Questions tagged [descent]
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132 questions
6
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effective descent of coherent sheaves
I am new to stacks and algebraic spaces. I have the following question:
Let $X$ be a scheme and $G$ be a group scheme action on $X$. Then $X/G$ exists as an algebraic space. Let $\pi: X \to X/G$ be ...
- 613
3
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0
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122
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Weil restriction of a bunch of points or more general disjoint unions
$\DeclareMathOperator\Spec{Spec}$For a finite extension of fields $k'/k$, let $R_{k'/k}$ denote the Weil restriction functor from quasiprojective $k'$-schemes to quasiprojective $k$-schemes, defined ...
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0
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175
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What is the equivalent of Artin gluing for quasicoherent sheaves?
Given a topological space or locale $X$ and an open $j : U \hookrightarrow X$ with closed complement $i : K \hookrightarrow X$, the inverse image functor $\langle i^*, j^* \rangle : \textbf{Sh} (X) \...
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7
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What are the intermediate semisimple groups of type A?
Background: The first examples one sees of reductive groups over a field $k$ are $\text{GL}_n$, $\text{SL}_n$, and $\text{PGL}_n$. We all know the definitions of $\text{GL}_n$ and $\text{SL}_n$, and ...
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449
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Descent vs effective descent for morphisms of ring spectra
Define a homomorphism $\varphi : A \to B$ of commutative discrete rings or commutative ring spectra to be a (effective) descent morphism if the comparison functor from $\mathsf{Mod}_A$ to the category ...
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109
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The Barr-Kock lemma for regular 2-categories
There is a nice result for regular 1-categories, which I quote from page 441 of Borceux & Bourn's textbook "Mal'cev, Protomodular, Homological and Semi-Abelian Categories".
This is ...
- 121
2
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72
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Gradient descent over the set of complex symmetric matrices
In the course of my research (somewhat related to compressive sensing), I am trying to determine a complex, symmetric matrix $L$ (i.e. $L = L^T$) through the following optimization formulation:
$$ \...
- 21
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Gluing together the moduli stacks of elliptic curves over Z[1/2] and Z[1/3]?
I have a long-running desire to understand what is the "global" moduli stack of elliptic curves, as a stack over $\mathrm{Spec}(\mathbb{Z})$.
Recently I was pointed to Katz and Mazur's book, ...
- 35.4k
2
votes
1
answer
265
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Gluing data for modules over a ring with idempotents
Let $A$ be a ring. If $e$ is an idempotent, then there is an abelian recollement involving the categories $A\text{-}\mathrm{Mod}$ and $eAe\text{-}\mathrm{Mod}$. This is Example 2.7 in Homological ...
- 1,071
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392
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Using comonadicity to prove faithfully flat descent
I have heard many times that faithfully flat descent could be reinterpreted via Beck's monadicity theorem; Deligne's paper "Catégories tannakiennes" even explains in section 4 how to do this ...
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21
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1
answer
797
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The derived category does not satisfy descent - example
One motivation for studying infinity categories is that the derived category does not satisfy Zariski descent, although the infinity categorical version does.
I would like to see an example of Zariski ...
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3
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1
answer
315
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Unitary involutions on a simple central algebra after a scalar extension
$\DeclareMathOperator{id}{id}$
Let $L/K$ be a quadratic separable extension of fields. Let $A$ be a central simple algebra over $L$ such that its norm $N_{L/K}(A)$ splits. Then we know that there ...
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1
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Faithfully flat descent in complex analytic geometry
A common technique for constructing objects (sheaves) and morphisms in algebraic geometry is faithfully flat descent. Roughly speaking this consists on constructing an object or a morphism "...
- 553
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498
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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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182
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Question regarding Galois descent of sections of a vector bundle
Let $\pi: Y\rightarrow X$ be a finite 'etale Galois morphism between two smooth projective varieties with Galois group $G$. Let $\mathcal{E}$ be a vector bundle on $X$. Then $\pi^*\mathcal{E}$ is a $G$...
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698
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Non-trivial automorphisms and descent
In this expository paper by Low it says:
Roughly
speaking, a topos in the sense of Grothendieck is the category of sheaves on a
kind of generalised space whose “points” may have non-trivial ...
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514
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Quasi-syntomic descent and prismatic F-crystals
I am reading Bhatt and Scholze's paper on F-crystals, and they seem to be using the following result in the proof of Theorem 5.6:
let $X \to Y$ be a quasisyntomic cover of formal schemes over $\...
- 461
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0
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194
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Hypercovers with sieves
Consider a covering family $\{Y_i \to X\}$ and the induced sieve $R \subseteq X$, the subpresheaf of all maps to $X$ that factor through some $Y_i$. The family gives me an induced Cech nerve $C_\...
