Questions tagged [descent]

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What is descent theory?

I read the article in wikipedia, but I didn't find it totally illuminating. As far as I've understood, essentially you have a morphism (in some probably geometrical category) $Y \rightarrow X$, where ...
30 votes
2 answers
3k views

Why are monadicity and descent related?

This question is probably too vague for experts, but I really don't know how to avoid it. I've read in several places that under mild conditions, a morphism is an effective descent morphism iff the ...
Arrow's user avatar
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30 votes
1 answer
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Cohomology of sheaves in different Grothendieck topologies

Suppose I have a sheaf $\mathcal{F}$ on the (small) étale site over $X$. By restriction, $\mathcal{F}$ is also a sheaf on $X$ (with the Zariski topology). When is it that the sheaf cohomologies (i.e. ...
Sam Derbyshire's user avatar
23 votes
3 answers
4k 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. ...
Friedrich Knop's user avatar
21 votes
1 answer
700 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 ...
Mathmop's user avatar
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21 votes
0 answers
1k views

Can we describe equivariant vector bundles of free group action in terms of descent theory (Barr-Beck theorem)?

It is well known that for a compact topological group $G$ acts (say, from the right) freely on a compact space $X$. Then the category of equivariant complex vector bundles on $X$, $\text{Vect}_G(X)$, ...
Zhaoting Wei's user avatar
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20 votes
1 answer
1k views

How is a descent datum the same as a comodule structure?

For a homomorphism of commutative rings $f:R\to S$, there are at least two notions of a descent datum for this map. One of these is to be an $S$-module $M$, with an isomorphism $M\otimes_R S\cong S\...
Jonathan Beardsley's user avatar
16 votes
6 answers
4k views

What is etale descent?

What is etale descent? I have a vague notion that, for example, given a variety $V$ over a number field $K$, etale descent will produce (sometimes) a variety $V'$ over $\mathbb{Q}$ of the same ...
David Hansen's user avatar
16 votes
1 answer
409 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 ...
Simon Henry's user avatar
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15 votes
1 answer
2k 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 ...
Arrow's user avatar
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15 votes
1 answer
4k views

Frobenius Descent

Let $S$ be a scheme of positive characteristic $p$ and $X$ a smooth $S$-scheme. Let $F:X\rightarrow X^{(p)}$ denote the relative Frobenius. A result by Cartier (often called Cartier descent or ...
Lars's user avatar
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14 votes
3 answers
3k views

Looking for reference talking about relationship between descent theory and cohomological descent

I am now taking a course focusing on triangulated geometry. The professor has formulated the Beck's theorem for Karoubian triangulated category. The proof is very simple. Just using the universal ...
Shizhuo Zhang's user avatar
13 votes
3 answers
992 views

Descent of functions along finite birational morphisms

Let $A\to B$ be a morphism of (unitary commutative) rings such that $B$ is module-finite over $A$ and there exists $f\in A$ which is a nonzerodivisor in $A$ and in $B$, with $A[1/f]\to B[1/f]$ an ...
Matthieu Romagny's user avatar
13 votes
1 answer
596 views

Les deux théorèmes d'existence en théorie formelle des modules

In Exposé 195 of the Séminaire Bourbaki, Grothendieck states the following two theorems of non-flat descent. Theorem 1. Let $\Lambda$ be a noetherian ring and $C$ the category of $\Lambda$-algebras ...
Matthieu Romagny's user avatar
12 votes
1 answer
1k views

Counter-example to faithfully flat descent

I am looking for a counter example to the fact that a faithfully flat morphism is an effective descent morphism for the category of quasi-coherent sheaves when one forgets the quasi-compact hypothesis....
Joël's user avatar
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12 votes
2 answers
484 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 ...
Arrow's user avatar
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11 votes
2 answers
2k views

What is descent data (of higher categories), conceptually?

First consider a scheme $X$ with an open cover $\mathcal{U}=\{U_i\}$. An object with descent data on $\mathcal{U}$ is a collection $(\mathcal{E}_i,\phi_{ij})$ where $\mathcal{E}_i$ is a quasi-...
Zhaoting Wei's user avatar
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10 votes
1 answer
746 views

Descent of sheaves under galois covering

Let $\pi: Y\rightarrow X$ be a finite Galois covering between normal projective varieties with Galois group $G$. Let $E$ be a coherent sheaf on $Y$ with a $G$-linearisation, i.e., there are ...
Diego Maradona's user avatar
10 votes
1 answer
1k views

Reinterpreting Galois descent over finite fields

This question is indirectly related to my previous question Is an elliptic curve that is isomorphic to its Frobenius conjugate defined over $\mathbb{F}_p$? Let $\mathbb{F}_{q^n}/\mathbb{F}_q$ be an ...
Lisa S.'s user avatar
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10 votes
1 answer
1k views

How does descent theory imply a sheaf is a scheme?

I've noticed that often authors will comment that "descent theory" shows that some sheaf in the étale topology is actually a scheme. I was wondering what result in descent theory actually implies ...
HNuer's user avatar
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10 votes
2 answers
1k views

Riccati differential equation and descent

I am currently trying to understand Euler's article E71 on the Riccati differential equation and its connection with continued fractions. Apparently Daniel Bernoulli had shown that the equation $$ y' ...
Franz Lemmermeyer's user avatar
10 votes
0 answers
579 views

A model category for descent?

