Hilbert series

Hilbert series (also called Poincare series) are important tools for studying algebra. They give a measure of size for certain vector spaces which have extra structure called a grading. In this video, we introduce the notion of graded vector spaces and their Hilbert series. We show how to compute this Hilbert series for polynomial rings with generators which are not necessarily in degree one. We also look at the ring of symmetric polynomials. We look at some basic properties of HIlbert series such as additivity and shifting which allows us to compute many Hilbert series. As an application, we look at the ordinary double point and show how Hilbert series can be used to identify its coordinate ring as a quotient of a polynomial ring.

[winatfail]

Thank you so much for this, this was a life saver.

[sachasinet]

Thank you Daniel, you made the evening of a physicist a bit less fastidious! :)

[GrapefruitGecko]

Your videos are always so helpful! Thank you! I am wondering if the Hilbert series is independent of the choice of grading of the vector space? If so, how would you show this?

[studentofspacetime]

Wonderful Daniel!
I have a question, suppose you show that two graded k-algebras have the same Hilbert series, is that enough to guarantee an isomorphism as graded k-algebras?

[wreynolds]

Towards the end you consider the polynomial ring k[u, v, w] but consider each generator as having degree 2. Thinking of k[u, v, w] as the coordinate ring of affine 3-space, do the degrees of the coordinates have any geometric significance/interpretation? Or is there no significance, perhaps because the ratio (2:2:2) is the same as the usual ratio (1:1:1)?

[generallecturesonmathemati]

Which book did you follow for these videos? I have Graded Syzygies by Irena Peeva

[sl]

Thank you Daniel! Around 11:00, I guess that x1 + x2 and x1x2 have algebraic relations or not depends on what the field you use?

[Orkiperson]

Ok so I love your stuff, I love learning me some math again. So this might be a really stupid question. But I don't see how you can make both the statements "we can make the elements of these variables on the left any degree we want" and "these groups have the same degree so we can show that they are injective maps and therefor this defines the kernel" in the 30-32min area. Would this imply that any group of elements can be defined as whatever degree you want and show it is therefor injective to, at the least, all other groups of degree an integer multiple of the original? The surjectivity could easily break down if you didn't do this right, and this might be both true and trivial, but it just struck me as significant and I wanted to clarify.