Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [descent]

Use for questions related to descent theory in topology.

Descent theory says that under certain conditions, homomorphisms between quasi-coherent sheaves can be constructed locally and then glued together if they satisfy a compatibility condition, while quasi-coherent sheaves themselves can be constructed locally and then glued together via isomorphisms that satisfy a cocycle condition.

78 questions
8
votes
1 answer

Do cokernels 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" on $X$. Namely, the second category consists of…
Akhil Mathew
  • 32,250
7
votes
0 answers

Group action on pullback sheaf.

I want to prove the following fact: If $G$ is a finite group scheme acting freely by $\mu$ on an abelian variety $X$ and $\pi \colon X \rightarrow X/G$ is the quotient map then for any coherent sheaf $\mathcal{F}$ on $X/G$ then there is a lift of…
Simon Cooper
  • 101
7
votes
1 answer

Determine the cocycle condition in Galois descent induced by faithfully flat descent

I initially asked this question on Mathoverflow as I thought it was to right place to do so. But it might not be so I will copy it here instead. I apologize for double posting and I will gladly erase the inapropriate one. It's the first time that I…
FelixBB
  • 73
7
votes
2 answers

combinatorial descents finding the number of permutations with criteria

I need help with the following: Define a descent of a permutation to be $j$ when $p_{j+1} < p_j$. Then the descent set of a permutation is the set of all descents. For example, the $5$-permutation: \begin{equation} 4, 3, 1, 5, 2 \end{equation} has…
CodeKingPlusPlus
  • 7,538
6
votes
1 answer

Question about limit of cosimplicial diagram associated with a sheaf

Let $F$ be a presheaf of sets on the usual topological site. Let $\left\{U_i\rightarrow U\right\}$ be covering of $U$. On the second page of Hypercovers and Simplicial Presheaves the authors write the equalizer of…
Arrow
  • 14,390
5
votes
1 answer

Using faithfully flat descent to prove representability of a functor in a simple case

Let $k$ be a field with a fixed separable closure $k_s$ and $G$ a finite type $k$-group scheme. Assume $F:(\mathrm{Sch}/k)^{opp}\rightarrow\mathrm{Set}$ is a contravariant functor whose restriction $F_{k_s}$ to schemes over $k_s$ is representable by…
Keenan Kidwell
  • 26,770
5
votes
1 answer

Can I check smoothness after a base-change

Let $X\to S$ be a flat morphism of noetherian schemes. I know that I can check smoothness on the geometric fibers to see whether $X\to S$ is smooth. Let $T\to S$ be a surjective morphism. Under what conditions can I check smoothness of $X\to S$ by…
901
  • 51
5
votes
1 answer

meaning of sentence that a "presheaf/K-theory satisfies descent on a Grothendieck site"

I'm reading a post about Nisnevich topology and I would like to clarify what the author means in Definition 1.5: We define $\mathrm{Spc}_S = L_{\mathrm{Nis}}\mathcal{P}(\mathrm{Sm}_S)$ to be the full subcategory of $\mathcal{P}(\mathrm{Sm}_S)$…
user267839
  • 9,217
5
votes
0 answers

Cech Nerve Good Cover

I'm trying to understand the notion of a Čech Nerve in simplicial presheaves. Suppose we have a site $C$, and we consider its category of simplicial presheaves $\mathsf{sPshv}(C)$. This is a simplicially enriched category, tensored and cotensored…
Emilio Minichiello
  • 1,154
5
votes
1 answer

Applications of fpqc descent of quasicoherent sheaves

I have been learning about fibered categories and stacks from Vistoli's notes. One of the main results in the notes is the statement that the fibered category of quasicoherent sheaves over a scheme $X$ is a stack in the fpqc topology on the category…
Alex K
  • 531
5
votes
0 answers

Galois Descent for representations of finite groups OR a question about block matrices where the blocks are Galois conjugates

The question is quite long since I give some background but really I am interested in some very concrete fact about matrix representations. Scroll all the way to the bottom for a self contained question, feel free to ignore the rest. Let $L/K$ be a…
Asvin
  • 8,489
5
votes
1 answer

The category of descent data

This is from Angelo Vistoli’s notes http://homepage.sns.it/vistoli/descent.pdf page $71$. Let $\mathcal{C}$ be a site and $\mathcal{U}=\{\sigma_i:U_i\rightarrow U\}$ be a covering of $U$. Let $\mathcal{F}$ be a fibered category over…
user537667
5
votes
2 answers

Confusion with definition of foliation

Below is the definition of foliation of a manifold appearing in the book Introduction to Foliations and Lie Groupoids by Moerdijk and Mrčun. Definition 1. Let $M$ be a smooth manifold of dimension $n$. A foliation atlas of codimension $m$ of $M$ is…
Arrow
  • 14,390
5
votes
0 answers

Relation between seemingly distinct notions of Beck-Chevalley condition?

(Terminology of pseudofunctors and fibrations is mixed throughout.) The Beck-Chevalley condition is defined on the nlab as a property of a quadruple of functors: starting from an invertible 2-cell of functors (a commutative-up-to-isomorphism…
Arrow
  • 14,390
5
votes
0 answers

Do the effective descent morphisms w.r.t the codomain fibration hint at the "right topology"?

An intuitive approach to basic descent theory for me started with open covers $ \left\{ U_i \right\}$, replaced them with a singleton covering $\coprod _iU_i\rightarrow X$, and then generalized to general morphisms $f:Y\rightarrow X$ using the Čech…
Arrow
  • 14,390
1
2 3 4 5 6