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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 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 …

There is a construction of a proper normal non-projective surface here . There is an example given by Nagata in his paper "Existence theorems for nonprojective complete algebraic varieties" in the I …

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 …

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 …

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 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 …

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 …

An example (Hartshorne Ch II Ex 3.7) where the condition at the generic point is not vacuous is: Suppose that f:X -> Y is a dominant finite type morphism of integral schemes such that Y is irreducibl …

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 …

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 …

The result in Hartshorne if I recall correctly only really uses the fact that affine locally a coherent sheaf on a scheme has a locally free resolution by finite rank projectives and that one can comp …

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.