The descent tag has no wiki summary.

**7**

votes

**2**answers

456 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 ...

**4**

votes

**0**answers

260 views

### Flat Connections on Ring Spectra

So first I'll try to give a really quick reminder of the classical description of these things when one is doing non-commutative descent theory. In the setting of discrete algebra, if we have a ...

**2**

votes

**0**answers

113 views

### Reference for cdh topology

Let $f:X\rightarrow Y$ be a proper surjective morphism over some base scheme $S$ of finite type, suppose $f$ restricts to an isomorphism over some open $U$ of $X$, we also suppose both $X$ and $Y$ are ...

**6**

votes

**0**answers

320 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" ...

**3**

votes

**1**answer

168 views

### Extending descent data from the special fiber of an extension of DVR's

My question is about the proof of Lemma D.3 on p. 147 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud. Namely, towards the end of that proof there is the sentence "That $\varphi$ ...

**0**

votes

**0**answers

131 views

### Descent datum for a line bundle

Let $\pi:C \to \mathbb P^1$ be a double cover branched at $r$ points. To understand the theory of descent better, I would like, if possible, to construct by hands the descent datum of a line bundle ...

**7**

votes

**2**answers

437 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 ...

**13**

votes

**3**answers

597 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 ...

**3**

votes

**0**answers

62 views

### Representable torsors on geometric groupoid

Let $(C,\tau,\mathbb P)$ be a geometric context, as defined by Toen and Vezzosi. Let $(X_1\rightrightarrows X_0)$ be a groupoid object in $C$ such that the source and target morphisms are in $\mathbb ...

**1**

vote

**1**answer

214 views

### Descend of etale morphism

I am not sure whether the title is appropriate for this question or not. I am sorry if there is anyone who is confused with the title and the contents.
What I want to ask is the following: let $k$ be ...

**7**

votes

**1**answer

226 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 ...

**4**

votes

**2**answers

377 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 ...

**6**

votes

**1**answer

349 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 ...

**5**

votes

**1**answer

232 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 ...

**9**

votes

**1**answer

415 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 ...

**1**

vote

**1**answer

138 views

### Pullback as a local property

Given a commutative square in a nice category, say, manifolds $Mfd$. Suppose all edges are submersions (I guess transverse should be OK), then the square is a pullback if and only if it locally is, ...

**6**

votes

**1**answer

352 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 ...

**7**

votes

**1**answer

438 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 ...

**1**

vote

**1**answer

190 views

### On divisorial correspondences between curves

Assume we are given two smooth curves $C_1$ and $C_2$ over an algebraically closed field $k$. It is known that divisorial correspondences between them correspond to homomorphisms between their ...

**1**

vote

**1**answer

307 views

### Galois descent for semilinear endomorphisms

Let $K \subset L$ be a finite Galois extension, $\sigma$ an automorphism of $L$ (not necessarily fixing $K$) and let $E$ be a finite-dimensional vector space over $L$ together with an $\sigma$-linear ...

**5**

votes

**1**answer

822 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 ...

**2**

votes

**0**answers

149 views

### cech cohomology in topos

Hi,
The following result seems to be well known, but I can't come up with a proof.
Suppose that $C$ is a topos and that $F\to G$ is an effective epimorphism in $C$. If $P$ is
any abelian sheaf on ...

**0**

votes

**0**answers

106 views

### descent of a complex of sheaves

Let X a projective variety over an algebraically closed field on which an abelian variety $A$ acts freely.
Then $X/A$ is a projective variety. Let $f:X\rightarrow X/A$
Let $K\in D_{c}^{\leq ...

**8**

votes

**1**answer

541 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 ...

**11**

votes

**0**answers

481 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 $X$, $\text{Vect}_G(X)$, ...

**2**

votes

**1**answer

458 views

### A small detail in Neron Models (Bosch-Lütkebohmert-Raynaud) on descent theory

My question is about a small detail on page 132 of the above-mentioned book.
Let $R'$ be a faithfully flat $R$ algebra and $M'$ a $R'$-module. Let $\varphi: p_1^* M' \cong p_2^* M'$ be a covering ...

