggest also the following references.

  • Chapter 11 of Atiyah-MacDonald's "Introduction to Commutative Algebra" develops the theory of dimension for graded and local rings and its relation with the Hilbert function. The prerequisites you mention should be enough for reading the book, but for Chapter 11 you may need notions developed in the previous chapters such as Noetherian rings and the length of a module.

  • Personally, I like a lot the approach of Bruns-Herzog's "Cohen-Macaulay Rings". Chapter 4 is dedicated to the Hilbert function. Although this book is more advanced and requires in several places a certain familiarity with homological algebra, I think that Chapter 4 is more accessible. So you may try to give a look also there.

  • Since you mentioned Gröbner bases, I recommend also Kreuzer-Robbiano's "Computational Commutative Algebra 2". They follow a more computational approach (as the title suggests) and they dedicate a very dense Chapter (the fifth) to Hilbert functions of graded and multigraded modules. The introduction of that chapter is nice and gives some simple examples of the geometric importance of the Hilbert function.

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  • $\begingroup$ I think this answer is very helpful, I will see which material suits my needs best. Thank you. $\endgroup$
    – user129588
    Commented Oct 25, 2019 at 19:37
  • $\begingroup$ You're welcome. I am glad I could be of help. $\endgroup$
    – Alessio
    Commented Oct 28, 2019 at 11:06
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I think for you specific purpose the most efficient and elementary presentation is Chapter 1 (in particular Section 1.3) of the new book "Invitation to Nonlinear Algebra" by Michałek and Sturmfels.

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  • $\begingroup$ Thank you, I will have a look at this too. $\endgroup$
    – user129588
    Commented Oct 25, 2019 at 19:36

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