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Questions tagged [derived-algebraic-geometry]

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Moduli stack of doubly periodic complexes?

Let $\mathcal{A}$ be an abelian category. In HAG II Toen and Vessozi built a higher derived stack $X$ whose category of perfect complexes is $\text{Perf}(X)\simeq D^b(\mathcal{A})$. So $X$ is a good ...
Pulcinella's user avatar
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Higher-order HKR theorems?

Recall that Hochschild-Kostant-Rosenberg -type theorems identify certain smoothness conditions guaranteeing an isomorphism between the cotangent complex and (a shift of) the Hochschild homology of an ...
Tim Campion's user avatar
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189 views

Is the homotopy limit of derived schemes along affine maps a derived scheme?

The title question is true in the setting of ordinary limits and ordinary schemes; that is, given an inverse limit of schemes along affine maps, the limit still lives in the category of schemes. I'd ...
Eric's user avatar
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Formally étale maps of animated $k$-algebras

In Lurie's DAG, he defines what it means for a natural transformation $T:\mathcal{F}\to\mathcal{F}'$ of functors $\mathcal{F},\mathcal{F}':\mathcal{SCR}\to\mathcal{S}$ to be formally étale. Namely, it ...
Eric's user avatar
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About derived divided power envelope

Assume $A$ is a $\mathbb{Z}_{(p)}$-algebra with ideal $I$ and $A,A/I$ are $p$-torsionfree. In this survey, Akhil Mathew defines the derived divided power envelope $LD_I(A)$ in Construction 7.15, after ...
Yang Chen's user avatar
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Lie bracket on the unshifted tangent complex?

My problem is as follows: if $X$ is a derived scheme, or derived stack, or any kind of a space where tangent complex makes sense, I guess there should be a lie bracket on its tangent complex, ...
dpistalo's user avatar
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Structure sheaf of derived intersection

Everything is over a field $k$ of characteristic $0$. Let $X$ and $Y$ two closed dg subschemes over a dg scheme $Z$. I am trying to understand the structure sheaf of the derived intersection of $X$ ...
Federico Barbacovi's user avatar
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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 ...
Daniel Pomerleano's user avatar
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94 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 ...
GTA's user avatar
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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 ...
Dmitry Vaintrob's user avatar
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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 $A\in k\text-\mathsf{cdga}_{\leq0}$ ...
elidiot's user avatar
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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}(...
Exit path's user avatar
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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 $A$-...
John Doe's user avatar
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Linear $\infty$-categories $\mathrm{QC(X)}$ and $\mathrm{Perf(X)}$ of a "derived" stack $\mathrm{X}$

For each scheme or algebraic stack their $\infty$-category of quasicoherent sheaves (resp., perfect complexes) on it is $k$-linear for a commutative ring $k$. That is (by a recent result of L.Cohn), ...
jereckherr's user avatar
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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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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}((\...
a-w's user avatar
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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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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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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 $...
Zhaoting Wei's user avatar
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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 ...
Paulo Rossi's user avatar
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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. ...
cdsb's user avatar
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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 ...
Yang's user avatar
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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....
Yang's user avatar
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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 ...
Yang's user avatar
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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 ...
cdsb's user avatar
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Fourier-Mukai transform is the derived functor

In Mukai's paper Duality between $D(X)$ and $D(\hat{X})$ with its application to Picard sheaves, Nagoya Math Journal, 1981, there is one sentence that puzzles me. Let $X$ be an abelian variety over an ...
Doug Liu's user avatar
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Computing the cotangent complex of morphisms of perfect complexes

In Lurie's Spectral Algebraic Geometry the cotangent complex of $\textbf{Perf}$ is computed as $ \Sigma^{-1}( \mathscr{F} \otimes \mathscr{F}^\vee)$ for some universal $\mathscr{F} \in \text{Qcoh}(\...
Anette's user avatar
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A question about relative deformations of the structure sheaf of the diagonal

Let $X$ be a smooth proper algebraic variety over $\mathbb{C}$. Let us consider an associative algebra $${p_2}_*{\mathcal{H}{om}}_{X\times X}(\mathcal{O}_\Delta, \mathcal{O}_\Delta) \in \text{Alg}_\...
Grisha Konovalov's user avatar
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202 views

Is the cotangent complex sensitive to truncation?

$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\Spec{Spec}$If $V$ is a dg vector space (in positive degrees), we can view it as a derived scheme $V = \Sym V^*$. It has (co)tangent complex $V$ (and $...
Pulcinella's user avatar
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Two definitions of cotangent complex

I have reading a paper by Professor Pridham(https://arxiv.org/abs/0905.4044v4). Page 47-48 contains a comparison of the two definitions of the cotangent complex, but there is a part I don't understand....
Walter field's user avatar
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281 views

Étale homotopy type of (derived) loop space

A feature of derived algebraic geometry is that we have internal homs. Furthermore, we can think of $B\mathbb{Z}$ as the derived algebraic geometric analogue of $S^1$. Thus we have an analogue of the ...
curious math guy's user avatar
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202 views

Schlessinger criterion and finiteness of tangent space

Schlessinger's criterion allows us to study whether or not a functor $\mathcal{F}:C_{\Lambda}\rightarrow \text{Set}$ from the category of local Artin $\Lambda$-algebras to sets is representable. One ...
curious math guy's user avatar
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137 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 ...
Walter field's user avatar
1 vote
0 answers
185 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 ...
Walter field's user avatar
1 vote
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148 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 ...
Merik's user avatar
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How can we construct a derived scheme as a gluing of derived schemes?

More precisely, consider a Segal groupoid $X_*$ in an infinity category of derived schemes : dSch In Toen's note, 'Derived Algebraic Geometry', he defines a 1-Artin stacks as a homotopy colimits of ...
keaton's user avatar
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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 ...
math no more's user avatar
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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\...
prochet's user avatar
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1 vote
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300 views

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, ...
ahar's user avatar
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Cone of morphism induced by Serre duality

For a smooth projective variety $X$, Serre duality gives an exact autoequivalence on the derived category : $$ S_X : D^\flat(X) \to D^\flat(X), \hspace{3em} S_X(-) = - \otimes \omega_X[\dim X] $$ ...
cdsb's user avatar
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What means "extended concepts of symmetry"?

Where could one find a short description oft: "two mathematical extensions of the symmetry - to moduli spaces of sheaves and to derived categories", found here? Happen there interesting things like ...
Thomas Riepe's user avatar
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