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This question is motivated by this recent question. Suppose $R$ is commutative, Noetherian ring and $M$ a finitely generated $R$-module. Let $FD(M)$ and $PD(M)$ be the shortest length of free and proj …

Let $k$ be a field. I am interested in sufficient criteria for $f \in k[x,y]$ to be irreducible. An example is Theorem A of this paper (Brindza and Pintér, On the irreducibility of some polynomials in …

In general $e_{HK}(R) \leq e(R)$. Most of the time the inequality is strict. The case $e(R)=2$ and $\dim R=2$ is studied carefully in the paper by Yoshida-Watanabe "Hilbert-Kunz multiplicity of two-di …

(EDIT 07/06/11: although the question has not been settled definitely, Sándor's excellent answer and the comments by Angelo and ulrich have highlighted many potential obstructions to the constructions …

It depends on what do you mean by "direct". There is no elementary proof, as far as I known, even for polynomials when $d=2$, see page 8 of this note. When $d=1$, the statement is an easy exercise: on …

In general there are many such functions and the non-trivial ones are pretty interesting. I will focus on the commutative case, as that's what I know best. So let $R$ be a commutative noetherian ring …

If $R$ is Noetherian then they are equal. For $n=0$ one can use the fact that any Boolean ring has weak dimension $0$ (any module is flat), but a free Boolean ring on $\aleph_n$ generators have global …

Write the right-hand side as $Hom_B(P/P^2,B)$. If the map you are interested in is surjective, then the preimage of the trace ideal of $P/P^2$ in $B$ must be contained in the the trace ideal of $P$ in …

(To supplement Alberto's example) If $V$ is projective, then the gap between being locally c.i and c.i is quite big. In particular, any smooth $V$ would be locally c.i., but they are not c.i. typicall …

Let $I$ be the annihilator of $M$, by assumption $I=(\pi^i)$ for some $i$. One can view $M$ as an $R/I$ module and furthermore, embed $0 \to R/I \to M$. But $R/I$ is also principal artin local, so it …

A source available online is this paper "Examples of bad Noetherian rings" by Marinari (example 2.1). The reason many of these types of construction work is because of the following vague and counter- …

Craig Huneke told me about this paper: "On the smoothness of blow-ups" (MR1446135, Zbl 0878.13004, by O'Carroll and Valla). The title alone seems to suggest it might be useful for you.

Let $R$ be an excellent Noetherian ring. A property $P$ is said to be open if the set $\{q \in \operatorname{Spec}(R) \ | \ R_q \ \text{satisfies} \ (P)\}$ is Zariski open. Examples of open propertie …

Let $f: R\to S$ be a map between two commutative Noetherian rings. Let $G_0(R)=K_0(mod R)$ be the Grothendieck group of finite generated modules over $R$. It means $G_0(R)$ is the quotient of the free …

Here is a summary of what I learned from a nice expository account by Eisenbud (written in French), can be found as number 27 here.1 First, one studies a more general problem: Let $A$ be a Noetherian …