**13**

votes

**1**answer

463 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 ...

**1**

vote

**1**answer

336 views

### Descent of Morphisms of Sheaves

While reading Brylinski I am trying to understand the descent of morphisms of sheaves.
In trying to form a new definition of a presheaf $A$ over a space $X$, we associate to each surjective local ...

**4**

votes

**0**answers

358 views

### classify \mu_n torsors

Recently I read in Milne's book "etale cohomology" that the set $H^1(X,\mu_n)$ ($X$ a scheme, $n$ a nature number, the cohomology is flat cohomology) can be described as the set of pairs $(L,\phi)$, ...

**5**

votes

**1**answer

716 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 ...

**2**

votes

**1**answer

294 views

### Classification of principal G-bundles over a differentiable stack

According to "Notes on differentiable stacks" by Heinloth,
the classifying stack will also classify $G$-bundles on stacks. (Remark 2.13)
(Here $G$ is a Lie group.) My questions are:
(1) What ...

**3**

votes

**1**answer

292 views

### Noetherian descent extension for a given ring

For a homomorphism of rings $R \to S$, the following are equivalent:
a) $(-) \otimes_R S : \mathrm{Mod}(R) \to \mathrm{Mod}(S)$ reflects isomorphisms
b) $R \to S$ satisfies effective descent with ...

**1**

vote

**0**answers

146 views

### Galois cohomology of generic points of formal completions (of components of a hypercovering of the subvariety): examples or general statements?

Let $Y$ be a closed smooth subvariety in a (smooth) affine variety $X$. What can one say about the etale cohomology of the generic points of the formal completion of $X$ along $Y$ i.e. about the ...

**5**

votes

**1**answer

400 views

### Weil's descent criteria for covers from the critereon for varieties?

I have read several articles which use a version of the Weil decent criterion for covers, but the reference is always to Weil's original paper (1956 - The field of definition of a variety). I would ...

**2**

votes

**0**answers

127 views

### Intersections of components of 'simple' ('local") Zariski coverings

I would like to study the ordered Cech cohomology with respect to a Zariski covering of a variety. I can pass to the limit with respect to refinements; the components of the 'limit covering' will be ...

**0**

votes

**0**answers

239 views

### Ordered Cech(-like) complexes that compute etale cohomology (of fields!)

It is well known (cf. Equivalence of ordered and unordered cech cohomology. ) that for 'usual' topologies one can compute the cohomology of sheaves either using unordered Cech complexes or ordered ...

**6**

votes

**1**answer

400 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, ...

**7**

votes

**1**answer

616 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 ...

**4**

votes

**2**answers

496 views

### Twisted forms and $\check{H}^1$

I am reading Milne's Étale cohomology, III.4.
A twisted form of an object $Y$ (a scheme, a sheaf of modules, of algebras...) over a scheme $X$ is an object $Y'$ such that there exists a covering in ...

**1**

vote

**1**answer

570 views

### when a section descends?

Let $C$ be a (reduced, possibly reducible, complex) projective singular curve. Let $\nu: C'\to C$ a finite surjective birational morphism. (For example the normalization, but could be some ...

**9**

votes

**1**answer

330 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 ...

**9**

votes

**1**answer

441 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 ...

**10**

votes

**2**answers

787 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' ...

**3**

votes

**1**answer

406 views

### Do coequalizers in RingSpc automatically lead to descent?

I'm currently interested in the following result:
Let $f: X \to Y$ be a fpqc morphism of schemes. Then there is an equivalence of categories between quasi-coherent sheaves on $Y$ and "descent data" ...

**24**

votes

**2**answers

4k views

### 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 ...

**18**

votes

**1**answer

1k views

### 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. ...

**3**

votes

**1**answer

413 views

### Simplifying the definition of a geometric context using sieves?

On Pages 1-3 of Cours 2 of Toën's Master Course on Stacks, he defines the notion of a Geometric context with a rather extensive list of axioms (they take up about two pages over and above the ...

**13**

votes

**3**answers

2k 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 ...

**9**

votes

**1**answer

1k 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 ...

**5**

votes

**2**answers

458 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, ...