Questions tagged [descent]
The descent tag has no usage guidance.
107
questions
1
vote
0answers
146 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^{\...
6
votes
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
votes
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
votes
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
votes
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.
...
1
vote
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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
0
votes
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
votes
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
votes
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{...
1
vote
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
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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-...
1
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
1
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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
votes
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
votes
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
votes
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 ...
