Multiplicity of $Ext^{d-t}(M,\omega_R)$, ($d=\dim R, t=\dim M$)

Question

Let $R=\bigoplus_{i \geq 0} R_i$ be a Cohen-Macaulay graded ring ($R_0$ is a field and $R$ is generated by $R_1$) of dimension $d$ with canonical module $\omega_R$, and $M$ a graded Cohen-Macaulay $R$-module of dimension $t$. Assume that we know the Hilbert series, Hilbert polynomial and all Betti numbers of $M$. What can be said about the multiplicity of $\textrm{Ext}^{d-t}(M,\omega_R)$? Can we compute it from the given data?

Answer 1

It is equal to the multiplicity of $M$. In fact, you do not need graded or even Cohen-Macaulayness of $M$. Let $N= \textrm{Ext}^{d-t}(M,\omega_R)$. Let $S(M) := \{P \in \textrm{Supp}(M), \dim R/P = t\}$. Then we have the so-called associativity formula:

$e(M) = \sum_{P \in S(M)} \textrm{length}_{R_P}(M_P)e(R/P)$

Now we just need to observe that $S(M) =S(N)$ and for each $P\in S(M)$, $\textrm{length}(M_P)= \textrm{length}(N_P)$. Both are consequence of Grothendieck Local Duality applied to the ring $R_P$.