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Questions tagged [descent]

Use for questions related to descent theory in topology.

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Equivalences of categories of descent data

I want to figure out when two categories of "descent data" are equivalent when we have equivalences on each "chart" commuting with the "restriction functors" up to an ...
Sergey Guminov's user avatar
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What are the advantages to fibered categories over pseudofunctors?

I read https://arxiv.org/pdf/math/0412512.pdf and I am left a bit lost: why did Grothendieck develop a theory of fibered categories? It seems like it was pitched as "you want to study functors ...
oggledog's user avatar
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Exercise about efficient step size

I have this exercise that I really don't know how to solve it. Can someone help me? Let $n ∈ \mathbb{N}$, $Q ∈ \mathbb{R}^{n×n}$ symmetric and positive definite, $b ∈ \mathbb{R}^n$, $c ∈ \mathbb{R}$ ...
MarcoDJ01's user avatar
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conditions on glueing group scheme

Say let $S=\mathbb{P}^1_k=\mathop{\mathrm{Proj}}k[x_0,x_1]$ be the base scheme where $k$ is a field. Let $\pi:G\to S$ be a morphism of $S$-schemes s.t. $G_i:=G|_{D_+(x_i)}$ is a group scheme over $D_+(...
Z Wu's user avatar
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1 answer
81 views

some total question about line search methods

can somebody tell me about these questions: 1-why we use line search? what kind of advantages have rather than others? 2-can introduce a good source about usage of line search in big data, please? 3-...
fatemeh-g's user avatar
2 votes
1 answer
96 views

Asymptotic behavior of the integral $\int_{0}^{\infty} e^{-\rho \cosh(R)}\cosh(Rq)dR$

Which method can I use to study the asymptotic behavior as $\rho \to \infty$ of the integral for $q \geq 0$? $$\int_{0}^{\infty} e^{-\rho \cosh(R)}\cosh(Rq)dR$$ I wish to study this behavior to ...
Victor's user avatar
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Induced group action on tangent bundle commutes with structure group?

I am trying to understand how the free and proper action of a discrete group $\Gamma$ on a manifold $X$ by automorphisms changes the structure group of the tangent bundle $\mathcal{T}_X$ of $X$. Let $\...
Aaryaman Patel's user avatar
1 vote
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49 views

Descent Methods (Line Search)

I am trying to understand descent methods I understand that we move towards the minimum taking steps let's call it $t^{(k)}$ at the current iteration. So the update equation becomes this: $x^{(k+1)} = ...
RGB's user avatar
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1 vote
1 answer
162 views

Pullback $p^*$ reflects isomorphisms implies $p$ is universal quotient map

$\newcommand{\im}{\operatorname{im}}$I'm trying to understand the implication $(i) \implies (ii)$ in the first theorem of "How algebraic is the change-of-base functor by Janelidze and Tholen, and ...
Jo Mo's user avatar
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Is a surjective morphism $V \to U$ quasi-compact and quasi-separated, if $V, U$ are quasi-compact and $U$ is affine?

In Olsson's Algebraic spaces and stacks there is the following lemma: Lemma 1.1.6 Let $f: X \to Y$ be a flat morphism of locally noetherian schemes that is locally of finite type (or more generally, $...
red_trumpet's user avatar
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What is the relationship between Fermat's method of descent and UFD?

I am reading a survey Reciprocity laws and Galois representations: recent breakthroughs. On page 5, it mentions that Theorem 2.1.5 shows that a prime $p \neq 2, 5$ divides an integer of the form $x^2+...
Umbrade's user avatar
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What is a fiber of the category of arrows?

Let $\mathcal C$ be a category. Grothendieck[1] defines the category of arrows in $\mathcal C$ to be the category of functors $$\Delta^1 \to \mathcal C,$$ where $\Delta^1$ is the category consisting ...
red_trumpet's user avatar
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3 votes
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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 ...
user267839's user avatar
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A morphism $S' \to S$ is a morphism of descent for $(Sch \to Sch)$ iff it is a universal effective epimorphism for $(Sch)$?

