Questions tagged [derived-algebraic-geometry]
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276 questions
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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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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 ...
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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 ...
user74900
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What is the original source for the Goerss-Hopkins-Miller-Lurie theorem on tmf?
The central basic theorem of topological modular forms states that the structure sheaf of $\widehat{\mathcal{M}}_{ell}$ lifts to a sheaf of complex-oriented $E_{\infty}$-rings whose formal groups are ...
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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 ...
user74900
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Explanation for Lurie's SAG Remark 25.1.3.7
I am trying to understand the theory of simplicial commutative rings or animated rings. I just find a remark in Lurie's book Spectral Algebraic Geometry:
Remark 25.3.1.7. Let $f : R[x_1,\ldots ,x_n]\...
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Defining ind-coherent sheaves and their singular support
Q1: My first question is about defining the category $\text{IndCoh}(S)$ for a $DG$ scheme $S$. So in page $18$ of this paper, they are defined as being the ind-completion of the category $\text{Coh}(S)...
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How to characterize $E_n$ morphisms from $\mathrm{Mod}(A)$ to $\mathrm{Mod}(B)$?
Suppose $A$ and $B$ are $E_{\infty}$ rings, then $\mathrm{Mod}(A)$ and $\mathrm{Mod}(B)$ are $E_{\infty}$ monoidal categories (left modules over those rings). We can ask about $E_n$ colimit-preserving ...
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In the not necessarily abelian cat setting, is there a Grothendieck spectral sequence for computing the homotopy of a composition of derived functors?
Recall the Grothendieck spectral sequence which computes the homology groups of a composition of left derived functors $F$ and $G$ on abelian categories:
\begin{align*}
E_{p,q}^{2}(A)=L_{p}G\circ L_{q}...
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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....
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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 $...
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Coherent objects in a hypercomplete $\infty$-topos
In Lurie's "Spectral Algebraic Geometry", Proposition A.6.6.1 (2) shows that for $\mathcal{X}$ an $\infty$-topos that is both locally coherent and hypercomplete, the full subcategory $\...
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Every Spectral Deligne-Mumford stack satsifies fpqc descent?
In SAG Remark 6.3.3.8, Lurie asserts that if we have a representable (by Spectral Deligne-Mumford stacks) natural transformation $X\to Y$ where $Y$ is a functor satisfying fpqc descent, then so too ...
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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 ...
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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 $\...
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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 $R$-...
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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, ...
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Milnor excision for algebraic stacks
Recall that a commutative square of commutative rings
$$\begin{matrix}
A&\to&B\\
\downarrow &&\downarrow\\
A^\prime&\to&B^\prime\end{matrix}$$
is called a Milnor square if the ...
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If $A\in \mbox{Rings}\subset E_\infty\mbox{-rings}$, what is the equivalence between objects of $\mathcal{D}(\mbox{Mod}_A)$ and $A$-module spectra?
In Lurie's "A Survey of Elliptic Cohomology", he writes on page 14 that if $A$ is an ordinary commutative ring considered as an $E_\infty$-ring, then $A$-module spectra are the same thing as objects ...
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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 $...
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D-modules as ind-coherent sheaves over positive characteristics?
There is an interpretation of D-modules over "sufficiently nice" prestacks $X$ (read: various finiteness conditions apply, perhaps even smoothness) by Gaitsgory and Rozenbylum (see chapter I....
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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 ...
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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 ...
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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 ...
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$\infty$-categorical enhancement of $\mathsf{D}_\mathsf{B}(\mathsf{A})$
In this question, it is asked why we like to consider $\mathsf{D}_\textrm{qc}(X)$ rather than $\mathsf{D}(\mathsf{QCoh}(X)).$ Professor Cisinski answers rather convincingly that the $\infty$-...
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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-...
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What's the definition of a microlocal sheaf?
I'm slowly becoming familiar with what microsupport of a sheaf is, but none of the references I've seen give a definition of what a microlocal sheaf should be in general.
In this paper of ...
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Relationship between different definitions of the Hochschild homology
Throughout the literature, one can find many definitions of the Hochschild homology of various objects. However, the precise relationship between these definitions is not always so clear, at least to ...
