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The answer is yes, assuming by $(f_1,\ldots, f_n)$-torsion you mean that each element of each cohomology group of $F$ is killed by some power of this ideal. There are several ways to see this. The mo …

For starters is worth noting that in the case of Jacobson rings (and more generally Jacobson schemes) (http://en.wikipedia.org/wiki/Jacobson_ring for instance has a definition) that the spectrum of ma …

The answer is yes. Indeed, over a perfect field the notions of smooth and regular coincide so it follows from the fact that base change and composition preserve smoothness.

You might also want to check out this paper by Landsberg and Teitler on getting bounds for the Waring rank (and border rank) of symmetric tensors using geometry. The point here is that the projecti …

If you make the statement Fix an algebraically closed base field k and let X be a scheme of finite type over k. Then X/k is proper iff for all smooth quasi-projective curves C/k and all maps f: C\c …

One uses the following trick. By the projection formula we have $${p_2}_*(\mathcal{O}_{X\times Y_1}) \cong {p_2}_*(p_2^*M_1^{-1} \otimes L_1) \cong M_1^{-1} \otimes {p_2}_*(L_1)$$ and since $X$ is co …

Given schemes $X,Y$ and $Z$ such that $Z$ is a closed subscheme of both $X$ and $Y$ the pushout exists in the category of schemes. So in particular one can glue schemes along a closed point. A referen …

The answer is yes. In general one can define a normal morphism of schemes $f:X \rightarrow Y$ to be a flat morphism such that for every $y \in Y$ the fibre over $y$ is geometrically normal. Then we …

Hartshorne has lots of problems and while a lot of them I probably wouldn't say are fun there are certainly a lot of them which are worth doing. But now that I have given the standard answer I can't …

Hopefully I can go some of the way toward addressing 2 and 3 without getting carried away and getting too technical/newbie unfriendly, although I have a feeling I'm going to fail at this last part. T …

So we know that such an adjunction exists for a closed immersion or for an open immersion where we get $(f_*, f^!)$ the pushforward and subsheaf with supports and $(f_!, f^*)$ the extension by zero an …

The answer is no - the point is that finitely generated projective modules are locally free but not necessarily globally so. For instance take a Dedekind domain $A$ which does not have unique factori …

I'm not sure if this is the sort of thing you are after but one can say the following. Suppose we work over a base field $k$. If $X$ is proper over $k$ and the $G$-linearized invertible sheaf $L$ is …

This is not an answer to your question about Zariski-local projectivity, but it is relevant to being locally free and you might be interested. One can get away with finitely generated rather than fin …

I thought I'd offer a high-tech alternative for certain varieties. If $X$ is smooth and projective over a field $k$ then Bondal and van den Bergh give a proof in Generators and representability of fun …