In SGA 1, VIII there is the following theorem (roughly translated by me) Theorem 5.2 Let $\mathcal F$ be the fibred category of arrows over the category of schemes (i.e. objects are morphisms $X \to ...
red_trumpet's user avatar
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If $A\otimes_R S\cong \frac{S[x]}{(x^p)}$ then $A\cong \frac{R[y]}{(y^p-c)}$ where $R\to S$ is an fppf map

This question generalizes my previous answered question. Conjecture: Let $R$ be a commutative ring of characteristic $p$, $R\to S$ be an fppf ring map (i.e. faithfully flat and of finite presentation)...
Z Wu's user avatar
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If $A_{\overline{k}}\cong \frac{\overline{k}[x]}{(x^p)}$, show that $A\cong \frac{k[x]}{(x^p-c)}$ where $k$ is a field of characteristic $p$

Let $p$ be a prime number, $k$ be a field of characteristic $p$ and $A$ be a $k$-algebra. Assume $A_\overline{k}=A\otimes_k \overline{k}$ is isomorphic to $\frac{\overline{k}[x]}{(x^p)}$, we want to ...
Z Wu's user avatar
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Hypercovers via Resolution of Singularity

I'm trying to understand the proof of Thm 4.16 in B. Conrad's notes on Cohomological Descent. A special case of this theorem is the following form: If $k$ is a field of characteristic zero and $S$ is ...
curious math guy's user avatar
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262 views

fppf/ etale Cohomology calculate with Cech cohomology

Let $R$ be a commutative ring with one (so living in standard commutative algebra setting) and let $\phi: R \to S$ a faithfully flat. Then the so called Amitsur complex $R \to S^{\otimes \bullet +1}$ ...
user267839's user avatar
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4 votes
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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 ...
Emilio Minichiello's user avatar
1 vote
1 answer
259 views

Cech nerve and descent data

When generalizing from sheafs on a site to 2-sheafs or stacks, it is useful to first rephrase the descent data for ordinary (pre)sheafs in terms of the Cech nerve of a coverage (e.g. https://ncatlab....
NDewolf's user avatar
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Fpqc-locally constant if and only if étale-locally constant?

Also in MO. 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)...
Z Wu's user avatar
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5 votes
1 answer
283 views

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 $X$...
Alex K's user avatar
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min max problem with eigenvalues

How can I find $$\text{s} = \inf_{a\in R}\left\{ \max_{λ\in[λ_n,λ_m]}|1-αλ|\right\}$$ I guess that λ_n and λ_m are eigenvalues of some descent method but there are no more clues in the question
GiorgosPap31's user avatar
7 votes
0 answers
434 views

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 ...
Simon Cooper's user avatar
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1 answer
206 views

Gradient descent methods

$f(x,y)=x^2+xy+y^2+yz+z^2-4x-8y-8z-1.$ Can we use gradient methods to determine the minimum of $f(x, y)$? What's the most interesting gradient methods can be used to determine the minimum of $f(x, y)...
Steve Boss's user avatar
7 votes
1 answer
475 views

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 ...
FelixBB's user avatar
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3 votes
0 answers
42 views

Descent theory: Transition morphisms which are normalized and satisfy the cocycle condition are isomorphisms

In the middle of the first paragraph of page 295 of the Elephant, Johnstone writes that normalized transition morphisms satisfying the cocycle condition are necessarily isomorphisms. The definitions ...
Arrow's user avatar
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1 answer
638 views

Find a descent direction at a saddle point

question: (Descent directions at stationary points). Assume that $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is $C^2$ with $\nabla f(x) = 0$ and $\nabla^2f(x)$ indefinite (with both positive and ...
ecjb's user avatar
  • 1,005
0 votes
0 answers
293 views

Proposition 1.6.6, Etale Cohomology theory, Lei Fu

I have difficulties in understanding a gap of the proof of proposition 1.6.6 in Etale Cohomology Theory written by Lei Fu. $\mathrm{Proposition} 1.6.6$ Let $g:S'\rightarrow S$ be a quasi-compact ...
Hardy's user avatar
  • 21
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1 answer
47 views

What is “distinguished cross section” (Solomon’s descent algebra proof)?

I’m attempting to read the proof of Solomon’s descent algebra from A Mackey Formula in the Group Ring of Coxeter Groups though I know nothing about Coxeter groups and Mackey formulas. I’m interested ...
Ahmbak's user avatar
  • 675
2 votes
0 answers
115 views

Non split real form of projective space

On the complex projective space $\mathbb{P}^1_\mathbb{C}$ we have an involution $z\mapsto -\frac{1}{\bar{z}}$. Using this as descent datum we should end up with a real form, which is not split (this ...
Notone's user avatar
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5 votes
0 answers
269 views

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 ...
Asvin's user avatar
  • 8,269
5 votes
1 answer
355 views

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 $\...
user avatar
5 votes
2 answers
543 views

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 $...
Arrow's user avatar
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3 votes
1 answer
331 views

Examples of modules with non-effective descent data

Let me first summarise descent data for modules, following the Stacks project: Given a ring homomorphism $R \to A$ we can extend scalars for $R$-modules $V$ to get $A$ modules $A \otimes V$ ($\otimes ...
user50948's user avatar
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5 votes
0 answers
259 views

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 ...
Arrow's user avatar
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0 votes
2 answers
214 views

(When) Is the category of descent data equivalent to the essential image of the base change?