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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 ...
user138661
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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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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 ...
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If we replace the spectrally ringed space in the definition of a spectral scheme with an arbitrary infinity-topos, what objects do we get?
I'll phrase this in terms of spectral AG, but I'm curious about the same question in the classical context.
We define a nonconnective spectral Deligne-Mumford stack to be a spectrally-ringed topos ...
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Higher commutators in E_n algebras and the Maurer--Cartan equation
Let $A$ be an associative algebra in $dgVect_k$. Then the commutator $[\cdot,\cdot]:A\otimes A\to A$ defined by $[x,y]=xy-(-1)^{|x||y|}yx$ gives $A$ the structure of a (dg-)Lie algebra. The Maurer--...
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Basic example of derived descent
I've been trying to understand the Adams spectral sequence, and I've gotten myself confused about how derived descent is supposed to work, so I would like to understand a simple example.
Given a ...
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What's the relation between analytic stacks and higher complex/non-archimedean analytic stacks?
Recently Clausen and Scholze have developed a theory of analytic stack to unify different analytic geometries. Actually I do not know many details about it. It says an analytic stack is a sheaf $\...
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Which morphisms of ring spectra are of effective descent for modules?
There is a well understood bifibration of $\infty$-categories over the $\infty$-category of commutative ring spectra whose fiber over a ring $R$ is the category of $R$-module spectra. This is in ...
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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 critical locus
I am looking for discussion in the literature that properly formalizes the heuristic idea that a BV-BRST complex is a model for the "derived critical locus of a function on an $\infty$-Lie algebroid".
...
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Kapustin-Witten branes and the derived moduli stack of Higgs bundles
A lot has been discussed on overflow regarding geometric Langlands and the physics of Kapustin and Witten's groundbreaking paper https://arxiv.org/abs/hep-th/0604151. I would like to add my two cents ...
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Canonical comparison between $\infty$ and ordinary derived categories
This question is a follow-up to a previous question I asked.
If $\mathcal{D}(\mathsf{A})$ is the derived $\infty$-category of an (ordinary) abelian category $\mathsf{A},$ then the homotopy category $h\...
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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 ...
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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 $\...
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Derived (non-commutative) geometry, geometric constructions in explicit form
I'm interested in the following construction. Start with derived category of coherent sheaves witch equivalent to derived category of representations of some dg-algebra. Quasi-isomorphic dg-algebras ...
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The De Rham Stack and $\text{LocSys}$
Q1: Here, on pg $103$, the stack $\text{LocSys}_G(X)$ is defined (for an affine algebraic group $G$, a fixed DG scheme $X$, and a test scheme $S \in \textbf{DGSch}$):
$Maps(S, LocSys_G(X)) := \text{...
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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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Somewhat general question that includes: "Do quasi-isomorphic cdgas have quasi-isomorphic spaces of derivations?"
Question: Given two quasi-isomorphic dg commutative algebras (over a field of characteristic zero, if you like), to what extent do their various homological geometric data agree?
Example: Given a dg ...
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Small objects in categories
I would like to pick out small objects from a category. I would like to find such a notion which
Dream 1. Picks out the schemes of finite type over $k$ from the category of $k$-schemes. Or at least ...
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What is a good reference (preferably thorough) for the Derived Category of a scheme/orbifold/stack?
I've sort of circled around the idea of derived categories a few times, read a few introductory papers ("Derived Categories for the working mathematician", e.g.), and feel now that this is something ...
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Reference: the category of derived affine schemes is extensive
The category (that is, $(\infty, 1)$-category) of derived affine schemes is the opposite category of the localization of simplicial commutative rings in weak equivalences.
See extensive category. Does ...
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2-shifted Poisson bracket on Lie algebra cohomology
Let $\frak{g}$ be a semisimple Lie algebra, and let $({-},{-})$ be an invariant inner product on $\frak{g}$. The Chevalley–Eilenberg complex $C^*(\frak{g})$ has a natural Poisson bracket of degree $-2$...
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