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Let $MV_F$ be the $\mathbf{A}^1$-homotopy category over the field $F$. Let $H$ be the homotopy category of spaces, and let $H_{/S^1}$ be the homotopy category of spaces over the circle. When $F = \ma …

No. Counterexamples were first constructed by Winkelmann, as quotients of $\mathbb A^5$ by algebraic actions of $\mathbb G_{\text{a}}$. I learned this from Hanspeter Kraft's very nice article availa …

Given an integrable system on a Kahler manifold X, is there a way to associate a toric degeneration of X whose Milnor fibers are related to the fibers of the integrable system? An integrable system i …

Let $C$ be a stable $\infty$-category. Is there any categorical construction $C \mapsto T^* C$, where $T^* C$ is another stable $\infty$-category, that specializes to the following? When $C$ is the …

Let $F$ be a quasi-coherent sheaf on a scheme $X$. To check that $F$ vanishes it suffices to check that all the stalks of $F$ vanish. I would like to know whether it suffices to check that all the f …

I'm inspired by Yuhao's question. The functor that takes a scheme S to the set of k-dimensional vector subbundles of C^n x S (understanding "subbundle" to mean that the quotient by it is another vect …

The set of 5-tuples of lines in $\mathbf{P}^3$ is parametrized by the 20-dimensional product of Grassmannians $G(2,4)^{\times 5}$. The set of cubic surfaces is parametrized by a 19-dimensional projec …

If you triangulate your space refining the stratification, a constructible sheaf is given by the data of a vector space $V_{\sigma}$ (a stalk at the barycenter, say) on each simplex $\sigma$ and a res …

Ariyan and Kevin Lin have asked about the problem of extending vector bundles defined on an open subvariety across the rest of the variety. There can be subtle commutative algebra obstructions, as in …

Here is a sufficient condition. If a space has finitely many points, or more generally has the property that the intersection of even an infinite number of open sets is itself open, then it will have …

If $L$ is a complex line bundle on a topological space $X$, let $c_1(L)$ denote the image of its Chern class in $H^2(X;C)$. A complex manifold structure on $X$ [ok which is also compact and say algeb …

Principal Ga-bundles on a scheme X, in any of the Zariski, etale, or flat topologies, are classified by the coherent cohomology group H^1(X,OX). For a smooth complex projective variety, this is the a …

If you take a product of finite fields of infinitely many characteristics and divide by a maximal ideal, the result is called a pseudo-finite field. This has characteristic zero and a commutative Gal …

Since a good picture for a length $n$ scheme set-theoretically supported at a point is some kind of limit of $n$ different reduced points getting closer together, let's try to understand what's going …

If etale pi_1 classifies obstructions to trivializing finite flat unramified Z-algebras, it would be nice if the whole etale homotopy type classified obstructions to trivializing simplicial commutativ …