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Questions tagged [derived-algebraic-geometry]

3
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0answers
104 views

Homotopy colimit description of stacks

Let $F$ be an Artin stack. If $p: X \to F$ is an atlas for $F$, can we express $F$, in the $\infty$-category ${\rm Shv}^{\acute{et}}(k)$ of higher stacks, as a homotopy colimit over the simplicial ...
9
votes
1answer
303 views

Power operations from a Tate construction

In an action-packed three pages of Lurie's DAG-XIII: Rational and p-adic Homotopy Theory, section 2.2: Power Operations on $\mathbb{E}_{\infty}$-algebras, one finds a construction of the power ...
2
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0answers
93 views

Compact generation of quasicoherent sheaves on mapping stack

Let $k$ be a field of characteristic $0$, and let $\mathcal{C}= \mathbf{Vect}_k^{\leq 0}$ be the $\infty$-category of vector spaces concentrated in degrees $\leq 0$. Consider the category $\mathbf{Pr}(...
7
votes
1answer
338 views

Koszul-Tate Resolution for Subvarieties of $\mathbb P^n$

All varieties appearing below are assumed smooth projective over $\mathbb C$ and all vector bundles, sections etc are assumed to be algebraic/holomorphic. We use the word resolution to mean quasi-...
3
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0answers
161 views

Tannaka duality for $DG$ indschemes

In Lurie's paper on Tannaka duality for geometric stacks he proves that there is a natural isomorphism $$\operatorname{Hom}(X,Y) \cong \operatorname{Hom}(\mathbf{QC}(Y), \mathbf{QC}(X)$$ where $X$ and ...
5
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0answers
118 views

Specific Example of a Morphism of Schemes for which the Push-Pull Morphism is not an Isomorphism

Consider a Cartesian diagram of schemes as follows: $\require{AMScd} \begin{CD} X \times_Z Y @>{\tilde{\pi}}>> Y\\ @VV{\tilde{\phi}}V @VV{\phi}V\\ X @>{\pi}>> Z \end{CD}$ From the ...
8
votes
1answer
525 views

What's special about elliptic cohomology?

Apologies for any basic mistakes in this question; I'm a beginner to this theory and don't have anyone at my institution to consult for advice. What I mean is, if you take an elliptic curve $E$ over $...
5
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1answer
140 views

Derived Morita equivalence of associative algebras

An associative algebra $A$ is said to be Morita equivalent to another one $B$ if there is an equivalence $$\mathsf{Mod}_A\simeq \mathsf{Mod}_B$$ between its corresponding abelian categories of modules....
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217 views

Comparing derived categories of quasi-coherent sheaves in the lisse-etale and the big etale toplogy on an algebraic stack

I am trying to understand the proof of Proposition 1.4.2. in "A study of derived algebraic geometry Volume 1" by Gaitsgory-Rozenblyum. http://www.math.harvard.edu/~gaitsgde/GL/QCohBook.pdf, page 8. ...
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132 views

Why does the following construction describe the Serre functor?

In the book "Spectral Agebraic Geometry" that Jacob Lurie is currently writing, he gives a construction (11.1.5.1), which describes the Serre functor: it has already been shown that any proper $A$-...
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187 views

Moduli stacks of flat bundles and of local systems are not algebraically equivalent?

Let $G$ be a complex reductive algebraic group, $X$ be a smooth compex projective curve. In the remark 1.1 here, it's claimed that the derived moduli stack of principal $G$-bundles equipped with a ...
5
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135 views

Applications of spectral Artin representability?

The spectral Artin representability theorem says that a functor $X:CAlg^{cn}\rightarrow S$ from the $\infty$-category of connective $E_{\infty}$-rings to the $\infty$-category of small topological ...
19
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1answer
855 views

Spectral algebraic geometry vs derived algebraic geometry in positive characteristic?

Let $R$ be a commutative ring. Then there is a forgetful functor from the $\infty$-category of simplicial commutative $R$-algebras to the $\infty$-category of connective $E_{\infty}$-algebras over $R$....
9
votes
1answer
277 views

Why does passage to DG categories cure non-locality of derived categories?

In the famous book 'Residues and duality', the author notes that one of the principal difficulties in constructing the exceptional inverse image functor $f^{!}$ is that the derived category of ...
10
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0answers
316 views

What is a derived Kähler manifold?

