Questions tagged [descent]

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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
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1answer
480 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}\...
3
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0answers
153 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 ...
5
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1answer
415 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 ...
19
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3answers
2k 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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0answers
147 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
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1answer
409 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
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0answers
35 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 $...
2
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0answers
117 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
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1answer
97 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
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1answer
316 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 ...
2
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0answers
55 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
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2answers
269 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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1answer
219 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
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1answer
372 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
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1answer
199 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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0answers
95 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
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1answer
257 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 ...
2
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0answers
246 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....
2
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1answer
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 \...
1
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0answers
102 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}}...
4
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1answer
316 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
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0answers
158 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
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1answer
160 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 ...
7
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3answers
383 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
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1answer
212 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
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1answer
198 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 ...
4
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0answers
202 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,...
5
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0answers
170 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 ...
3
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0answers
354 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 $N$ ...
1
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0answers
119 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 ...
6
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0answers
170 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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0answers
156 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 ...
8
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4answers
591 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 ...
8
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0answers
824 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 $Y=X\times _K L$ ...
9
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2answers
379 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 ...
2
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0answers
378 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 ...
2
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0answers
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 ...
4
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1answer
388 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 ...
2
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0answers
117 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 ...
1
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0answers
56 views

Notation for largest universal subclass and class of arrows “locally in” a given class of arrows

Let $\mathcal M$ be a class of arrows in a category $\mathsf C$. I would like suggestions for good notation for the following two classes. The smallest universal (pullback stable) subclass $\mathcal ...
6
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1answer
684 views

Geometric intuition for the condition of Galois descent

Continuing in my attempts to understand bits and pieces of Borceux and Janelidze's Galois Theories, I've just realized that I don't have any geometric intuition for the most convenient ...
2
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2answers
388 views

For a universal covering morphism $p:E\rightarrow B$, how to prove $E$ connected implies $B$ connected?

Definition. An arrow $\alpha:A\rightarrow B$ in $\mathsf C=\mathsf{Fam}(\mathsf A)$ is said to be a covering morphism if there exists an effective descent morphism $p:E\rightarrow B$ that splits it, i....
5
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1answer
336 views

Categorification of covering morphisms

Given a category $\mathsf{A}$, let $\mathsf{Fam}(\mathsf{A})$ be its free coproduct cocompletion (which is always extensive). This means every object has a unique up to iso presentation as a coproduct ...
2
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0answers
129 views

When does effective descent of modules hold?

Let $A$ be a commutative ring with identity. I denote by $\Delta_{\leq 1}$ the full subcategory of the simplex category $\Delta$ with objects $[0]$ and $[1]$. Let $B_{\cdot} : \Delta_{\leq 1}^{\mathrm{...
6
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0answers
111 views

Expressing the stack of sheaves with 1-limits

Zhen Lin's answer to the MSE question mentions that $\mathsf{Sh}(-)$ is a stack for the canonical topology on the site of open subsets of a space. One of the comments asks whether the stack descent ...
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0answers
172 views

$\mathbb E$-descent maps in topological spaces in terms of different sites?

The paper Facets of Descent I by Janelidze and Tholen defines $\mathbb E$-descent maps as those for which $\Phi^p:\mathbb EB\longrightarrow \mathsf{Des}_\mathbb{E}(p)$ is an equivalence of categories. ...
12
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2answers
389 views

Which simplicial objects are Čech nerves?

In 1-categories, a regular epimorphism is a coequalizer of some parallel pair. An effective epimorphism is one which coequalizes its kernel pair. In the presence of kernel pairs, regular and ...
13
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1answer
972 views

Difficulties with descent data as homotopy limit of image of Čech nerve

Apologies if this question is inappropriate for MO. It is not a research level question in any of the topics it addresses, I just don't see how a novice can go about answering it alone (I've tried ...
15
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1answer
333 views

Descent of Higher categorical structures along geometric morphisms

Let $f: \mathcal{E} \rightarrow \mathcal{T}$ be a geometric morphism between two (Grothendieck) toposes (or maybe more generally a bounded geometric morphism between elementary toposes). It is well ...