Questions tagged [derived-algebraic-geometry]
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276 questions
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Derived algebraic geometry: how to reach research level math?
I know the question "how to study math" has been asked dozens of times before in many variations, but (I hope) this one is different.
My goal is to study derived algebraic geometry, where derived ...
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2
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Connections between various generalized algebraic geometries (Toen-Vaquié, Durov, Diers, Lurie)?
As far as I know, there are four possible ways to generalize algebraic geometry by 'simply' replacing the basic category of rings with something similar but more general:
$\bullet$ In the approach by ...
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What is the cotangent complex good for?
The cotangent complex seems to be a pretty fundamental object in algebraic geometry, but if it's treated in Hartshorne then I missed it. It seems to be even more important in derived algebraic ...
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What is an infinite prime in algebraic topology?
The links between algebraic topology (stable homotopy theory in particular) and number theory are nowadays abundant and fruitful. In one direction, there is chromatic homotopy theory, exploiting the ...
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Connes–Consani's absolute geometry and Lurie's spectral algebraic geometry
Alain Connes and Caterina Consani seem to be currently working on "absolute algebraic geometry", which is a kind of "algebraic geometry over the sphere spectrum" (https://arxiv.org/...
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What is a simplicial commutative ring from the point of view of homotopy theory?
Let $k$ be a field. There are two natural categories to consider:
The category of simplicial commutative $k$-algebras.
The category of connective $E_\infty$ $k$-algebras (i.e., chain complexes of $...
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What is the relationship between connective and nonconnective derived algebraic geometry?
"Derived algebraic geometry" usually means the study of geometry locally modeled on "$Spec R$" where $R$ is a connective $E_\infty$ ring spectrum (perhaps with further restrictions). Why "connective", ...
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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 $R$....
user74900
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Which properties of a variety are detected by its derived category of coherent sheaves?
Context: I'm giving an informal seminar/reading group collection of talks on derived categories, following on from earlier talks giving the abstract definition. I am starting to talk about $\mathcal{...
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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 ...
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Spectral sequences as deformation theory
I believe that running the spectral sequence of a filtered complex / spectrum $ \cdots \to F_n \to F_{n+1} \to \cdots$ can be viewed as doing deformation theory in some very primitive "derived ...
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What is the status of a result of Kontsevich and Rosenberg?
In their influential paper Noncommutative Smooth Spaces (https://arxiv.org/abs/math/9812158), Kontsevich and Rosenberg define the notion of a noncommutative projective space. In Section 3.3 they ...
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Why do people say DG-algebras behave badly in positive characteristic?
It seems to be a common wisdom in derived algebraic geometry that commutative DG-algebras are not, in general, a good model for derived rings, with the stated reason that they behave badly in positive ...
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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 ...
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Motivation and potential applications of spectral algebraic geometry
Nowadays there is a lot of talk about derived algebraic geometry, but not so much about the related subject of spectral algebraic geometry.
Now I'm curious what future is there for spectral algebraic ...
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Why does elliptic cohomology fail to be unique up to contractible choice?
It is often stated that the derived moduli stack of oriented elliptic curves $\mathsf{M}^\mathrm{or}_\mathrm{ell}$ is the unique lift of the classical moduli stack of elliptic curves satisfying some ...
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Deligne's letter to Millson
The deformation theoretic principle that any reasonable deformation problem should be governed by a dg-Lie algebra seems to come from a letter of Deligne to Millson. It is clear how the Maurer-Cartan ...
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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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List of known Fourier Mukai partners?
I'm familiar with some examples of pairs of derived equivalent varieties, for example an abelian variety and its dual, a K3 surface and certain moduli schemes on it, or the Pfaffian-Grassmannian ...
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Is it always possible to write a scheme as a colimit of affine schemes?
My question is: Is it possible to write any scheme as a (1-categorical) colimit of a diagram of affines? If no, what are some examples?
I ask this question because I have read that one can write any ...
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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 ...
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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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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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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 ...
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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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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 ...
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$\infty$-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 ...
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Why do we say IndCoh(X) is analogous to the set of distributions on X?
$\DeclareMathOperator\IndCoh{IndCoh}\DeclareMathOperator\QCoh{QCoh}$I've seen it written (for example, in Gaitsgory–Rozenblyum) that for a scheme $X$, the category $\IndCoh(X)$ is to be thought of as ...
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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 ...
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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 $\...
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Solution spaces of algebraic differential equations and derived geometry
We consider potentially non-linear differential equations on the formal one dimensional disc $\Delta$. Such equations are given by expressions $$P(z,f,f',f'',...)=0,$$ where $P$ is an element of the ...
user108998
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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 ...
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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 ...
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1
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Reference for symplectic structures on schemes?
My original goal was to read the PTVV paper Shifted Symplectic Structures https://arxiv.org/pdf/1111.3209v4.pdf. I was quickly humbled!
Being told the theory ought to generalize symplectic structures ...
user100272
14
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2
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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}(...
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$E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras
I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral ...
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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’...
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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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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$-...
user138661
12
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1
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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 ...
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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's a holonomic D-module from the point of view of de Rham spaces?
Let $X$ be a smooth algebraic variety over $\mathbb{C}$. We can consider its de Rham space $X_\text{dR}$ as the sheaf on $(\textsf{Sch}/\mathbb{C})_\text{ét}$ defined by $X_\text{dR}(S):= X(S_\text{...
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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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1
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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 ...
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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 & \...
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1
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(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 \...
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Thom Spectra and Hopf-Galois Extensions of Ring Spectra
So I've been fiddling with this for a long time, so apologies to anyone that's already heard me talk about this ad nauseum. I haven't been able to get anywhere with it, and it seemed that as such, it ...
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11
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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 ...
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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 ...
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$\infty$-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 ...
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