Questions tagged [derived-algebraic-geometry]
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276 questions
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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 ...
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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 ...
- 439
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2
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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", ...
- 64k
4
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0
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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-...
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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 ...
- 1,423
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0
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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-...
- 431
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268
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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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9
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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 ...
- 7,799
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3
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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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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 $\...
- 131
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Derived stack 2-perfect complexes and derived equivalences
Let $X$ be a scheme of finite type over $\mathbb{C}$. Toën and Vaquié construct the derived stack of perfect complexes on $X$, which I will denote $\mathcal{DP}er(X)$. They prove that this derived ...
- 7,320
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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
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Why is the stabilization of augmented $\mathbb{E}_\infty$-algebras equivalent to $k$-module spectra?
(I have already asked this on Math.SE, but it didn't draw much attention there, so I am reposting it here.)
Example 1.1.4 of Jacob Lurie's DAGX says that the stabilization $\operatorname{Stab}((\...
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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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0
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classifying space of algebraic groups
Let $G$ be a connected reductive group over $\mathbb{C}$. We fix a Borel pair $(B,T)$.
Let $BG$ be the classifying space of $G$.
Can we say that $BG$ is the homotopy colimit of all $BP$ for $P$ a ...
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286
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Exterior tensor of derived categories of coherent sheaves
Let $X, Y$ be Noetherian perfect derived stacks over $S$ a regular perfect derived stack. Consider the exterior tensor functor
$$\text{DCoh}(X) \otimes_{\text{DCoh}(S)} \text{DCoh}(Y) \rightarrow \...
- 1,423
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3
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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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21
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1
answer
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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 ...
- 211
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370
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Three examples of $S^1$-actions on derived loop spaces
Let $X$ be a derived stack. There is a $S^1$-action on the derived loop space $\mathcal{L}(X) = \text{Maps}(S^1, X)$. In particular, $\mathcal{O}(\mathcal{L} X)$ should be quasi-isomorphic to a ...
- 1,423
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Is there a derived geometric interpretation of morse functions?
Given a smooth affine scheme $X = \mathbb{V}(g)$ over a field of characteristic 0, let $f:X \to \mathbb{A}^1$ be a morphism of schemes. Then, the critical locus is given by $\pi_*(dg \cap df)$ for $\...
- 1,716
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1
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438
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Twisted derived Morita theory of schemes
It has been proved by Toën and Lunts-Schnürer that the dg category $\mathrm{L}_{qcoh}(X\times Y)$ of quasi-coherent sheaves over the product of two quasi-compact, quasi separated (and flat over a ...
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0
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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\...
- 3,472
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1
answer
851
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Bar/Cobar Adjunction Between Modules and Comodules
There is a pretty well known, and widely written about, adjunction between augmented algebras and coaugmented coalgebras given by taking the bar construction on algebras and the cobar construction on ...
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4
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0
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232
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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 ...
- 15.4k
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225
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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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2
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2k
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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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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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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 $\...
- 3,017
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1
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859
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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 ...
- 3,017
5
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1
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334
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Reference request: category of sheaves of O-modules with coherent cohomology
Suppose $X$ is a smooth algebraic variety (say, in characteristic $0$). It's a folklore result that $D^b\text{Coh}(X)$ is equivalent to the derived category of complexes of sheaves of $\mathcal{O}_X$-...
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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 ...
14
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0
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930
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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 ...
- 2,227
4
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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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Derived global functions on (derived) stacks $BG$ and $G/G$
In Toen's Affine Stacks, he computes that $\mathcal{O}(B\mathbb{G}_a) = k[\epsilon]$ with $|\epsilon| = 1$ and trivial differential (where here $\mathcal{O}$ is computed in a derived sense, and we ...
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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:
...
- 2,074
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Direct proof that the model category of cdgas is left proper
Let $k$ be a field of characteristic $0$. The projective model structure on the category $cdga$ of commutative differential graded $k$-algebras is proper. Since this model structure is transferred ...
- 15.5k
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On a theorem of Hopkins-Neeman-Thomason on generators of thick subcategories of perfect complexes
Notations and background. Let $R$ be a commutative noetherian local ring and let $D(R)$ denote the derived category of the category of R-modules. A strictly perfect complex on $R$ is a bounded complex ...
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1
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Monoidal Forgetful/Free Adjunction for $E_2$-algebras
Suppose I am given two $E_2$-ring spectra $A$ and $B$ and a morphism of $E_2$-rings $\phi:A\to B$. Then I have $E_1$-monoidal categories of modules $LMod_A$ and $LMod_B$. Moreover I have morphisms $F:...
- 10.5k
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1
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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: \...
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Integral transform on noncommutative spaces
In their paper "Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry" the authors show that for perfect stacks $X$ and $Y$ over $k$, and their $k$-linear $\infty$-categories of ...
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0
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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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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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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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1
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Hopf-algebras in associative ring spectra
I'm interested in a definition of cocommutative Hopf-algebra objects in the $\infty$-category of associative (read: $A_\infty$) ring spectra. One thought I had was to think of cocommutative Hopf-...
- 10.5k
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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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400
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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
2
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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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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 ...
- 4,780
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166
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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
11
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1
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650
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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 ...
- 10.5k