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I am not sure if you are only really interested in properly stacky things, but it is perhaps worth pointing out that the result you mentioned from Hartshorne is true in significantly greater generalit …

As Hailong said in his comment this only happens in the Gorenstein case; here is a sketch of an argument. Suppose $X$ is a quasi-compact quasi-separated scheme with a dualising complex $D$ and let us …

The answer to (1) is yes for any local abstract hypersurface $S$ whose singular locus is closed (which is barely a hypothesis, and free in the case of interest). Let us write $\mathrm{Sing} \;S$ for t …

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 …

I'm not completely sure if this is the sort of thing you are after, but the telescope conjecture (conjecture isn't a great word as it is known to be false for some categories) springs to mind as somet …

Depending upon how strict you are with your definition of torsion theory a good source of examples is the theory of semi-orthogonal decompositions. A really nice example of this is the appearance of s …

The category of sheaves of $\mathbb{Q}$ vector spaces on $M$ is a Grothendieck abelian category. It follows that the derived category of such, $D(M)$ in your notation, is a well generated triangulated …

The problem with respect to applications of the abelianization is that the abelian categories one produces are almost uniformly horrible. More precisely they are just too big to deal with. So using th …

The original action takes the form of an additive functor $A \times B \to B$ with notation as in the question (and appropriate coherence conditions giving compatibility with the monoidal structure on …

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'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 …

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 …

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 …

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 …