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Edit: I gave a proof of a false statement earlier. It can be salvaged to the folowing much weaker statement: the point $(x,y)$ is in the dominant irreducible component if we have $df|_x(T_x)+dg|_y(T_y …

Edit The original diagonalization argument function I gave didn't satisfy the inequality. The following answer works. A diagonalization argument works here to show there aren't! Take an enumeration $X …

Well, unless I am mistaken, whenever you have a cdg ring given by dg generators and relations, then the cotangent space at zero is simply the cone of the differential $T^*\text{Spec}(S^*{\text{generat …

I don't think K-homology will satisfy your requirements. The reason people think of G-theory as dual to K-theory is because for proper $X$, the graded space $Ext^*(F,Q)$ is finite-dimensional for $F$ …

Just follow your nose. Functoriality of cohomology gives a map $\operatorname{Aut}(X_\bar{\mathbb{Q}})\to \operatorname{Aut}\big(H^q_{\text{ét}}(X_{\bar{\mathbb{Q}}}, \mathbb{Q}_\ell)\big)$ (actually, …

Yes, what you are saying is true, at least over an algebraically closed base $k$ of characteristic $0$. In fact, all you need is that $S$ is affine and that the normal bundle $I/I^2$ is trivial (as a …

This is a terminology question about a class of log varieties. Given an fs (fine and saturated) log variety $(X, M)$ (for $M$ the defining sheaf of monoids), any geometric point $x\in X$ has a finitel …

A stack is an object that mixes the notions of (algebraic) space and group. The key insight of stack theory is that most things you would want to do with spaces you can do with stacks: namely, you hav …

There are several notions of weak homotopy equivalence for topological spaces. The standard one can be formulated as follows: a map of spaces $X\to Y$ is a homotopy equivalence if the map of simplicia …

Suppose $\mathcal{C}$ is a dg category (over some base) with all colimits. We say that $X\in \mathcal{C}$ is a generator if $\mathcal{C}$ is equivalent to $\operatorname{End}_\mathcal{C}X$-modules (vi …

I realized the answer is almost certainly "no", so I asked a better version of this question at How much of a variety can be reconstructed from codimension-zero data?. I think that if you take two hy …

This is a follow-up to question Locally toric resolutions of compactifications, answered by Jason Starr. In a series of papers (see https://arxiv.org/abs/math/9904076), Jaroslaw Wlodarczyk proves th …

I would guess that the intuition is that separable extensions extend uniquely over "infinitesimal thickenings", i.e. deformations over a nilpotent base. This makes their classification problem "rigid" …

I'll go ahead and turn my comment into an answer. It does form a monad, but (probably) not a very interesting one. Namely, first note that any pair of adjoint functors $L:\mathcal{C}\leftrightarrows \ …

I want to offer a possibly heretical opinion based on conversations I've had with people who do algebraic geometry, especially Joe Harris. I think that it is not necessary to know very much commutativ …