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**16**

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260 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**

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771 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**

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324 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**

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229 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**

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111 views

### flatness and derived completion

Let $A$ be a local ring of maximal ideal $\mathfrak{m}$. Let $\hat{A}$ be its completion.
If $A$ is noetherian , then we know that $A\rightarrow\hat{A}$ is faithfully flat.
If $A$ is not noetherian, ...

**6**

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514 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

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551 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 ...

**4**

votes

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181 views

### On a theorem of Hopkins-Neeman-Thomason on generators of thick subcategories of perfect complexes

Notations and background. Let $R$ be a commutative noetherian local ring and let $D(R)$ denote the derived category of the category of R-modules. A strictly perfect complex on $R$ is a bounded complex ...

**3**

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334 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

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89 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**

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97 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 ...

**1**

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162 views

### Monoidal Forgetful/Free Adjunction for $E_2$-algebras

Suppose I am given two $E_2$-ring spectra $A$ and $B$ and a morphism of $E_2$-rings $\phi:A\to B$. Then I have $E_1$-monoidal categories of modules $LMod_A$ and $LMod_B$. Moreover I have morphisms ...

**1**

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149 views

### Classifying Spaces and Eilenberg-Maclane objects in the category of simplicial rings

[Skip down to the bottom for a correction] Let's work over a field k, assume it is as nice as you need it to be.. Suppose I have an ordinary (edit: commutative) affine group scheme G = Spec(A) over k, ...

**0**

votes

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163 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 ...