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As the link in Erica's comment shows, you can find this in SGA3 Exp. 2, but it is not so easy to extract from the very general language. Here is a rough guide: From Definition 3.9.0, the Lie algebra …

I suspect the expected structure of $Rep(V)$ common to all vertex algebras $V$ is something like "abelian pseudomonoidal category" and I don't think you can say much else. The abelian structure follo …

Recall from the beginning of the proof of the lemma that $R$ is defined to be $U \times_F U$, so the surjections $U \to F \to U/R$ induce a canonical étale sheaf map $U \times_{U/R} U \to U \times_F U …

The Max-Cut problem for a graph asks for a subset $S$ of vertices such that the number of edges between $S$ and the complement of $S$ is maximized. This problem is NP-hard. In fact, Håstad showed th …

One historical reason for considering $\ell$-adic cohomology, not completely disconnected from the example you introduce, is that for a curve over a field, we get a natural Galois representation by ta …

I believe Nash originally chose the term "arc" to mean a short path, rather than a circle. The ring of functions on the completion of an algebraic curve over $k$ at a smooth $k$-point is isomorphic t …

Let $(G,T,M)$ be a split reductive group (over say, the integers), with Lie algebra $(\mathfrak{g}, \mathfrak{t})$, and let $R$ be a commutative ring. When $R$ is an algebraically closed field, it is …

The universal map is not a map of formal groups without some extra condition. An easy class of counterexamples comes from completions of an affine group along a closed subscheme that does not contain …

This follows from Theorem 6 in Steinberg's "Some consequences of the elementary relations in $SL_n$" in "Finite groups — coming of age", Contemporary Math. 45 Amer. Math. Soc. (sorry, I could only fin …

The condition that the pullback $e^*(\mathcal{F})$ be naturally identified with the pushforward $p_*(\mathcal{F})$ can be tautologically interpreted as saying that for any open set $U$ in $X$, any sec …

For any algebraic stack $X$, there is a groupoid in schemes whose fppf stackification is equivalent to $X$. You can construct such a groupoid following the stacks project, by choosing a smooth presen …

I'm not sure how you have chosen to define "quasicoherent sheaf on the stack $\mathcal{G}$". One way to make a definition is to construct the fibered category $QCoh$ over affine schemes, whose object …

Since you are looking for moduli interpretations, we may either work with stacks, or assume $d$ is large enough that $Y(d)$ is representable (i.e., at least 3). Perhaps the easiest answer is that in …

As Piotr Achinger suggested in a comment, your log moduli space is the direct product of $M_{g,n}$ with the log point $\operatorname{Spec}(\mathbb{N}^n \to \mathbb{C})$ given by the monoid map $(x_1,\ …

The main advantage of chiral algebras over vertex algebras is that they admit "very functorial" definitions, and this helps more general concepts and constructions appear naturally. The usual example …