Questions tagged [derived-algebraic-geometry]

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4
votes
2answers
163 views

The correct homotopically relevant notion of ideals of dg-algebras (or $\mathbb E_1$-rings)

I'm trying to figure out what an ideal of a, say, dg-algebra (or, if you prefer, $\mathbb E_1$-ring) $R$ is in a homotopically relevant fashion, but I can't actually figure it out. I can assume that $...
13
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1answer
366 views

When does QCoh have 'enough perfect complexes'?

Let $X$ be a derived fpqc stack on the $\infty$-category of connective spectral affine schemes $\mathbf{Aff}^{\mathrm{cn}}=(\mathbf{Ring}^{\mathrm{cn}}_{E_\infty})^{\mathrm{op}}$, that is to say, a ...
1
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0answers
74 views

Open problems for shifted symplectic structures

I am now interested in shifted symplectic structures. What are the open problems of shifted symplectic structures regarding the moduli space of sheaves ? Especially now I am interested in moduli ...
1
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0answers
131 views

Is there a stacky definition of irreducible symplectic manifold?

I am now interested in studying symplectic structures in the field of stacks. In particular, is there a stacky definition of irreducible symplectic manifold ? I'm also interested in similar things in ...
1
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0answers
154 views

Two results about (shifted) symplectic structures

I am now interested in shifted symplectic structures. I found Zhang's results about symplectic structures (2011, p.3-4, arXiv link, Comm. Anal. Geom. 2017) and Pantev–Toen–Vaquié–Vezzosi's results ...
2
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0answers
65 views

Linearity of a dg category $C$ over $HH^0(C)$

Let $C$ be a pre-triangulated dg-category over a field $k$ whose Hochschild cohomology groups $\operatorname{HH}^*(C)$ are concentrated in non-negative degree (cohomologically). Is $C$ Morita ...
3
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0answers
141 views

Reference request: Derived structure on the moduli stack of Higgs bundles

I am reading arXiv:1708.08124. When talking about the moduli stack of Higgs bundles on a projective curve $X$. It is said on page 59, first paragraph that It is often better to put derived ...
8
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1answer
390 views

Connectedness, loops and formal moduli problems

Let $k$ be an algebraically closed field of characteristic zero. Formalizing a classical folk concept, Pridham and (in a different way,) Lurie defined a formal moduli problem (over $k$) to be a ...
0
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0answers
30 views

Extension-closed subcategory $P(I)$ defined by stability condition $(Z, P)$ and an interval $I \subset \mathbb{R}$

Let $D$ be a triangulated category, and let $\sigma = (Z, P)$ be a Bridgeland stability condition on $D$. Let $I \subset \mathbb{R}$ be any interval (open, closed, or half-open). The category $P(I)$ ...
3
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1answer
247 views

Is the category of spectra on $\mathbb{P}^1$ a module category?

I cannot really state my question in an incredibly precise way as I'm very new to this area, but I hope what I'm asking will be clear. Let $\mathcal{C}$ be the infinity category of sheaves of quasi-...
2
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0answers
62 views

Derived manifold and real virtual dimension

In https://arxiv.org/pdf/1504.00690.pdf, it seems like the "derived manifold structure" given on a certain complex analytic space seems to have the real virtual dimension the same as the complex ...
19
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2answers
1k views

What (or how) are the new spaces of derived algebraic geometry?

I am a beginner in derived algebraic geometry and I am trying to develop some visual and geometrical intuition about derived schemes (and stacks), or more precisely about the new geometrical phenomena ...
4
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0answers
194 views

Derived category of a fiber product

Let $X = Y \times_Z W$, where $X,Y,Z,W$ are Noetherian schemes, and consider the pullback diagram associated to $X, Y, Z, W$. We have a diagram $$ \require{AMScd} \begin{CD} D(Z) @>>> D(Y)\\ @...
2
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1answer
128 views

Jordan–Hölder sequence for $\mu$-semi stable sheaves

Let $X$ be a smooth variety over $\mathbb{C}$, and let $\omega \in \operatorname{Pic}(X)_\mathbb{R}$ be an ample class. I would like to know if any $\mu_\omega$-semistable sheaf $E \in \operatorname{...
15
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0answers
440 views

What is the relationship between Artin and Lurie representability?

