Questions tagged [derived-algebraic-geometry]
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276 questions
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Proper Model Category
Let R be a commuative ring. Consider the category of simplicial R-modules with the projective model stucture. Can someone give me a precise reference which proves that this model category is proper? ...
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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 ...
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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, i....
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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 $...
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On a question motivating Lurie's treatment of formal moduli problems
Lurie, in his ICM 2010 proceedings paper Moduli Problems for Ring Spectra (pdf), says that one motivating problem (for him, I presume, and possibly others) for thinking about formal moduli problems is ...
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Several simple questions on the geometry of higher stacks
I'm trying to understand definition/work out some examples. So, there are some simple questions about higher stacks.
For the simplicity assume that we are working with higher DM (Deligne-Mumford) ...
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Which morphisms of ring spectra are of effective descent for modules?
There is a well understood bifibration of $\infty$-categories over the $\infty$-category of commutative ring spectra whose fiber over a ring $R$ is the category of $R$-module spectra. This is in ...
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Small objects in categories
I would like to pick out small objects from a category. I would like to find such a notion which
Dream 1. Picks out the schemes of finite type over $k$ from the category of $k$-schemes. Or at least ...
- 13.5k
6
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A construction of Kähler differentials and Illusie cotangent complex as colimit over embeddings
Let $\Bbbk$ be a field, $X$ affine scheme of finite type over $\Bbbk$. Let $\mathcal C_X$ be the category of closed embeddings of $X$ into (say affine) smooth $Y$'s of finite type over $\Bbbk$, ...
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Relation between hypercompleteness and the property that Cech cohomology calculates sheaf cohomology
Let $C$ be a small site with fibre products. The (injective) Čech model structure on simplicial presheaves $\operatorname{sPre}(C)$ on $C$ presents an $(\infty,1)$-topos and one may ask if this $(...
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Higher commutators in E_n algebras and the Maurer--Cartan equation
Let $A$ be an associative algebra in $dgVect_k$. Then the commutator $[\cdot,\cdot]:A\otimes A\to A$ defined by $[x,y]=xy-(-1)^{|x||y|}yx$ gives $A$ the structure of a (dg-)Lie algebra. The Maurer--...
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Why should we study deformations of perfect complexes
What are the advantanges of studying deformation of perfect complexes over the classical theory of deformation of coherent sheaves? Any references which elaborates on the applications on deformation ...
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Pushout schemes/stacks
I would like to know in what generality do we have pushouts for schemes/stacks/derived stacks. More precisely, let $f : X \rightarrow Y$ be a proper flat surjective morphism of schemes of finite type ...
CommunityBot
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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 $...
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Blowing up a derived scheme
Is there a sensible notion of blowing up in any of the available frameworks for derived algebraic geometry? I am happy to remain in the affine setting, where I think the right question to ask is "what ...
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What is a good reference (preferably thorough) for the Derived Category of a scheme/orbifold/stack?
I've sort of circled around the idea of derived categories a few times, read a few introductory papers ("Derived Categories for the working mathematician", e.g.), and feel now that this is something ...
CommunityBot
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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 ...
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When does the cotangent complex vanish?
The question is already in the title. Less succinctly, let's call a map $f:X \to Y$ of schemes $L$-trivial if its cotangent complex is quasi-isomorphic to $0$. Such maps have striking deformation-...
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question about higher geometric stacks
I have a naive question I am asking. Given a higher geometric stack X in the sense of Simpson, Toen etc is it true that there is an affinization Spec Gamma(O_X) such that Hom(X, Spec(A))= Hom(A,Gamma(...
- 24.1k
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Defining ind-coherent sheaves and their singular support
Q1: My first question is about defining the category $\text{IndCoh}(S)$ for a $DG$ scheme $S$. So in page $18$ of this paper, they are defined as being the ind-completion of the category $\text{Coh}(S)...
- 24.1k
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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)) := \text{...
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If $A\in \mbox{Rings}\subset E_\infty\mbox{-rings}$, what is the equivalence between objects of $\mathcal{D}(\mbox{Mod}_A)$ and $A$-module spectra?
In Lurie's "A Survey of Elliptic Cohomology", he writes on page 14 that if $A$ is an ordinary commutative ring considered as an $E_\infty$-ring, then $A$-module spectra are the same thing as objects ...
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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 D^b(...
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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 ...
- 10.8k
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685
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Somewhat general question that includes: "Do quasi-isomorphic cdgas have quasi-isomorphic spaces of derivations?"
Question: Given two quasi-isomorphic dg commutative algebras (over a field of characteristic zero, if you like), to what extent do their various homological geometric data agree?
Example: Given a dg ...
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can a common mortal understand why the affine line is not smooth in brave new algebraic geometry?
In the introduction to HAGII Toen and Vezzosi write that in brave new algebraic geometry (that is, algebraic geometry over the category of symmetric spectra) Z[T] is not smooth over Z.
I am told that ...
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