Search Results

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 …

I don't know of a counterexample but I can tell you some more situations in which it is true. Ballard has extended Orlov's result (in Equivalences of derived categories of sheaves on quasi-projective …

In the answer to this question we saw that there exists a nonsingular quasi-projective threefold over a field with non-finitely generated global sections. I was talking about this previous questi …

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 …

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 …

Provided that $X$ is quasi-compact and separated and $f$ is separated then what is true is that $Rf_\ast \colon \operatorname{D}(X) \to \operatorname{D}(Y)$ has a right adjoint $f^!$ where these are 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 …

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 …

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 …

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 …

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 …