Castelnouvo-Mumford Regularity

Question

I'm learning Castelnouvo-Mumford regularity of associated graded ring. As u see in 2 pics below, Lemma 3.3.

• $(A,\mathfrak{m})$ is a Noetherian ring local, $\dim(A)=1$; $\mathfrak{q}=(x)$ is a parameter ideal of $A$ ($\mathfrak{q}$ is a parameter ideal mean that $\mathfrak{q}$ is $\mathfrak{m}$-primary ideal of $A$ and generated by $n=\dim(A)$ element(s)).
• $G_\mathfrak{q}(A)=A/\mathfrak{q}\oplus \mathfrak{q}/\mathfrak{q^2} \oplus \mathfrak{q^2}/\mathfrak{q^3} \oplus \cdots$ is the associated graded ring of $A$ respect ideal $\mathfrak{q}$; $\operatorname{reg}(G_\mathfrak{q}(A))$ is the Castelnouvo-Mumford regularity of $G_\mathfrak{q}(A)$; $p(G_\mathfrak{q}(A))$ is the postulation number of $G_\mathfrak{q}(A)$.

In this proof, because $\operatorname{depth}(A) \leq \dim(A)$ and $\dim(A)=1$ so we have $\operatorname{depth}(A)=0$ or $\operatorname{depth}(A)=1$.

• If $\operatorname{depth}(A)=1$ we will prove that $\operatorname{reg}(G_\mathfrak{q}(A))=0$ and $p(G_\mathfrak{q}(A))=-1$
• If $\operatorname{depth}(A)=0$ we will prove that $\operatorname{reg}(G_\mathfrak{q}(A))=p(G_\mathfrak{q}(A))$

So my questions are:

1. Why $\oplus (x^n,L)/(x^{n+1},L)=\oplus (x^n)/(x^{n+1},x^nL)$? and same with $K$. Are these equations from a theorem,...? Moreover please explain about maps in the exact sequence.
2. Where do I can find the equations about $\operatorname{length}$ as you see in $H_{\mathfrak{q}}(n-1)$?
3. In this proof, with $M$ is a $A$-module, $I=(x)$ is a submodule of M can I replace $G_\mathfrak{q}(A)$ by $G_I(M)=\oplus I^nM/I^{n+1}M$ and we have: $\operatorname{reg}(G_I(M))=p(G_I(M))+\operatorname{depth}(M)$?

Thanks for your regarding!

Answer 1

1. Actually $(x^n,L)/(x^{n+1},L)$ is isomorphic to $(x^n)/(x^{n+1},x^nL)$. This isn't hard to prove: take the map $(x^n)\hookrightarrow (x^n,L)\to(x^n,L)/(x^{n+1},L)$ whose kernel is $(x^n)\cap(x^{n+1},L)$. Use that $L:x=L$ in order to prove that $(x^n)\cap(x^{n+1},L)=(x^{n+1},x^nL)$.
   Similar arguments work for $K$.
   About the exact sequence: the $n$th graded piece of $G_q(A)$ is $(x^n)/(x^{n+1})$ and we have a surjection $(x^n)/(x^{n+1})\to(x^n)/(x^{n+1},x^nL)$ whose kernel is $(x^{n+1},x^nL)/(x^{n+1})$.

2. The first equality is the definition, the second comes from the short exact sequence $0\to(x^n,L)/(x^n)\to A/(x^n)\to A/(x^n,L)\to 0$, for the third use the isomorphism theorems for rings, and the last is obvious.

3. I suppose that you want to extend Lemma 3.3 to modules. Well, first of all $x\in A$ should be $M$-regular and $\dim M=1$. Now remark that the regularity behaves well for modules, too (see my first comment below). Finally I think you have chances to generalize the lemma to modules, but I leave you to check all the details.