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I would like to know if there are some interesting partial orders defined on the isomorphism classes of vector bundles on $\mathbb P^n_k$ (you can assume $k$ is $\mathbb C$ if that helps). Motivat …

This answer of mine briefly discusses Hartshorne conjecture and some related questions about smooth subvarieties of $\mathbb P^n$ of small codimensions. It links to Hartshorne's original paper, which …

You need to refine the question to get better answers, but here are some thoughts: 1) Over a normal variety, you can think of line bundles as divisors, and "intersect" them. 2) A vector bundle can b …

That is a pretty terse proof! Let me give an outline of a proof that I know. First, one could deduce the statement from a more general: Theorem 1: Let $R$ be a regular local ring, $E$ be a reflexive …

Actually the implication should be reversed: a separated regular Noatherian scheme has enough locally frees (this is Exercise 6.8, Chapter III Hartshorne). So the hypothesis is certainly needed for t …

To complement Angelo's answer (which you should accept, as it answers your original question): If $\mathrm H^1(E^\vee \otimes E) = 0$ then $E$ must be homogeneous, see for instance Theorem 3 this pap …

Locally, the depth increases along syzygies, so there are always counter examples as long as there is a closed point whose local ring has depth at least 3. For instance, let $R=k[x_1,...,x_n]$ for $n\ …

Let $X$ be a connected scheme. Recall that a vector bundle $V$ on $X$ is called finite if there are two different polynomials $f,g \in \mathbb N[T]$ such that $f(V) = g(V)$ inside the semiring of vect …

Let $E,F,G$ be algebraic vector bundles over $\mathbb P_{\mathbb C}^n$. My general question is: Assume $E\otimes G \cong F\otimes G$, under what conditions can one conclude that $E\cong F$? Some ea …

Since you asked for reference, here are some references that may be helpful (the question is local): 1) (EDITED: Thanks to BCnrd for keeping me honest here!) If $(R,m)\to (S,n)$ is a map of Cohen-M …

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 …

One modern account can be found in this thesis of Majidi-Zolbani, especially Chapter 1 and Appendix A. The point is that if $E$ is a vector bundle on $U$, then the global sections $\Gamma_U(E)$ is a f …

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 …