Questions tagged [derived-algebraic-geometry]
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276 questions
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Operations on filtered / graded vector spaces via $\mathbb{A}^1 / \mathbb{G}_m$
A well known fact (e.g. Moulinos - The geometry of filtrations) is that one can describe the category of filtered vector spaces as $\operatorname{FilVect} \simeq \operatorname{QCoh}(\mathbb{A}^1/\...
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Is the geometric realization of simplicial functors interesting?
While studying a completely unrelated problem, I have proved something on the following line: given a diagram of simplicial sets $X: C \to \textrm{sSet}$, some deformations of the geometric ...
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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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The assignment of derived infinity category of étale sheaf is an infinity functor?
Consider the ordinary category of schemes $Sch$, for $X\in Sch$, consider the abelian category of étale sheaf with coefficient $\wedge$ as $Mod(X_{ét},\wedge)$, then we can form the derived infinity ...
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Derived Koszul complex
Let $X$ be a projective variety over $\mathbb{C}$ and $V$ be a vector bundle over $X$. Let $\pi: V\to X$ be the natural projection.
Let $i: X\to V$ be the zero section map. Let $V^\vee$ be the dual ...
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Construction of smooth projective space in Spectral Algebraic Geometry
In section 19.2.6 of Lurie's "Spectral Algebraic Geometry," he constructs the smooth projective space, which represents the derived version of [the dual of] the usual functor of points ...
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Status of motives in higher category theory: motives and algebraic cycles through a higher categorical perspective
A while ago this interesting question was asked Derived Algebraic Geometry and Chow Rings/Chow Motives.
Primary question:
Have there been any recent developments/advances on the above question? If not,...
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Étale morphisms of derived schemes and stacks
Conventions: In the below,
unless otherwise stated, terms regarding derived algebraic geometry will follow the conventions of Yaylali.
an algebraic stack will be a stack $\mathscr{S}$ over a base ...
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'Naive cotangent complex' as 1-truncation of cotangent complex
In the stacks project, there is a 'naive version' of cotangent complex $NL_{S/R}$ for a ring morphism $R\rightarrow S$, given by the chan complex $(I/I^{2}\rightarrow \Omega_{R[S]/R}\otimes_{R[S]}S)$ ...
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full strong exceptional collection
I am wondering whether, if a triangulated category $\mathcal{D}$ has a full strong exceptional collection (infinite), it is triangle-equivalent to the bounded derived category of finitely generated ...
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Geometric stability conditions on calabi-yau's fibred over Fano always identical to geometric stability conditions on Fano
I apologize in advance for the long title. This question is motivated primarily by [2], with the explicit example of $\mathbb{P}^2$ and $\omega_{\mathbb{P}^2}$ computed in [3] and [1], respectively.
...
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$\varinjlim_{k}\Omega^{k}\circ \Sigma^{k}$ as 1-excisive approximation of the identity functor
In chapter 6 of HA by Lurie, for any functor $F:C\rightarrow D$ between 'nice' categories (like differentiable infinity categories), there is an $n$ excisive approximation to this functor behaving ...
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Square zero extension in the derived setting
Here we take the infinity category of simplicial ring $SCRing=Fun^{\prod}(Poly^{op},Spc)$ and follow the construction 25.3.1.1 in DAG by Lurie, where we extend the construction of square zero ...
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Infinite suspension is cotangent complex
In Higher Algebra by Lurie, we define the absolute cotangent complex $L_{A}$ through the composition $C\stackrel{\triangle}{\longrightarrow} Fun(\triangle^{1},C)\stackrel{F}{\longrightarrow}T_{C}$ ...
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The spectrum object in the $\infty$ category $CAlg_{R}$
Consider the infinity category of simplicial rings $CAlg=Fun^{\prod}(Poly^{op},Spc)$, and the under-category over $R$: $CAlg_{A/}$ is equivalent to $CAlg_{A}=Fun^{\prod}(Poly^{op}_{R},Spc)$ by 25.1.4....
