Questions tagged [derived-algebraic-geometry]
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312 questions
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Quillen's conjecture
I was reading about cotangent complexes in derived algebraic geometry. I saw the following two surprising conjectures by Quillen (from "Cohomology of commutative rings").
My question is: are ...
- 1,463
7
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0
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602
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Defining the cotangent complex via equivalence classes of derived étale curves
In the classical setting, the tangent space $T_p X$ at a point $p$ of a smooth scheme $X$ can be defined using equivalence classes of morphisms from the affine line $\mathbb{A}^1$. To avoid the issue ...
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15
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1
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What is an example of locally finitely presented algebra which is not finitely presented?
Given an animated ring A, Lurie defines in DAG 3.1 an algebra over A to be finitely presented if it lies in the smallest subcategory of A-Algebras containing A[x] and stable under finite colimits.
On ...
- 761
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435
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Analytic de Rham stacks of Fargues-Fontaine curves?
Earlier this year, Johannes Anschütz, Arthur César Le Bras, Guido Bosco, Juan Esteban Rodríguez Camargo and Peter Scholze published a preprint titled “Analytic de Rham stacks of Fargues-Fontaine ...
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Grothendiek duality and dualizing sheaf
Let $i:Z\to X$ be a closed immersion and quasi-smooth map between derived algebraic stacks. Let $\omega$ be the dualizing sheaf on $X$. Then is the following isomorphism true? Here $i^!$ is the right ...
- 1,463
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A description of morphism $\beta: \bar{Q}_{l,BG_m}\to \bar{Q}_{l,BG_m}[2]$ in $ Shv(BG_m)$?
The follwing question is about Example 4.1.12 in Revisiting Mixed Geometry by Quoc Ho and Li Penghui.
Given a scheme $X$ over $\bar{Q}_l$, considering the derived category of constructible l adic ...
- 1,010
3
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178
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Definition of a stack equipped with a group scheme $G$ action
$\newcommand\op{^\text{op}}\newcommand\Sch{\mathbf{Sch}}\newcommand\dSch{\mathbf{Sch}}\newcommand\Stk{\mathbf{Stk}}$Given a group scheme $G$ (for simplicity, we can take $G=G_m$), I have seen two ways ...
- 1,010
3
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0
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178
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What does the spectral functor of points of a classical Deligne-Mumford stack look like? Is it valued in 1-groupoids?
Let $A$ be an ordinary ring, and let $R$ be a connective $\mathbb{E}_{\infty}$-ring. Maps $R\to A$ can be described easily: any such map will factor uniquely through the truncation map $R\to\pi_0 R$. ...
- 2,275
4
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0
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Local structure of derived $n$ stacks
I am reading Toen's note 'higher stacks — a global overview'. For theory of stacks I read Alpers note 'Stacks and Moduli'. My question is the following:
For Artin stacks with some nice properties we ...
- 1,463
3
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1
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659
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Is derived Kähler space useful for researching Kähler manifolds?
After this question, I looked into various things and discovered Eita Haibara's paper. Haibara's derived Kähler space feels very novel, but since the paper only provides one example, I'm unsure if ...
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2
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262
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Are these categories stable $\infty$-categories?
I was reading the paper Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry. In section 2.2, the authors stated that the category of modules over an algebra object in a monoidal $\...
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5
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244
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Why is it important to understand derived complex analytic spaces?
I’m currently reading Lurie’s DAG IX. I understand the definition of derived complex analytic spaces, but I don’t see their necessity. Can someone provide motivation for derived complex analytic ...
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7
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543
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Derived schemes vs dg-schemes
I am reading about derived schemes from Lurie's Thesis. There is also a notion of dg-schemes. It is there in the literature that in characteristics zero case these two notions are equivalent, i.e. the ...
- 1,463
5
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1
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525
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Is the quotient of an algebra by a module defined in derived algebraic geometry?
For a commutative $R$-algebra $A$, surjections $A \twoheadrightarrow B$ are in natural bijection with ideals $I \hookrightarrow A$ (in fact, this is an isomorphism of complete lattices).
Passing to ...
- 8,070
6
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1
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Are varieties over $\mathbb{E}_{\infty}$-rings automatically spectral algebraic spaces?
In Jacob Lurie's book Spectral Algebraic Geometry, as Definition 19.4.5.3, he defines a variety over a connective $\mathbb{E}_{\infty}$-ring to be a spectral algebraic space which is proper, locally ...
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