Questions tagged [derived-algebraic-geometry]
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276 questions
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Motivic Interpretation of Rationally Trivial Cycles
The Chow groups are defined by taking groups of cycles modulo rationally trivial cycles. One then has a cycle class map to etale cohomology (over the base field), and for a number field, one expects ...
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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(...
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Given a map of classifying spaces, can the target be described as a groupoid quotient of the source mod some action of some (co)kernel?
Let $H \to G$ be a homomorphism of affine algebraic groups (over characteristic $0$, if it matters). The case I care most about is when $H \to G$ is an inclusion. There is a corresponding map $f: \...
- 54.6k
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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 ...
- 54.6k
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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 ...
- 1,126
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Does the Amitsur complex have a universal property?
The question is essentially the title. In other words, is there some universal property that the Amitsur complex for a morphism of rings $\phi:A\to B$ satisfies as a cosimplicial ring, or cosimplicial ...
- 10.5k
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Derived equivalent varieties with differing integral Mukai-Hodge structures?
For a smooth projective complex variety $X$ of dimension $n$, let $H^i(X)$ denote its integral Hodge structure of weight $i$. Define $\tilde{ H^0}(X) = \bigoplus H^{2i}(X)\otimes \Bbb Z(i)$ and $\...
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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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Is there a relation between Projection formula and Verdier duality
For suitable settings, $f\colon X\to Y$, $F,G$ we have projection formula and Verdier duality:
Projection formula: $Rf_!(F\otimes^\mathbb{L}f^{-1}G)\cong Rf_!F\otimes^{\mathbb{L}}G$
Verdier Duality:...
user39380
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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? ...
- 1,180
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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 ...
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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 ...
- 19.9k
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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, ...
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Flat resolutions of DG-schemes
Recall that a DG-scheme is a pair $(X,\mathcal{O}_X)$,
where $(X,\mathcal{O}^0_X)$ is a scheme, $\mathcal{O}_X$ is a sheaf of commutative DG-algebras over $(X,\mathcal{O}^0_X)$, and each $\mathcal{O}^...
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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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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, ...
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derived invariants, perversity and modular coefficients
Let $\pi:X\rightarrow Y$ a Galois cover of finite type schemes over $\mathbb{C}$ of group $\Gamma$.
Let $n$ an integer such that it is not prime with the order of $\Gamma$.
Then $\pi_{*}\mathbb{Z}/n\...
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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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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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Derived Deformations of associative algebras
Let $k$ be a field (if necessary for my question, we can assume its characteristic to be zero). In the non derived context, we can then define deformations of an associative algebra $S$ as follows:
...
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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{...
- 2,216
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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 $...
- 9,019
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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 ...
- 789
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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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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(...
- 1,191
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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 ...
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