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The space of virtual Cartier divisors on a classical scheme over a closed immersion is discrete
I am currently reading the paper Virtual Cartier divisors and blow-ups where the virtual Cartier divisor on an $X$ scheme $S$ over a quasi-smooth closed immersion $Z\rightarrow X$ is defined to be the ...
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Derived b-calculus and logarithmic tangent sheaves
Melrose's b-calculus provides a powerful framework for analyzing elliptic operators on manifolds with boundary. In the context of log geometry, log smooth manifolds offer a natural generalization of ...
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Construction of Weil restriction in the derived setting
I am currently reading 19.1,2 in SAG by Lurie about Weil restrictions. In the classical case, Weil restriction is the right adjoint to the pullback functor along some morphism $f:X\rightarrow Y$, and ...
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Existence of Kan extension for the functor with codomain a complete infinity category
I am currently reading this paper on derived blow up, in definition 2.4.1, I am faced with such situation:
if we denote the infinity category of simplicial ring as $Alg$ and the 1 category of ...
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Does derived tensor product preserve fiber sequence？
In lemma 3.1.5 of this paper I read, there is a fiber sequence of the underlying spaces of simplicial commutative rings $A\stackrel{f}{\longrightarrow} A\rightarrow A/\!\!/f$. Here we define the "...
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The exact sequence for a derived zero locus
For a locally free sheaf of rank one $L$ on a derived scheme and a morphism $s:L\rightarrow O_{X}$, we consider the derived zero locus of $s$ defined by the following derived fiber product
$$\require{...
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Classical schemes as derived schemes are discrete valued
$\newcommand\Spc{\mathrm{Spc}}\newcommand\SCRing{\mathrm{SCRing}}\DeclareMathOperator\Map{Map}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Hom{Hom}\...
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What is the Isomorphism subspace of the mapping space in an infinity category
When $E$ is a locally free sheaf of rank n on a classical scheme $X$, there is a sheaf $Isom$ on the category $Sch_{X}$ defined as $(S\rightarrow X)\rightarrow Isom_{O_{S}}(O_{S}^{n},E)$. And this ...
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Divided power structure on $E_\infty$-algebras?
Let $A$ be a simplicial commutative ring, then it is known that the ideal of elements of degree $\ge1$ in the associated CDGA has a "DG divided power structure," which induces a divided ...
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Reference: the category of derived affine schemes is extensive
The category (that is, $(\infty, 1)$-category) of derived affine schemes is the opposite category of the localization of simplicial commutative rings in weak equivalences.
See extensive category. Does ...
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What do we know about effective epimorphisms of derived affine schemes/manifolds?
By default, all terms are understood in the infinity sense (“category” means “(∞,1)-category”, etc.)
Recall that the morphism $X \to Y$ is an effective epimorphism if the Čech diagram
$$ ... \to X \...
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Does every Artin $\infty$-stack have a formally étale cover by semi-free non-connective CDGAs?
The title essentially says it all. Feel free to assume as many finite generation conditions as you want. For example, I'm pretty sure the Chevalley-Eilenberg complex $\simeq \wedge^\bullet \mathfrak g^...
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Animated rings and $\mathbb E_{\infty}$-rings
Let $R$ be a discrete commutative ring. The Proposistion 25.1.2.4 in Lurie's Spectral Algebraic Geometry says that the natural functor from the $\infty$-category of animated $R$-algebras to the $\...
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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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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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Derived $\infty$-category of quasi-coherent sheaves on schemes
Let $X$ be a scheme. On the one hand, we have the derived $\infty$-category constructed from the abelian category of quasi-coherent sheaves on $X$. On the other hand, we can define the stable $\infty$-...
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Pushforward of exceptional vector bundle is spherical for local P^2
I've been reading through a bit of the literature on stability conditions, and one of the models that has come up is the 'local projective plane'. Explicitly, this is the total space of the canonical ...
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New investigations on Homotopical Algebraic Contexts
Homotopical algebraic context are models that allows Toën and Vezzosi to do derived geometry. It have been defined in their seminal paper Homotopical Algebraic Geometry II.
These are general abstract ...
