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The authors reduce to the case of $R$ complete with infinite residue field and use them implicitly at a couple of places in the proof. This is a fairly standard practice. For instance, to assert that …

We use the fact that in an Artinian Gorenstein ring, any ideal contains the socle. The assumption tells us that the socle of $A$ is $\mathfrak m^2$, which is principal. Let $I\neq (0)$ be a non-maxima …

When $d\geq 3$, these are isolated hypersurface singularities of dimension at least $4$, so are UFD by the Grothendieck's local Lefschetz Theorem. When $d=2$ and the field has characteristic $0$, the …

For the equivalence you need two more assumptions: (a) $M$ to have finite projective dimension and (b) $R$ to be Gorenstein. (a) is implicitly required in the second condition, otherwise you don't hav …

Yes, for any ideal in a Noetherian local ring. See: this paper.

Here is a counter-example when $A$ is regular local of dimension $4$ and $n=3$, the first non-trivial case. Let $A = k[[x,y,z,t]]$, and $P$ be a prime ideal of height $3$, say $P=(x,y,z)$. Let $M=\Ome …

The answer is no. Because locally, you can have a module over $\Lambda$ which is mCM over $R$ but not free. There are examples of singular $R$ with a module $M$ such that $\Lambda= End_R(M)$ has finit …

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\ …

Very interesting question! I do not have a full answer, but here are a few cases when one can say something not completely trivial. I will assume as in your last paragraph that $X = {\rm Spec (S/(f))} …

A useful way to think about this issue is to consider $J=I^{(n)}$, the $n$-symbolic power of $I$, which by definition is the $I$-primary component of $I^n$. When $R$ is a polynomial rings over $\mat …

This is true if $A$ is local but fails in general. First, a counterexample. Let $A=\mathbb Z[X]$ and $M= A/p\oplus A/q$ with $p=(2)$, $q=(3,X)$. Since any maximal ideal $m$ of $A$ can not contain bo …

Edit: the following does not answer Ben's question. It gives an example of the subring fixed by $G_m$ being not CM, while the question asked about the subscheme of fixed points, see the comments for m …

An example was given in Section 3 of this paper by Cutkosky: "A new characterization of rational surface singularities." (The scheme $Z$ in the last page, which is a blow up of some $m$-primary ideal …

Hi Alex, I think this fails for $n=2$. Start with a polynomial ring $S$ and height $2$ prime $P$ such that $S/P$ is not Cohen-Macaulay (for example let $P$ be the kernel of the map $S=k[a,b,c,d] \to …