Artin's representability theorem gives conditions for a functor from commutative rings to sets (or groupoids) to be representable by an algebraic space (stack). The conditions are largely expressed ...
1
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0answers
117 views

Perfect complexes on affine schemes

I'm reading a paper on algebraic stacks and in some part is stated the following: Let $X$ be an algebraic stack and let $P\in D_{qc}(X)$ be a perfect complex. Then, for every $x\in |X|$, there ...
6
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1answer
323 views

Derived Category of the derived critical locus, is it the category of Matrix Factorizations?

Let $W \in \mathbb{C}[x_1, \dots, x_n]=R$ be a polynomial with an isolated critical point at the origin. A Matrix Factorizations for $W$ consists a $\mathbb{Z}/2\mathbb{Z}$-graded finite free $R$-...
1
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1answer
166 views

Computing units in a dg-algebra

Let $\mathbb{G}_m= Spec(k[z,z^{-1}])$ be the usual multiplicative group over a field $k$ viewed as a discrete commutative dg-algebra, and let $A$ be some arbitrary commutative dg-algebra concentrated ...
4
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0answers
104 views

Does the ∞-category of Derived/Spectral schemes admit all colimits over constant diagrams?

In the case of ordinary schemes, all coproducts exist, so given any constant diagram $D_S:C\to \operatorname{Sch}$, the colimit over $D_S$ is isomorphic to the coproduct of $S$ over the connected ...
9
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1answer
379 views

Descent properties of topological Hochschild homology

Question: What is the finest topology on $\mathrm{CAlg}$ (commutative ring spectra) for which THH (Topological Hochschild Homology) satisfies descent? Adaptations of the arguments appearing in ...
2
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0answers
188 views

Do dg schemes have derived points?

Working over a base field $k$ of characteristic $0$, say $K$ is a field (over $k$) and $X$ is a ("nice" if necessary) dg scheme in the sense of Toen-Vezzosi and others, and say $X^0$ is the reduced ...
8
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0answers
323 views

The virtual fundamental class as derived intersection

Say $X$ is a smooth projective variety and $\beta\in H_2(X)$ is a class. Then there is a finite-type proper scheme (or in general, stack) $SM : = \overline{\mathcal{M}}_{g,n}(X,\beta)$ of stable maps ...
6
votes
1answer
504 views

Derived base change in étale cohomology

Given a commutative square of ringed topoi $$\begin{array}{ccc}X'\!\! & \overset{f'}\to & Y'\!\! \\ \!\!\!\!\!{\small g'}\downarrow & & \downarrow{\small g}\!\!\!\! \\ X & \...
9
votes
1answer
581 views

Proj construction in derived algebraic geometry

The question My question is easy to state: Is there a Proj construction in derived geometry, that produces a derived stack from a “graded derived algebra”? Given the vagueness of the question, you’...
2
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0answers
121 views

When is $C\text-\mathsf{dg\text-mod}$ determined by the connective base changes?

I'm using cohomological gradings. For $C\in k\text-\mathsf{cdga}$ (where $k$ can be taken of characteristic 0), a morphism $C\to A$ to a connective dg-algebra $A\in k\text-\mathsf{cdga}_{\leq0}$ ...
1
vote
1answer
212 views

Is the pushforward of a closed immersion of spectral Deligne-Mumford stacks conservative?

Let $ X \hookrightarrow Y$ be a closed immersion of (connective) spectral Deligne-Mumford stacks, is $ i_* : Qcoh(X) \rightarrow Qcoh(Y)$ conservative? Somehow I couldn't find the statement in SAG...
7
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0answers
125 views

Is it always possible to write a derived manifold (in the sense of Spivak) as a homotopy colimit of principal derived manifolds?

