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7
votes
1answer
149 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
2answers
268 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 ...
4
votes
1answer
299 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
1answer
189 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
1answer
316 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
1answer
130 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, ...
7
votes
1answer
319 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 ...
6
votes
1answer
279 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
1answer
133 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
1answer
277 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 ...
4
votes
1answer
657 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
0answers
131 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
0answers
96 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
1answer
484 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 ...
10
votes
0answers
370 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
1answer
374 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
1answer
436 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
1answer
308 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
0answers
327 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
1answer
655 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
1answer
271 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
1answer
272 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
0answers
135 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
1answer
335 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
0answers
125 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
0answers
213 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
1answer
383 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
1answer
595 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
2answers
447 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
1answer
547 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
1answer
307 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
1answer
418 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
2answers
723 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
1answer
377 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" ...
21
votes
2answers
3k 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 ...
16
votes
1answer
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
1answer
327 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
3answers
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 ...
8
votes
1answer
820 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
2answers
433 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, ...