Consider a family of continuous maps $(U_i\to U)$. For a family of bundles $(X_i\to U_i)$, TFAE: The family is a pullback of a single bundle $X\to U$; There exist transition isomorphisms satisfying ...
Arrow's user avatar
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1 vote
1 answer
228 views

Question about remark in Vistoli's notes on descent

Section 4.1.1 of Vistoli's Notes on Grothendieck topologies, fibered categories and descent theory motivates the formalism of stacks via topological spaces. Proposition 4.1 says that given an open ...
Arrow's user avatar
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3 votes
0 answers
180 views

Is the definition of fpqc topology in SGA 4.5 different from usual?

I am confused about Theorem 1.4.5, Example 1.6.4(b) in SGA 4.5, because here a covering of $U$ is a finite family $\{U_i\to U\}$ of flat morphisms such that $\coprod U_i\to U$ is surjective $\...
Yifeng Huang's user avatar
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1 vote
1 answer
205 views

Why does the cocycle condition on fiber isomorphisms imply a fiber bundle is trivial?

While reading about connections I stumbled on this entry of the encyclopedia of math. In the second paragraph of the comments section it's written that if a fiber bundle admits coherent isomorphisms ...
Arrow's user avatar
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2 votes
0 answers
82 views

Newton algorithm not finding minimum of $100(y-x^2)^2 + (1-x^2)$

Edit: Solved. I just forgot to update the inverse of the Hessian. I have updated the code now in case someone needs it. I am implementing some optimization algorithms (steepest descent, newton's ...
user3528810's user avatar
4 votes
0 answers
98 views

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 ...
Arrow's user avatar
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1 vote
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135 views

Question Janelidze and Tholen's 'Beyond Barr Exactness: Effective Descent Morphisms'

I have some questions about Janelidze and Tholen's paper Beyond Barr Exactness: Effective Descent Morphisms. First of all, they remark at the end of the introduction: Throughout this chapter $\...
Arrow's user avatar
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6 votes
1 answer
249 views

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 ...
Arrow's user avatar
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4 votes
1 answer
183 views

Proposition 4.1 in Vistoli's notes on descent

Proposition 4.1 of Vistoli's notes says $\mathsf{Top}^2$ is a stack over $\mathsf{Top}$. The proof starts by looking at the fibered product $\coprod_iU_i \times _U\coprod_iU_i$, however this object ...
popo's user avatar
  • 49
0 votes
2 answers
242 views

Silly question about descent

Most sources say descent is defining an object over $S$ using objects over $U_i$ for some cover $\left\{ U_i \right\}$ of $S$. If I replace the covering family with a single arrow $\coprod _i U_i\...
pop's user avatar
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1 vote
0 answers
50 views

$\mathsf{Top}^2$ is a stack over $\mathsf{Top}$?

In section 4.1 of Vistoli's notes he starts by showing we may locally construct arrows in $\mathsf{Top}^2$, i.e arrows over $U$. Then, the author says that we can moreover do the same for spaces, ...
pop's user avatar
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0 votes
0 answers
118 views

Asymptotic expansion using method of steepest descents

I am trying to find the first term in the asymptotic expansion of $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{1}{s^2}e^{t(s-m\sqrt{s^2-1})} ds $$ where $0<m<1$, $c<1$, as $t$ ...
meanderingthroughmath's user avatar
2 votes
1 answer
3k views

why use a small learning rate in gradient descent

I am new to neural networks and recently found out about gradient descent. Something does not sit right with me. x←x−λ∇fk(x) Why does this formula work? Wouldn'...
aceminer's user avatar
  • 357
2 votes
1 answer
130 views

Reference request: convergence property of continuous gradient descent?

Does anyone know of a text that treats the problem of gradient descent from a continuous perspective instead of a discretized perspective? For example, most text investigates the numerical properties ...
Fraïssé's user avatar
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