Is it always possible to write a derived manifold as a homotopy colimit of principal derived manifolds (i.e. zero sets of smooth functions)? This is true for schemes and derived schemes, so it seems ...
11
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0answers
481 views

How far can Spectral Algebraic Geometry be developed over $\mathbb{E}_2$-rings (instead of $\mathbb{E}_\infty$-rings)?

Jacob Lurie has extensively developed derived algebraic geometry in the setting of $\mathbb{E}_\infty$-ring spectra [SAG]. The resulting theory of Spectral Algebraic Geometry (SAG) gives (in ...
2
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0answers
212 views

Confusion about DAG terminology

This question refers to higher and derived algebraic geometry as developed by Toen-Vezzosi, not by Lurie. I have seen two expository documents by Toen. In the first text, there is a definition: A ...
2
votes
1answer
206 views

Local quotient covers for derived Deligne-Mumford geometric stacks of Toen-Vezzosi

Let $\mathcal{X}$ be a separated Deligne-Mumford stack, and $X$ its coarse moduli space. A well-known lemma establishes an etale covering $X_{\alpha} \rightarrow X$, such that for each $\alpha$, there ...
10
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2answers
1k views

Results relying on higher derived algebraic geometry

Are there any results in number theory or algebraic geometry whose statement does not involve either higher categories or any derived structures but whose most natural (known) proof uses derived $n$-...
3
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0answers
316 views

DAG applied to homotopy theory: how to reach research level?

It is my dream to do research on applications of spectral algebraic geometry in homotopy theory one day. Specifically, giving a more uniform treatment for the results proved via scary computations (of ...
8
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0answers
230 views

Motivating derived stacks via Euclidean geometry

Here (see Section 3) triangles in Euclidean plane are used to motivate the notion of DM stack (an equilateral triangle has more symmetries than a generic triangle). Can something similar be done to ...
4
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0answers
98 views

Examples of non-hypercomplete sheaves on affine schemes

Let $A$ be a commutative ring and let $\mathcal{O}$ be a sheaf of $E_{\infty}$-ring spectra on $\mathrm{Spec} A$ such that $\pi_0\mathcal{O} = \mathcal{O}_{\mathrm{Spec} A}$. Lurie provides a ...
4
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0answers
228 views

Examples of Lurie tensor product computations

I am interested in examples of computing the Lurie tensor product. For example, if $A$ and $B$ are connective DG algebras (over $\mathbb{Z}$, say), then I think there is an equivalence $A\text{-mod} \...
3
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0answers
276 views

DAG vs Classical algebraic geometry

I have a very vague question, but also a fairly specific wish. Namely, I'm wondering what the similarities and differences are between the theory of ordinary schemes on the one hand, and the theory of ...
7
votes
1answer
429 views

$\infty$-categorical understanding of Bridgeland stability?

On triangulated categories we have a notion of Bridgeland stability conditions. Is there any known notion of "derived stability conditions" on a stable $\infty$-category $C$ such that they become ...
0
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1answer
466 views

Metrics on derived smooth manifolds

Derived geometry explains how to remove the transversality condition and make sense out of a nontransversal intersection. For example, if $X$ and $Y$ are embedded submanifolds of a manifold (or ...
4
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0answers
931 views

How should one approach reading Spectral Algebraic Geometry by Lurie?

A question posed at the nForum asked for a roadmap to learn Lurie's Higher Topos Theory. This MathOverflow question asked for a roadmap to Lurie's Higher Algebra. Still another question asked for a ...
4
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0answers
147 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 ...
11
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1answer
459 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
111 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
537 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
185 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 ...
6
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0answers
142 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 ...
7
votes
1answer
725 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
votes
1answer
187 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....
11
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0answers
248 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. ...
2
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0answers
144 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 $A$-...
6
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0answers
181 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 ...