Recall that an $(\infty,1)$-category $C$ is said to have descent if for any small diagram $X:I\to M$ with (homotopy) colimit $\overline{X}$, the adjunction between $C/\overline{X}$ and "equifibered" $...
Mike Shulman's user avatar
9 votes
1 answer
1k 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 ...
Kim's user avatar
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9 votes
5 answers
945 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 ...
Arrow's user avatar
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9 votes
1 answer
2k views

what is Deligne's cohomological descent (and what are some examples)

As far as I understand Deligne's far reaching generalisation of Čech cohomology is called cohomological descent and is used to endow any variety with a (mixed) Hodge structure. Again, AFAIU, the idea ...
Jacob Bell's user avatar
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9 votes
2 answers
657 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 ...
user481494's user avatar
9 votes
1 answer
584 views

Descent of closed subschemes over finite fields

Let $p$ be a prime number, and $k$ a finite field with $q=p^n$ elements. Let $X$ be a scheme over $k$ and denote by $X'$ its base change to an algebraic closure $\bar k$ of $k$. Denote by $F_X:X\...
unknown's user avatar
  • 91
9 votes
1 answer
393 views

Homotopical descent information contained in the Dwyer-Kan function complexes of a presheaf category?

Recall that the category of sheaves on some site $C$ equipped with a grothendieck topology $\tau$ is equivalent to the localization of the category of presheaves $W^{-1}Psh(C)$ at $W$ where $W$ is the ...
Harry Gindi's user avatar
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9 votes
2 answers
928 views

modular forms, invertible sheaves, and quotients

I'm very confused about some contradicatory statements, and I hope someone can help me clarify this. Let $\Gamma$ be a congruence subgroup. It is well known that modular forms of weight $k$ for $\...
Nadim Rustom's user avatar
9 votes
2 answers
429 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 ...
Martin Brandenburg's user avatar
8 votes
2 answers
1k views

Pure morphisms which are not faithfully flat

Joyal and Tierney proved that morphisms of rings which are of effective descent are exactly those morphisms $\phi:R\to S$ such that $\phi$ presents $S$ as a pure $R$-module. Grothendieck had ...
Jonathan Beardsley's user avatar
8 votes
1 answer
1k views

Schemes do not form a stack in the etale topology?

As I understand, one of the reasons for "bootstrapping" to the category of algebraic spaces before constructing the category of Artin stacks is that algebraic spaces form a stack in the etale (at ...
Akhil Mathew's user avatar
  • 25.3k
8 votes
1 answer
489 views

Are associated bundles representable in schemes?

I have seen the following claim without proof in more than one paper, but it is sufficiently general that I suspect it is stated too strongly to be true: Let $G$ be an affine group scheme (say, ...
S. Carnahan's user avatar
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8 votes
1 answer
1k views

References to SGA 8 and descent theory

In Geometric Invariant Theory, by Mumford, Fogarty, and Kirwan, if there is a mention of descent theory, it almost always comes along with a reference to SGA 8, Theorem 5.2 (see the end of the proof ...
rghthndsd's user avatar
  • 419
8 votes
0 answers
1k 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 $Y=X\times _K L$ ...
Anonymous Coward's user avatar
7 votes
1 answer
387 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 ...
Jake McNamara's user avatar
7 votes
3 answers
643 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 ...
Mikhail Borovoi's user avatar
7 votes
1 answer
501 views

Which morphisms of ring spectra are of effective descent for modules?

There is a well understood bifibration of $\infty$-categories over the $\infty$-category of commutative ring spectra whose fiber over a ring $R$ is the category of $R$-module spectra. This is in ...
Jonathan Beardsley's user avatar
7 votes
1 answer
599 views

fpqc stackification

I am reading Lurie's Tannakian paper (http://www.math.harvard.edu/~lurie/papers/Tannaka.pdf) and I am confused about one point. At the end of page 3 he defines a stack-hom in any topos, which is ...
John Salvatierrez's user avatar
7 votes
1 answer
323 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 "...
G. Gallego's user avatar
7 votes
0 answers
246 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 ...
Michael's user avatar
  • 71
6 votes
2 answers
1k views

Does projectiveness descend along field extensions?

Background: Properness is a much more robust notion than projectiveness. For example, properness descends along arbitrary fpqc covers (see, for example, Vistoli's Notes on Grothendieck topologies, ...
Anton Geraschenko's user avatar
6 votes
1 answer
387 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 ...
happymath's user avatar
  • 167
6 votes
1 answer
993 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 ...
Arrow's user avatar
  • 10.3k
6 votes
1 answer
694 views

Why does the first Cech cohomology classify twisted forms?

Suppose I have a faithfully flat cover of schemes $\phi:X\to Y$, and a sheaf $F$ on $Y$. I might be interested in so-called ``twisted forms for $F$." That is, sheaves $F'$ on $Y$ such that $\phi^\ast(...
Jonathan Beardsley's user avatar
6 votes
1 answer
498 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}\...
Stabilo's user avatar
  • 1,479
6 votes
1 answer
543 views

Higher descent cohomology

Descent cohomology for a comonad is defined at degrees 0 and 1 by Mesablishvili in his paper "On Descent Cohomology" (as well as by many other authors in many other contexts). For a comonad $\bot$ on ...
Jonathan Beardsley's user avatar
6 votes
0 answers
423 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 $\...
Martin Ortiz's user avatar
6 votes
0 answers
437 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 [...]". ...
Gabriel's user avatar
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6 votes
0 answers
183 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-...
Saal Hardali's user avatar
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