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The criterion for dimensional conjecture for universal Galois deformation rings
I’m writing to ask a question about Mazur’s dimensional conjecture in Lemma 7.5 of the paper [Galatius S, Venkatesh A. Derived Galois deformation rings. Advances in Mathematics. 2018 Mar 17;327:470-...
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“Geometric” vs Homotopical completion
There are two notions of completions of slightly different nature, and I am wondering if there is a precise statement relating them.
The first one is the “homotopical” (or maybe it should be called ...
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Quasicompact quasiaffine classical schemes are nonconnectively-affine
In this answer to What is the relationship between connective and nonconnective derived algebraic geometry? I learned that any quasicompact open subscheme of an affine scheme is affine in the sense of ...
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Is a derived scheme determined by classical + formal points?
Say we have a derived scheme over an algebraically closed field $X/k$, viewed as a functor $X : \operatorname{Aff}_k^{\operatorname{op}} \to \infty\operatorname{-Grpd}$ and we know its formal ...
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Resolutions of semi free (or almost free) commutative dg algebras with finitely generated cohomology
Let $A^{\bullet}:=\{ \cdots \rightarrow A^i \overset{d^i}{\rightarrow} A^{i+1} \rightarrow \cdots \rightarrow A^{-1} \rightarrow A^0 \rightarrow 0 \rightarrow \cdots \}$ be a non-positively graded ...
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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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Derived flat bundles
I am looking for a notion of derived flat bundles over a surface $X$. Flat vector bundles may be thought of in terms of surface representations $\pi_1(X)\rightarrow\text{GL}(V)$. Is there a notion of ...
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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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Does "derived" make anything constant in non-flat families?
This is an extremely basic (and surely amateurish) question that might be about derived geometry.
In usual algebraic geometry, if we have a flat projective morphism $f:X \to S$ with $S$ integral, and ...
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Formal neighborhood of isolated singularity via DAG
I work over a field of characteristic $0$, denoted $k$. Let $f:\mathbf{A}^{d+1}\rightarrow\mathbf{A}^{1}$ have an isolated singularity at $0$, and let $\widehat{Z}$ denote the formal neighborhood of $...
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Algebraic Fukaya categories and mirror symmetry
Dominic Joyce and collaborators have outlined a programme to construct algebraic Fukaya categories on an algebraic symplectic manifold (“Fukaya categories” of complex Lagrangians in complex symplectic ...
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Dualizing sheaf for classifying stack and duality
For an algebraic group $G$ there should be an equivalence $\operatorname{Rep}(G) \simeq \operatorname{IndCoh}(BG)$. I'm trying to understand what the dualizing sheaf (or complex) of $BG$ is. Here's ...
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What is the k-linear structure on the derived infinity category of quasi-coherent sheaves?
Let $f : X \overset{f}{\rightarrow} Y \overset{g}{\rightarrow} \mathrm{Spec} (k)$ be morphisms of schemes (feel free to add any hypothesis necessary). Let $\mathrm{QCoh}(Y)$ denote the derived (stable)...
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Computing Grothendieck group of (unnodal) Enriques surface
Let $X$ be an unnodal Enriques surface together with an isotropic 10-sequence $\{ f_1, \dots, f_{10}\} \subset \operatorname{Num}(X)$, and let $F_i^\pm \in \operatorname{NS}(X)$ denote the two ...
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resolution property and perfect stacks
Recall that for a scheme $X$, it has the resolution property if every coherent sheaf $E$ on $X$, is the quotient of a finite locally free $\mathcal{O}_X$-module.
On the other hand, Ben-Zvi-Nadler-...
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Examples when algebraic 1-stack = derived enhancement?
Are there any examples where a usual algebraic 1-stack $X$ and the corresponding derived stack enhancement $\mathbb{R}X$ coincide?
Let me take an example from notes of Bertrand Toen, page 41 of https:/...
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Formal loop space in algebraic geometry
Does anyone have a reference or an explanation about the relationship between the formal loop space defined for affine schemes via $LX\left(R\right) = X\left(R\left(\left(t\right)\right)\right)$ (or ...
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