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I don't think this is true even when $X$ and $V$ are schemes if you only require that the map $V\to X$ is an embedding of functors (rather than a locally closed embedding). Example: Take $X=Spec(k[x]) …

As far as I understand, this is false. Here is an example (familiar to $D$-module people): $A=k[x,y]$; $M=k[a,b]$ on which $x$ (resp. $y$) acts as $\frac{d}{da}$ (resp. $\frac{d}{db}$). Since the acti …

I think the explicit description that you suggest can be wrapped up as follows. For every $i,j$, let $C_{ij}$ be a family of indexes such that $$U_i\cap U_j=\bigcup_{a\in C_{ij}} W_a,$$ with $W_a$ b …

There are standard ways of constructing this kind of objects, but I can't immediately think of a reference, so here it goes: Let $p:Y\to X$ be a projective map (in your case, $Y=S\times X$), let $F$ …

Let us try to come up with a criterion for $L+C\ne A$. Let us assume both $L$ and $C$ are irreducible. (Otherwise consider separate irreducible components.) Let us also shift both $L$ and $C$ so that …

There is also the following (probably unhistorical) point of view (it is a version of Hailong Dao's answer). Namely, you don't have to work with flat families at all, so if you want, you can just decl …

This question is about a class of commutative algebras that is (potentially) a little wider than locally complete intersection, but should still have reasonable properties. Fix a ground field $\Bbbk$ …

Let me try to give a counterexample. (I don't know whether it is 'nice'). First, let us rewrite your properties for an affine scheme $X=Spec(A)$. Connectedness for $A$ means $A$ has no nontrivial ide …

Vague question. Is there anything special about degenerations of smooth projective varieties (separating them from arbitrary projective schemes)? Precise setup. Let $f:X\to Y$ be a projective flat m …

I think it is clear that $R\hat{S}$ is not an equivalence on the categories of all $O$-modules. Indeed, if it were an equivalence, it would send product to product, and also preserve quasi-coherence. …

As discussed in comments, the claim holds in the derived world; here is a counterexample to the naive statement. As I was writing it, I realized that I looked for a counterexample in the classical top …

Here is one approach using the Fourier transform for $D$-modules on an abelian variety due to Laumon. Let $A^\flat$ be the moduli space of rank one local systems on $A$, it is the universal extensio …

I don't think so (finite etale covers cannot be localized in smooth topology in the sense that you describe). Say, $\mathcal{X}$ is a point, and $X$ is a smooth variety with non-trivial fundamental gr …

I also wanted to mention a `high-technology' answer to (1). Namely, if $C$ is a smooth algebraic curve, its $n$-th symmetric power coincides with the variety of all degree $n$ effective divisors on $C …

Here is an explanation why connectedness is important. Let's work over ${\mathbb C}$. The Theorem of the Cube can be stated as follows: If $s:X\to X$ is a shift by a fixed element $g\in X$, then $s^*L …