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votes
1
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226
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An analogy of product formula for homogeneous space?
$\DeclareMathOperator\Sel{Sel}$Let $E$ be an elliptic curve defined over a number field $K$ with full $2$-torsion. The classical complete $2$-descent method tells that the $2$-Selmer group $\Sel_2(E/K)...
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Examples of descent in basic algebraic geometry
I'm studying descent theory and I recall that there were multiple instances before where I heard something like "we can prove this as follows, but this is just descent applied to [...]". ...
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273
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Ind-etale vs weakly etale
In this article Bhatt and Scholze consider ind-etale and weakly etale maps of affine schemes. We have two (easy) statements, proven in Prop.2.3.3(1) and (5):
-- any ind-etale map is weakly etale,
-- ...
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Category of coherent sheaves on blow-ups or resolution of singularities
Let $X$ be a scheme and $Y$ a closed subscheme. I would like to know if there is a good relation between the category of coherent sheaves on $X$ and the category of coherent sheaves on the blow-up $\...
- 4,386
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73
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Would the iterated finite abelian descent obstruction equality hold for curves?
Let $X$ be a smooth projective geometrically integral variety over a number field $k$. We begin with some established notions in the theory of descent obstruction to the local-global principle, ...
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303
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2-descent on elliptic curves, and units modulo squares of units
Setup: Let $p$ be a prime, let $f(x) \in \mathbb{Q}_p[x]$ be a separable monic cubic polynomial cutting out the maximal order $\mathcal{O}_{K_f}$ in the etale algebra $K_f := \mathbb{Q}_p[x]/(f(x))$, ...
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254
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Does a field extension define an effective descent morphism for locally ringed spaces?
Let $K'/K$ be an extension of fields and set $X=\operatorname{Spec}(K)$ and $X'=\operatorname{Spec}(K')$. As the category of locally ringed spaces has fibre products (see arXiv:1103.2139 or here) we ...
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641
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fppf/ etale Cohomology calculate with Cech cohomology
Let $R$ be a commutative ring with one and $S$ commutative faithfully flat $R$-algebra (that is there is a faithfully flat ring map
let $\phi: R \to S$). Then the so called Amitsur complex
$R \to S^{\...
- 5,998
6
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1
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499
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If $M\otimes_S T$ is an $A$-module, is $M$ an $A$-module?
Let $\mathbb{C}$ be the field of the complex numbers. Let $R=\mathbb{C}[x]$, $T=\mathbb{C}\langle x\rangle$ be the ring of entire series with convergence radius at least $1$, and let $S=\mathbb{C}\...
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459
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Does isomorphism on local rings imply the global isomorphism for the sheaf of spectra?
Let's assume we have a sheaf of spectra on some scheme. As an example I will assume that we are working with the $K$-theory sheaf. There are certain local to global spectral sequences, like descent ...
- 5,901
11
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1
answer
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How do $\infty$-categories allow us to do descent on the derived level?
I have heard that one application of $\infty$-categories is that they allow us to formulate a meaningful theory of descent for derived categories (say of sheaves on a scheme). While I'm sure the ...
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3
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What is Barr-Beck?
This is a question about a naming convention. The Barr-Beck theorem (or simply Barr-Beck) is used a lot in descent theory over the past 30 years, almost invariably without a reference, like folklore.
...
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2
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262
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Zariski descent of algebraic $K$-theory on formal schemes
This question is highly related to some other questions that I've previously asked, especially to this one. In this problem we have a scheme $X$ and a closed subscheme $Z$ the formal completion $X_Z$. ...
- 5,901
2
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1
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616
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Fpqc-locally constant if and only if étale-locally constant?
Also in SE.
Let $\mathcal{F}$ be sheave over $S_\mathrm{fpqc}$. We say $\mathcal{F}$ is a fpqc-locally constant sheaf (of finitely generated abelian groups) if there exists a fpqc covering $(S_i\to S)...
- 452
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50
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Complex analytic descent along G-actions
Let $G$ be a complex Lie group acting on a complex analytic space $X.$ To be clear, I don't require $X$ to be reduced. Let $f: Y\rightarrow X$ be a smooth morphism such that the $G$-action lifts to $...
- 3,020
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340
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Characterization of effective descent morphism
A faithfully flat morphism of commutative rings $A \rightarrow B$ is an effective descent morphism. So is a regular monomorphism (right?). What is a characterization of effective descent morphisms?
...
user30211
3
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1
answer
133
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Is there any relation between two pseudofunctors associated to two different cleavages of the same fibered category?
It is well known that given a Fibered category $P_F: E \rightarrow C$ with a cleavage $K$ we can construct a pseudofunctor $F_K: C^{op} \rightarrow Cat$. Now if one chooses a different cleavage $L$ ...
