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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 $n\geq 3$ be an integer and $0<\alpha_1, \dots ,\alpha_{n-2}<1$. Let's say a tuple of positive numbers $(e_1,\dots, e_n)$ is nice if there is a convex $n$-gon $A_1\dots A_n$ such that $\hat A_i=\ …

(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 …

Here is a reference that may be relevant (you may know it already): Auslander-Rim has a paper called "Ramification index and multiplicity" . Even though they mostly discussed normal rings, their defin …

Let $X$ be a smooth projective variety over $\mathbb C$ with $d = \dim X \geq 1$. Let $CH(X)$ denotes the total Chow group of (cycles modulo rational equivalences of) $X$ and $CH(X)_{\mathbb Q} = CH( …

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 …

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 …

Well, if you read on to Chapter 2, exercise 6.3, then it is stated that: $$Cl(A) \cong Cl(X)/\mathbb Z[H]$$ here $[H]$ represents the hyperplane section. So the answer is yes. There is a less well-kno …

To expand on Emerton's answer: Using the excision sequence, Cartan's result in the algebraic case boils down to showing the following: Let $R$ be a regular local ring, and $I$ and ideal of height at l …

It may be a bit unfair to compare $X=\mathbb P^1 \times \mathbb P^1$ to $\mathbb P^1$. EDIT: I removed a too optimistic statement about restricting vector bundles on $\mathbb P^3$ to a smooth hypersur …

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 …

Pete's answer gave a very good reason, and indeed contained what I planned to say at the beginning of this post. However, let me provide some mildly positive news, namely some cases when we are forced …

A cool application which I can somehow appreciate is Van den Bergh's proof of dimension $3$ case of the Bondal-Orlov conjecture that two birational smooth Calabi-Yau varieties $X,X'$ have equivalent d …