From what I understand, there exists a notion of derived $\mathbb{C}$-analytic space. Let $T_{an}$ be the pregeometry in the sense of Lurie whose underlying $\infty$-category is the category of open ...
5
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0answers
156 views

Resolution of Simplicial Commutative Rings

I have just started learning some derived algebraic geometry. I was told that (if $ \mathrm{char}(\mathbb{K})=0 $) using commutative differential graded algebras in negative degree (for short $ \...
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1answer
705 views

$\infty$-operads and $E_\infty$-algebras

I work in algebraic geometry. Lately, the answer to most of my questions seems to be "you should read Lurie's Higher Algebra." I took this advice seriously, however it turned out not to be an easy ...
5
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228 views

relative spectrum in derived algebraic geometry

I am trying to understand how much it is possible to extend the notion of spectrum of a qcoh sheaf of algebras to stacks. More precisely, given a scheme $S$ and a stack $F$ of cohomological cdga's ...
16
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1answer
586 views

A sheaf is a presheaf that preserves small limits

There is a common misconception that a sheaf is simply a presheaf that preserves limits. This has been discussed here before many times and I believe I understand it well enough. However when reading ...
5
votes
1answer
370 views

Derived completion of complexes

Suppose $K$ is a bounded above complex of free abelian groups, and take its derived $\ell$-adic completion $K^{\wedge,\ell} = R\lim (K/\ell^n)$ in the derived category, for $\ell$ a prime. If $K\to L$...
6
votes
1answer
341 views

Is every algebraic space a 1-geometric stack?

In many references (Toen, Higher and derived stacks: a global overview, Toen, Vezzosi, Homotopical algebraic geometry II, and so on), the definition of $n$-geometric stack appears. In the non-derived ...
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0answers
131 views

How can we construct a derived scheme as a gluing of derived schemes?

More precisely, consider a Segal groupoid $X_*$ in an infinity category of derived schemes : dSch In Toen's note, 'Derived Algebraic Geometry', he defines a 1-Artin stacks as a homotopy colimits of ...
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0answers
80 views

Global functions algebra of formal (infinity) groupoid associated to Lie (infinity) algebroid

I was wondering if there is a smooth (sophisticated) way to associate the algebra of global functions of formal groupoid associated to Lie-Rinehart algebra (considered as 1-stack) to its Chevalley-...
8
votes
3answers
287 views

Cofiber of the inclusion of an $E_0$-algebra $M$ into the free $E_k$-algebra generated by it

Let $\mathcal{C}$ be the $E_k$-monoidal $\infty$-category of left modules over a fixed connective $E_{k+1}$-ring spectrum $A$. Suppose that $M$ is an object of $\mathcal{C}$ which is an $E_0$-algebra, ...
5
votes
1answer
512 views

Elementary (English) reference for the cotangent complex?

I'm trying to understand cotangent complexes and their role in deformation theory, and later the statement that they're somehow natural in a derived scheme/stack. I understand that the standard ...
7
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2answers
573 views

Derived topological stacks?

I apologize for the vagueness of the following. Informally, in the site of commutative rings, one roughly get the notion of a derived stack by swapping out the commmutative rings with its subcategory ...
15
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0answers
669 views

How would you organize a cycle of seminars aimed at learning together some basics of Derived Algebraic Geometry?

This question is similar to this one because it's asking about a possible roadmap towards learning some derived algebraic geometry (DAG). But it's also different, because the goal is not to form a ...
15
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0answers
951 views

What to expect from spectral algebraic geometry

So I've been trying to learn some derived algebraic geometry, and I've chosen to approach the subject from the perspective of spectral or "brave new" algebraic geometry. Without having to go through ...
11
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2answers
495 views

Interpretation of the cohomology of compact lie groups and their classifying spaces in DAG?

I'll be using homological grading throughout this question. Let $G$ be a compact connected lie group. The following isomorphisms are classical and can be proven using several methods: $$H^{\bullet}(...
4
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0answers
332 views

The lisse-etale site and derived algebraic geometry

If one reads say Olsson's book on algebraic stacks or Laumon-Moret-Bailly. The lisse-etale topology is used to define quasi-coherent sheaves and the cotangent complex (or rather cutoff's of the ...
3
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156 views

vanishing of higher homotopy sheaves of cotangent complex

Let $X$ be a $\mathbb{C}$-scheme and suppose that there is an isomorphism (in the derived category of qc-sheaves on $X$) between the cotangent complex, $\mathbb{L}_{X}$, and its 0th homotopy sheaf, ...
8
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1answer
222 views

(Pre)orientation vs. formal completion

Let $\mathbb G$ be an abelian vatiety over an $\mathbb E_\infty$-ring $A$. That is to say, it consists of an abelian group object in the $\infty$-category of relative schemes $\mathbb G\to \...
11
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1answer
647 views

Can “ampleness” be detected inside the derived category?

Let $X$ be an algebraic variety (separated quasi-compact scheme of finite type) over a field $k$. One of the possible definitions of an ample line bundle goes as follows: Def 1: A line bundle $\...
4
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0answers
299 views

Why do motivic stacks make sense?