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1
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Descent for $K(1)$-local spectra
For odd primes, we have an equalizer diagram for the $K(1)$- local sphere given by
$$L_{K(1)}S \rightarrow K{{ \xrightarrow{\Psi^g}}\atop{\xrightarrow[i_K ] {}}} K$$
where $g$ is a topological ...
- 177
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123
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Seeking bijective proof of a recurrence for generalized Narayana numbers
Consider lattice paths in $d$ dimensions with the steps $X_1\mathrel{:=}(1,0,\dotsc,0)$, $X_2\mathrel{:=} (0,1,\dotsc,0)$,…, $X_d\mathrel{:=} (0,0,\dotsc,1)$. Let $\mathcal C(d, n)$ denote the set of ...
- 1,644
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2
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396
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Projective after fpqc base change
Let $S$ be a Noetherian affine scheme. Let $S'\to S$ be a flat surjective morphism of affine schemes. Let $X\to S$ be a morphism such that $X_{S'}\to S'$ is projective. Is $X\to S$ projective? It is ...
user145520
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Fundamental group of a smooth projective curve of char $0$
In this note of Akhil MATHEW, when he proves the fundamental group of a smooth projective curve over a algebraic closed field $k$ of characteristic $0$ admits $2g$ topological generators, there are ...
user141691
4
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1
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411
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Reducing the stack condition (descent condition) over an fpqc site to the case of single coverings
This is the lemma 4.25 of Vistoli's note
Let $S$ be a scheme, $\mathscr{F} \to \mathscr{S}ch/S$ a fibred category.
Then $\mathscr{F}$ is a stack over the fpqc site on $S$ iff
(1) $\mathscr{F}$ ...
- 1,364
4
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1
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286
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Descent a representation over finite field
Let $p$ be a prime integer, and $q$ a power of $p$. Let $\mathbb{F}_p$ and $\mathbb{F}_q$ be the corresponding finite fields. Suppose
\begin{equation}
\rho: G\longrightarrow GL_2(\mathbb{F}_q)
\end{...
- 461
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Is nefness preserved under base change
Let $f:X \rightarrow Y$ be a morphism between (geometrically normal) varieties over a field $k$, $\bar{k}$ be the algebraic closure of $k$ and $B$ be a Cartier divisor on $X$ which is $f$-nef, that is ...
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Local question and descent category for a quasi-coherent sheaf on $\mathbb{G}_m$-gerbe
Update: I removed what I thought was unecessary and tried to be more straightforward in the hope to get an answer.
Context:
Suppose I have a $\mathbb{G}_m$-gerbe $\mathcal{G}$ over a scheme $X$ with ...
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348
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Notions of algebraic/differential geometry of scheme/manifolds extended to algebraic/differential stacks
Given a manifold, one can associate a stack over the category of manifolds, which is a differential geometric stack. This gives a functor $\text{Man}\rightarrow \text{D.Stacks}$. This is an embedding....
- 6,023
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1
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51
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Descending central extensions to homogeneous spaces
Let $G$ be a Lie group (finite dimensional or Banach), and let $H$ be a Lie subgroup (in the Banach case we assume that $H$ is a submanifold which is also a Lie group). Let $\text{U}(1) \rightarrow \...
- 556
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134
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Completion in the non-noetherian case
Let $A$ be a non-noetherian, commutative $\mathbb{C}$-algebra and $X, Y$ be noetherian affine $\mathbb{C}$-schemes. Denote by $X_A:=X \times_{\mathbb{C}} \mbox{Spec}(A)$ and $Y_A:=Y \times_{\mathbb{C}}...
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Number Rings and (Galois) Descent
In algebraic number theory, one chooses for each finite étale $\mathbb{Q}$-algebra $K$ a finite $\mathbb{Z}$-algebra $\mathcal{O}_K$. Usually one simply speaks of the finite $\mathbb{Q}$-algebras ...
user30211
4
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0
answers
253
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Homotopy colimit description of stacks
Let $F$ be an Artin stack. If $p: X \to F$ is an atlas for $F$, can we express $F$, in the $\infty$-category ${\rm Shv}^{\acute{et}}(k)$ of higher stacks, as a homotopy colimit over the simplicial ...
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When can a scheme be recovered from its descent groupoid?
Suppose that $ Y $ is a scheme and $ f\colon X\to Y $ a covering of $ Y $ in some Grothendieck topology on the category of schemes (i.e. if $\{ U_i\to Y\}$ is a covering in the topological sense, then ...
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3
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Infinite Galois descent for finitely generated commutative algebras over a field
Let $k_0$ be a field of characteristic 0, and let $k$ is a fixed algebraic closure of $k_0$.
Write $G={\rm Gal}(k/k_0)$.
Let $A_0$ be a finitely generated commutative $k_0$-algebra with a unit.
Then ...
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