Cohen-Macaulay rings in GIT

Question

I'm starting to learn about Cohen-Macaulay rings in my commutative Algebra course, and my teacher mentioned its use in Geometric Invariant Theory. Having some basic knowledge on the subject and preparing my thesis on the subject, can anyone please clarify or give possibly some headlines about the role of CM rings in GIT ? that would help me smoothly orientate my research in the subject. Thanks

Answer 1

Cohen-Macaulayness is instrumental in GIT when it comes to counting invariants of a given degree.

Let $R=\bigoplus_{d\ge0}R_d$ be a finitely generated graded $\mathbb C$-algebra (for example a ring of invariants) such that all dimensions $a_d:=\dim R_d$ are finite dimensional. Then one would like to compute the generating function $\chi_R(t)=\sum_{d\ge0}a_dt^d$. For that one proceeds in two steps:

1. By the normalization theorem there exist algebraically independent homogeneous elements $x_1\in R_{e_1},\ldots,x_r\in R_{e_r}$ such that $R$ is a finitely generated module over $R_0:=\mathbb C[x_1,\ldots,x_r]$. These are called the primary invariants.
2. Now Cohen-Macaulyness comes in: $R$ is Cohen-Macaulay if and only if $R$ is a free $R_0$-module. In that case, there are homogeneous elements $y_1\in R_{f_1},\ldots,y_s\in R_{f_s}$ (the secondary invariants) which form an $R_0$-basis of $R$. Then $\chi_R$ can be computed as $$\chi_R(t)=\frac{t^{f_1}+\ldots+ t^{f_s}}{(1-t^{e_1})\cdots(1-t^{e_r})}$$

Cohen-Macaulayness of an invariant ring is the content of the celebrated Hochster-Roberts theorem. There exist very short proofs but all of them either use transcendental (Grauert-Riemenschneider) or characteristic $p$ (Frobenius) methods (at least, as far as I am aware).