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I am interested in the topology on schemes where surjective morphisms of finite presentation are coverings. In particular, I am interested in the topology on Noetherian schemes where surjective morph …

A counterexample showing that direct products in the category of quasi-coherent sheaves over the projective line $\mathbb P_k^1$ over a field $k$ are not exact functors can be found in the paper "The …

Let's consider coalgebras over a field rather than corings. There is a theorem that every (coassociative) coalgebra over a field is the union of its finite-dimensional subcoalgebras. So the category …

Sheaves of DG-algebras were studied by Hinich here, and he also gives some other references, but I cannot say whether it contains what you are asking about.

The short answer is that if you tried doing this, you would be getting into lots of (mathematical) trouble. There were a number of papers and preprints by Alexander Rosenberg devoted to this problem, …

I am trying to think about certain problems in the theory of motives without having a proper background in Grothendieck topologies and the like, hoping to teach myself the related techniques in the pr …

Let $f\colon X\to U$ be a morphism of Noetherian schemes such that the scheme $U$ is affine and the scheme $X$ is separated and, e.g., quasi-projective over affine. Let $U=\bigcup_\alpha U_\alpha$ be …

Attempting to answer your last question: given a (topological, Lie, algebraic, etc.) group G and a closed subgroup P in G, the category of G-equivariant vector bundles on X=G/P is equivalent to the ca …

Let $\mathcal M$ be a left $\mathcal D$-module over a smooth curve $X$. Then the Koszul duality functor assigns to $\mathcal M$ the DG-module $\Omega_X\otimes_{\mathcal O_X}\mathcal M$ over the de Rh …

Apparently $B$ should be finitely generated as a ring over $A$ and be a flat $A$-module, and the module of Kaehler differentials of $B$ over $A$ should vanish. When $A$ is a field, the characterizati …

It is hard to think of a general way to obtain an isomorphism of vector bundles from several isomorphisms of cohomology spaces. In particular, there can be moduli of vector bundles, I presume, even o …

You interpret your derivation $D_e$ as a distribution on $G$ supported in $e$, and then your derivation $D$ is the convolution with $D_e$ with respect to $m$. I.e., take your local function on $G$, c …

Let $R$ be a regular local ring (I am particularly interested in the case when $R$ is the local ring of a point on a smooth scheme of finite type over a field). Let $G$ be the etale fundamental group …

The answer to the question in the title is "no". Semisimplicity is an open condition; however, it is not a dense open condition. Indeed, the variety of Lie algebras is reducible. There is one equat …

To a filtered algebra $(A,F)$ one can assign its Rees algebra $R=\bigoplus_i F_iA$. It is a graded algebra containing the algebra of polynomials in one variable $\mathbb{C}[t]$ naturally embedded as …