# 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 ...
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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 ...
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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, $b ∈ \mathbb{R}^n$, $c ∈ \mathbb{R}$ ...
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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)...
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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 ...
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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 ...
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### 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 $R \to S^{\otimes \bullet +1}$ ...
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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 ...
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### 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....
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $\mathsf{Top}^2$ 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, ...
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### 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 $t$ ...
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### 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'...
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