Questions tagged [descent]
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138 questions
1 vote
0 answers
93 views
Compositions of Affine Torsors
Let $X_2\xrightarrow{f}X_1\xrightarrow{g}X_0$ be the composition of $k$-schemes of finite type, where $k$ is perfect of characteristic $p>0$. Assume that each $X_{i+1}\to X_i$ forms a $G_i$-torsor ...
- 325
1 vote
0 answers
121 views
Descent of vector bundles on Quot schemes?
Let $S$ be a scheme and $T : M \to N$ a morphism of locally free rank $n$ sheaves on $S$. Let $p : Q(T , r) \to S$ where $Q(T , r) := \mathrm{Quot}(\mathrm{CoKer}\,T , r)$ for $0 \leq r \leq$ minimal ...
- 526
5 votes
0 answers
334 views
Descent of formal affine morphisms
Suppose $\mathfrak{X}$ is a formal scheme over an affine scheme $S$ such that on a Zariski (alternatively étale/fppf/fpqc) cover $U_i$ of $S$ the base changed formal schemes $\mathfrak{X}|_{U_i}$ are ...
- 2,540
7 votes
0 answers
119 views
Is the pseudo-codescent object of discrete opfibrations a discrete opfibration?
This is a question about "descent" or "van Kampen-ness" of colimits in the 2-category $\mathcal{C}\mathsf{at}$ of categories. Because $\mathcal{C}\mathsf{at}$ has a discrete ...
- 165
5 votes
1 answer
370 views
Do forms of $\mathbb{G}_a$ split over perfect rings?
I'll try to pursue my understanding of forms of $\mathbb{G}_a^d$ over general bases. Let $R$ be a ring of positive characteristic $p$ and let $G$ be a smooth affine group scheme over $R$ that is a ...
- 1,573
4 votes
0 answers
101 views
Avoiding class/unit group computation when computing $p$-Selmer groups
Let $K$ be a number field, $S$ be a finite set of places of $K$, and $K(S,p)$ be the $p$-Selmer group of the $S$-integers of $K$, that is the set of nonzero elements of $K$ modulo $p$-th powers whose ...
- 273
6 votes
1 answer
398 views
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 ...
- 1,181
3 votes
0 answers
191 views
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 ...
- 824
4 votes
0 answers
248 views
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) \...
- 17.4k
7 votes
1 answer
347 views
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 ...
- 977
8 votes
0 answers
521 views
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 ...
- 448
7 votes
0 answers
151 views
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 votes
0 answers
103 views
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
4 votes
0 answers
377 views
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, ...
- 36.8k
2 votes
1 answer
304 views
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 ...
- 1,207
4 votes
0 answers
519 views
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 ...
- 803
21 votes
1 answer
958 views
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 ...
- 323
3 votes
1 answer
406 views
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 ...
- 93
7 votes
1 answer
512 views
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
7 votes
1 answer
679 views
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 ...
- 258
1 vote
0 answers
222 views
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$...
- 787
9 votes
2 answers
784 views
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 ...
- 91
6 votes
0 answers
661 views
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 $\...
- 481
3 votes
0 answers
279 views
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_\...
- 1,134
4 votes
1 answer
269 views
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)...
6 votes
0 answers
585 views
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 [...]". ...
- 1,033
3 votes
0 answers
327 views
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, -- ...
- 795
1 vote
0 answers
216 views
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 $\...
- 4,468
1 vote
0 answers
93 views
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, ...
- 915
5 votes
0 answers
346 views
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))$, ...
7 votes
0 answers
291 views
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 ...
- 71
2 votes
0 answers
766 views
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^{\...
- 3,914
6 votes
1 answer
536 views
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}\...
- 1,573
3 votes
0 answers
527 views
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 ...
- 6,063
12 votes
1 answer
2k views
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 ...
- 4,232
23 votes
3 answers
5k views
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. ...
- 16k
2 votes
0 answers
306 views
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$. ...
- 6,063
2 votes
1 answer
681 views
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)...
- 632
2 votes
0 answers
71 views
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 $...
- 3,060
3 votes
0 answers
420 views
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 votes
1 answer
173 views
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$ ...
- 1,235
6 votes
1 answer
456 views
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
3 votes
0 answers
145 views
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 ...
- 1,903
4 votes
2 answers
446 views
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
0 votes
1 answer
268 views
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 votes
1 answer
452 views
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 votes
1 answer
358 views
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{...
- 543
1 vote
0 answers
168 views
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 ...
- 121
2 votes
1 answer
448 views
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 ...
- 65
2 votes
0 answers
393 views
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....
- 6,203