Questions tagged [descent]

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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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4 votes
0 answers
194 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 $\...
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3 votes
0 answers
122 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_\...
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  • 924
4 votes
1 answer
191 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)...
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6 votes
0 answers
308 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 [...]". ...
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  • 1,317
3 votes
0 answers
174 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, -- ...
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1 vote
0 answers
125 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 $\...
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1 vote
0 answers
57 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, ...
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5 votes
0 answers
216 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 $K_f := \mathbb{Q}_p[x]/(f(x))$, ...
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7 votes
0 answers
236 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 ...
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  • 71
1 vote
0 answers
279 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^{\...
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  • 1,015
6 votes
1 answer
492 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}\...
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  • 1,243
3 votes
0 answers
238 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 ...
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  • 4,926
8 votes
1 answer
646 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 ...
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  • 3,686
22 votes
3 answers
3k 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. ...
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1 vote
0 answers
181 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 $X_Z$. ...
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1 vote
1 answer
483 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)...
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  • 291
2 votes
0 answers
38 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 $...
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  • 2,780
2 votes
0 answers
161 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? ...
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  • 4,283
2 votes
1 answer
99 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 $L$ ...
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6 votes
1 answer
339 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 ...
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  • 111
2 votes
0 answers
72 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 ...
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  • 1,064
4 votes
2 answers
296 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 ...
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0 votes
1 answer
222 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 ...
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4 votes
1 answer
381 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) $\mathscr{F}$ ...
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  • 1,556
3 votes
1 answer
212 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{...
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  • 433
1 vote
0 answers
96 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 ...
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  • 121
2 votes
1 answer
288 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 ...
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  • 55
2 votes
0 answers
287 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....
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2 votes
1 answer
46 views

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 \...
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  • 536
1 vote
0 answers
107 views

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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  • 2,046
4 votes
1 answer
362 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 ...
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  • 4,283
4 votes
0 answers
188 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 ...
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4 votes
1 answer
166 views

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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  • 255
7 votes
3 answers
490 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 ...
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3 votes
1 answer
250 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 ...
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  • 828
4 votes
1 answer
216 views

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 ...
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4 votes
0 answers
212 views

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,...
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5 votes
0 answers
182 views

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 ...
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3 votes
0 answers
382 views

"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 $N$ ...
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1 vote
0 answers
126 views

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 ...
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  • 19
6 votes
0 answers
174 views

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-...
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  • 6,927
1 vote
0 answers
157 views

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 ...
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  • 9,635
8 votes
4 answers
675 views

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 ...
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  • 9,635
8 votes
0 answers
979 views

Galois descent for schemes over fields

Let $K\subset L$ be a finite galois extension of fields (the case I have in mind is $K=\mathbb{R},L=\mathbb{C}$). Given a scheme $X$ over $K$ by pulling back to $L$ we get a scheme $Y=X\times _K L$ ...
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9 votes
2 answers
399 views

Pushouts of commutative pseudomonoids

Let $(\mathcal{C},\otimes)$ be a symmetric monoidal bicategory. Assume that $\mathcal{C}$ has bicategorical coequalizers which are preserved by $\otimes$ in each variable. My question is if then the ...
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2 votes
0 answers
413 views

Neat applications of Galois descent?

I'm enjoying reading about Janelidze's categorical Galois theory, which gives as a special case the usual theorems of Galois descent (along torsors). The approach I took was just with covering space ...
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  • 9,635
2 votes
0 answers
125 views

Characterization of torsors which are locally trivial in terms of descent

Let $\mathsf C$ be a category and $G$ an internal group. Suppose $\mathsf C$ is finitely complete, so that $\pi_2:G\times B\to B$ is an internal group in $\mathsf C_{/B}$ for every $B$. A $G$-bundle ...
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  • 9,635
4 votes
1 answer
405 views

Descent theory, fibrations, and bundles

In the very last page of Janelidze and Tholen's paper Beyond Barr Exactness: Effective Descent Morphisms, the authors relate the theory of fiber bundles (and covering spaces in particular) to descent ...
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  • 9,635
2 votes
0 answers
119 views

Descent for Dualizable Modules

It's known that a pure morphism of commutative rings $\phi:A\to B$ is of effective descent for the stack of modules. In other words if $\phi$ is pure one can recover $Mod(A)$ as the 2-limit of a ...
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