In the paper "Motivic model categories and motivic derived algebraic geometry", Yuki Kato, whose email-address I sadly couldn't find out, describes a procedure to "motivy" the objects of any $(\infty,...
8
votes
1answer
446 views

deformation theory in positive characteristic

The idea "Formal deformation theory in characteristic zero is controlled by a differential graded Lie algebra (dgla)" goes back to Goldman-Millson, Deligne, Drinfeld among others; see Lurie's ICM talk....
3
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1answer
163 views

Is there a notion of injective, projective, flat, dimension for a differential graded algebra?

Given a differential graded algebra $(A_\bullet,d)$, is there a well-defined notion of a K-injective, K-projective, K-flat dimension of a differential graded module, or even of the category of ...
5
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0answers
179 views

Derived tensor products and Tor of commutative monoids

Two commutative monoids $M,N$ have a tensor product $M\otimes N$ satisfying the universal property that there is a tensor-Hom adjunction for any other commutative monoid $L$: $$\text{Hom}(M\otimes N,L)...
3
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1answer
198 views

Who are the compact generators in the derived category of $\mathcal{D}_X$-modules?

Let $X$ be a smooth affine variety over $\mathbb{C}$ and let $\mathcal{D}_X$ be its algebra of differential operators. Consider $\mathcal{C}=\mathcal{D}_X$-$\text{mod}$, the stable $\infty$ category ...
5
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0answers
214 views

Infinity categories with an action of a simplicial group

Recent papers in derived algebraic geometry use a notion of $S^1$-actions on infinity categories. I think I understand what this "should" be and how to calculate with it; however, I can't find a much ...
3
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0answers
244 views

Linear $\infty$-categories $\mathrm{QC(X)}$ and $\mathrm{Perf(X)}$ of a “derived” stack $\mathrm{X}$

For each scheme or algebraic stack their $\infty$-category of quasicoherent sheaves (resp., perfect complexes) on it is $k$-linear for a commutative ring $k$. That is (by a recent result of L.Cohn), ...
2
votes
1answer
237 views

Closed symmetric monoidal structure on the derived category of modules whose unit is a dualizing complex?

Let $A$ be non-positively graded commutative DG-algebra almost of finite type over a field $k$ of characteristic $0$. Most of these assumptions (affine, commutative, characteristic, bound) are only to ...
8
votes
1answer
502 views

Derived noncommutative geometry includes derived, or spectral algebraic geometry?

Let $k$ be a commutative ring. In derived noncommutative (algebraic) geometry a "noncommutative space over $k$" is a $k$-linear $\mathrm{DG}$-category. This is motivated by the fact that homological ...
27
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2answers
1k views

What is the relationship between connective and nonconnective derived algebraic geometry?

"Derived algebraic geometry" usually means the study of geometry locally modeled on "$Spec R$" where $R$ is a connective $E_\infty$ ring spectrum (perhaps with further restrictions). Why "connective", ...
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0answers
139 views

Topological invariance of periodic cyclic homology of stacks

Goodwillie proved (in Cyclic homology, derivations, and the free loopspace) that the periodic cyclic homology of a connective dg algebra is that of its reduced classical ring. Preygel proved (in Ind-...
7
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1answer
218 views

Topological Hochschild homology and Hochschild homology of dg algebras

Topological Hochschild homology is a generalization of Hochschild homology from rings to $E_\infty$-ring spectra. On the other hand, there is a natural way to extend the notion of Hochschild homology ...
3
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159 views

What are projective morphisms in derived algebraic geometry

I was curious if someone could point me to a reference or tell me if there is any notion of projective morphism between derived schemes? What about a notion of Proj? Also I am interested in what blow-...
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0answers
192 views

Interesting examples of large, accessible, non-presentable $\infty$-categories?

What are some interesting examples of accessible $\infty$-categories which are not presentable and not small? By interesting I mean a category which comes up naturally in a certain context and in a ...
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393 views

Categorification of definitions in the context of the derived category of quasi-coherent sheaves

Let $SpecA=X$ be an affine noetherian scheme. Let $QCoh(X)$ denote the derived (stable $\infty$-)category of quasi-coherent sheaves on $X$. There are the following special full subcategories spanned ...
15
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3answers
832 views

Where does one go to learn about DG-algebras?

The theory of differential graded algebras (in char 0) and their modules has numerous applications in rational homotopy theory as well as algebraic geometry. I'm looking for a reasonably complete ...
10
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245 views

derived schemes and perfect obstruction theories

In a survey article of Toen's it is claimed that that there is forgetful $\infty$-functor between the $\infty$-category of derived schemes locally of finite presentation over a field $k$ and the $\...