# Questions tagged [descent]

The descent tag has no usage guidance.

123
questions

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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 ...

20
votes

1
answer

576
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 ...

3
votes

1
answer

205
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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 ...

7
votes

1
answer

252
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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 "...

7
votes

1
answer

307
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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 ...

1
vote

0
answers

119
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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 $G$...

9
votes

2
answers

595
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 ...

4
votes

0
answers

297
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 $\...

3
votes

0
answers

148
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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_\...

4
votes

1
answer

196
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

355
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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 [...]". ...

3
votes

0
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214
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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,
-- ...

1
vote

0
answers

140
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 $\...

1
vote

0
answers

66
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, ...

5
votes

0
answers

251
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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 $K_f := \mathbb{Q}_p[x]/(f(x))$, ...

7
votes

0
answers

241
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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 ...

2
votes

0
answers

448
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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^{\...

6
votes

1
answer

493
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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}\...

3
votes

0
answers

347
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 ...

9
votes

1
answer

865
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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 ...

23
votes

3
answers

3k
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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.
...

1
vote

0
answers

217
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 $X_Z$. ...

1
vote

1
answer

520
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)...

2
votes

0
answers

40
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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 $...

2
votes

0
answers

207
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?
...

2
votes

1
answer

119
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 $L$ ...

6
votes

1
answer

374
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 ...

3
votes

0
answers

105
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 ...

4
votes

2
answers

325
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 ...

0
votes

1
answer

233
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 ...

4
votes

1
answer

392
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) $\mathscr{F}$ ...

3
votes

1
answer

239
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{...

1
vote

0
answers

103
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 ...

2
votes

1
answer

319
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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 ...

2
votes

0
answers

305
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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....

2
votes

1
answer

47
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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 \...

1
vote

0
answers

114
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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}}...

5
votes

1
answer

381
views

### 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 ...

4
votes

0
answers

222
views

### 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 ...

4
votes

1
answer

173
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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 ...

7
votes

3
answers

593
views

### 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 ...

3
votes

1
answer

281
views

### Stack descent to sheaf descent via Grothendieck construction?

Let S be a Grothendieck site, the (either left or right adjoint to the) Grothendieck construction assigns to each groupoid fibration over S a presheaf valued in groupoids. The following feels it might ...

4
votes

1
answer

231
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### Descent of isomorphisms between irreducible closed subschemes

Let $S$ be an affine scheme, $X$ be a projective $S$-scheme, $W,Z\to X$ two reduced, irreducible closed $S$-subschemes, flat over $S$. Let $S'\to S$ be a faithfully flat map, with $S'$ affine.
Assume ...

4
votes

0
answers

259
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### Representable $\text{Hom}$ functors

Let $X, Y, S$ be noetherian schemes, $X$ flat and quasi-projective over $S$, $Y$ projective over $S$.
Is the hom-functor $T\mapsto\text{Hom}_T(X_T, Y_T)$ representable?
If $X$ is flat and projective,...

5
votes

0
answers

230
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### Descent along purely inseparable morphisms

What properties of $\mathbf{F}_p$-algebras descent along powers of the absolute Frobenius?
What properties of morphisms of $\mathbf{F}_p$-algebras descent along powers of the absolute Frobenius?
Is ...

3
votes

0
answers

417
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### "Frobenius Descent"

The following proposition is there in Pink's lecture notes on finite group schemes.
Let $k$ be an algebraically closed field of characteristic $p$. The category of finite length $W(k)$-modules $N$ ...

1
vote

0
answers

137
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### Does being big for a line bundle satisfy fpqc descent

Let $k$ be a field of characteristic zero, and let $L/k$ be a field extension. [Assume $k$ and $L$ are algebraically closed if necessary.]
Let $X$ be a variety over $k$ and let $\mathcal{L}$ be a ...

6
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0
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177
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### Algebraic model for the abelian category of descent data for modules in the non-affine case

Let $f: X \to Y$ be a morphism of schemes. I'd like to have a completely algebraic description of the belian category of descent data for modules along $f$. Here's my attempt:
The category of quasi-...

1
vote

0
answers

162
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### Choice free definition of category of descent data w.r.t a fibration?

Let $\mathsf C$ be a category and consider a pseudofunctor (non-strict 2-functor) $P:\mathsf C^{\text{op}}\to\mathsf{Cat}$. Given an arrow $f:X\to Y$ in $\mathsf C$, define the category of descent ...

8
votes

4
answers

778
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### English Reference for the Bénabou-Roubaud theorem

The Bénabou-Roubaud theorem links fibrational descent theory with monadicity. Particularly, it says that given a bifibration satisfying the Beck-Chevalley condition w.r.t some arrow $p$ in the base ...