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15
votes
0answers
226 views

Elliptic $\infty$-line bundles over $B G$

Theorem 5.2 in Jacob Lurie's "Survey of Elliptic Cohomology" (pdf) states the equivalence of two maps $$ B G \longrightarrow B \mathrm{GL}_1(A) $$ for $A$ an $E_\infty$-ring carrying an oriented ...
11
votes
0answers
629 views

Motivic derived algebraic geometry

In algebraic geometry one studies spaces locally modelled on commutative algebras, i.e. commutative algebra objects in the symmetric monoidal category Ab of abelian groups. Now supposedly in the ...
10
votes
0answers
284 views

Thom Spectra and Hopf-Galois Extensions of Ring Spectra

So I've been fiddling with this for a long time, so apologies to anyone that's already heard me talk about this ad nauseum. I haven't been able to get anywhere with it, and it seemed that as such, it ...
7
votes
0answers
222 views

Derived (non-commutative) geometry, geometric constructions in explicit form

I'm interested in the following construction. Start with derived category of coherent sheaves witch equivalent to derived category of representations of some dg-algebra. Quasi-isomorphic dg-algebras ...
6
votes
0answers
465 views

The De Rham Stack and $\text{LocSys}$

Q1: Here, on pg $103$, the stack $\text{LocSys}_G(X)$ is defined (for an affine algebraic group $G$, a fixed DG scheme $X$, and a test scheme $S \in \textbf{DGSch}$): $Maps(S, LocSys_G(X)) := ...
6
votes
0answers
509 views

Tensor product of structure sheaves

Let $\iota_A:A\hookrightarrow X$ and $\iota_B:B\hookrightarrow X$ be subschemes of a smooth ambient variety $X$. Then the derived tensor product $$\mathcal O_A\stackrel{L}{\otimes}\mathcal O_B\in ...
3
votes
0answers
305 views

Analysis of Eilenberg-MacLane Stacks

In a series of three papers from the fifties, Eilenberg and MacLane did a pretty exhaustive study of what we now call "Eilenberg-MacLane spaces" and used a lot of machinery to do it, e.g. Whitehead's ...
2
votes
0answers
76 views

quotient a scheme by a stratified vector bundle

Let $k$ be a field. Let $X$ be a $k$-scheme of finite type, normal and integral. We consider $f,g:R\rightarrow X$ an equivalence relation, surjective and such that it is a stratified vector bundle, ...
2
votes
0answers
87 views

How to understand $\mathcal{L}BG \simeq G/^{\text{ad}}G$ in term of simplicial sets?

First let $G$ be a topological group and $BG$ its classifying space. Let $\mathcal{L}BG=\text{Map}(S^1, BG)$ be the free loop space of $BG$. We can see that $\mathcal{L}BG$ has the homotopy type of ...
0
votes
0answers
160 views

What means “extended concepts of symmetry”?

Where could one find a short description oft: "two mathematical extensions of the symmetry - to moduli spaces of sheaves and to derived categories", found here? Happen there interesting things like ...