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Another and more "modern" approach would be using the construction of the complete cotorsion pair ("a half of an abelian model structure") generated by a generating set of objects in a Grothendieck ab …

I'd guess that if $X$ is a reasonable scheme or stack, then $\operatorname{QCoh}(X)$ is, at least, a Grothendieck abelian category. In particular, it has a generator $G$. If $X$ has resolution proper …

This isomorphism always holds, and no conditions are needed. Here I presume that the pro-algebraic group $G$ is pro-affine (as the context of the question seems to suggest). Let $G$ be a pro-affine p …

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 …

Let $A=k[x_1,\dots,x_m]$ be the algebra of commutative polynomials in $m$ variables over a field (or commutative ring) $k$, of arbitrary characteristic. Denote by $B=\bigwedge(x_1^*,\dots,x_m^*)$ the …

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 $f\colon X\to U$ 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 …

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 …

I don't know specifically about homotopies, but the notion of a curved $A_\infty$-algebra is generally problematic. In the conventional setting of algebras over a field, it is just trivial in the fol …

In this question, which attracted no responces so far, I've asked whether it is possible to extend the Beilinson-Lichtenbaum etale descent rule for motivic cohomology to singular varieties, in particu …

It seems to be important that the resolution of singularities be a proper map.

Let $X$ be a smooth variety over a field $K$ whose characteristic does not divide a positive integer $m$. Then the motivic cohomology of $X$ with coefficients in $\mathbb Z/m(j)$ can be computed in t …

Let $X$ be a "reasonable" scheme (I am particularly interested in smooth algebraic varieties over a field). Let $Zar_X$ denote the (small) Zariski site of (open subschemes of) $X$ and $Et_X$ denote t …

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 …

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 …