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Gradient descent over the set of complex symmetric matrices

In the course of my research (somewhat related to compressive sensing), I am trying to determine a complex, symmetric matrix $L$ (i.e. $L = L^T$) through the following optimization formulation: $$ \...
Shreyas B.'s user avatar
4 votes
0 answers
283 views

Gluing together the moduli stacks of elliptic curves over Z[1/2] and Z[1/3]?

I have a long-running desire to understand what is the "global" moduli stack of elliptic curves, as a stack over $\mathrm{Spec}(\mathbb{Z})$. Recently I was pointed to Katz and Mazur's book, ...
David Roberts's user avatar
  • 34.2k
2 votes
1 answer
252 views

Gluing data for modules over a ring with idempotents

Let $A$ be a ring. If $e$ is an idempotent, then there is an abelian recollement involving the categories $A\text{-}\mathrm{Mod}$ and $eAe\text{-}\mathrm{Mod}$. This is Example 2.7 in Homological ...
Sergey Guminov's user avatar
4 votes
0 answers
327 views

Using comonadicity to prove faithfully flat descent

I have heard many times that faithfully flat descent could be reinterpreted via Beck's monadicity theorem; Deligne's paper "Catégories tannakiennes" even explains in section 4 how to do this ...
Michael Barz's user avatar
21 votes
1 answer
722 views

The derived category does not satisfy descent - example

One motivation for studying infinity categories is that the derived category does not satisfy Zariski descent, although the infinity categorical version does. I would like to see an example of Zariski ...
Mathmop's user avatar
  • 313
3 votes
1 answer
277 views

Unitary involutions on a simple central algebra after a scalar extension

$\DeclareMathOperator{id}{id}$ Let $L/K$ be a quadratic separable extension of fields. Let $A$ be a central simple algebra over $L$ such that its norm $N_{L/K}(A)$ splits. Then we know that there ...
Haowen Zhang's user avatar
7 votes
1 answer
337 views

Faithfully flat descent in complex analytic geometry

A common technique for constructing objects (sheaves) and morphisms in algebraic geometry is faithfully flat descent. Roughly speaking this consists on constructing an object or a morphism "...
G. Gallego's user avatar
7 votes
1 answer
412 views

Basic example of derived descent

I've been trying to understand the Adams spectral sequence, and I've gotten myself confused about how derived descent is supposed to work, so I would like to understand a simple example. Given a ...
Jake McNamara's user avatar
1 vote
0 answers
168 views

Question regarding Galois descent of sections of a vector bundle

Let $\pi: Y\rightarrow X$ be a finite 'etale Galois morphism between two smooth projective varieties with Galois group $G$. Let $\mathcal{E}$ be a vector bundle on $X$. Then $\pi^*\mathcal{E}$ is a $G$...
Hajime_Saito's user avatar
9 votes
2 answers
667 views

Non-trivial automorphisms and descent

In this expository paper by Low it says: Roughly speaking, a topos in the sense of Grothendieck is the category of sheaves on a kind of generalised space whose “points” may have non-trivial ...
user481494's user avatar
6 votes
0 answers
442 views

Quasi-syntomic descent and prismatic F-crystals

I am reading Bhatt and Scholze's paper on F-crystals, and they seem to be using the following result in the proof of Theorem 5.6: let $X \to Y$ be a quasisyntomic cover of formal schemes over $\...
Martin Ortiz's user avatar
3 votes
0 answers
180 views

Hypercovers with sieves

Consider a covering family $\{Y_i \to X\}$ and the induced sieve $R \subseteq X$, the subpresheaf of all maps to $X$ that factor through some $Y_i$. The family gives me an induced Cech nerve $C_\...
Leo Herr's user avatar
  • 1,084
4 votes
1 answer
220 views

An analogy of product formula for homogeneous space?

$\DeclareMathOperator\Sel{Sel}$Let $E$ be an elliptic curve defined over a number field $K$ with full $2$-torsion. The classical complete $2$-descent method tells that the $2$-Selmer group $\Sel_2(E/K)...
Shenxing Zhang's user avatar
6 votes
0 answers
449 views

Examples of descent in basic algebraic geometry

I'm studying descent theory and I recall that there were multiple instances before where I heard something like "we can prove this as follows, but this is just descent applied to [...]". ...
Gabriel's user avatar
  • 1,039
3 votes
0 answers
245 views

Ind-etale vs weakly etale

In this article Bhatt and Scholze consider ind-etale and weakly etale maps of affine schemes. We have two (easy) statements, proven in Prop.2.3.3(1) and (5): -- any ind-etale map is weakly etale, -- ...
AlexIvanov's user avatar
1 vote
0 answers
168 views

Category of coherent sheaves on blow-ups or resolution of singularities

Let $X$ be a scheme and $Y$ a closed subscheme. I would like to know if there is a good relation between the category of coherent sheaves on $X$ and the category of coherent sheaves on the blow-up $\...
curious math guy's user avatar
1 vote
0 answers
71 views

Would the iterated finite abelian descent obstruction equality hold for curves?

Let $X$ be a smooth projective geometrically integral variety over a number field $k$. We begin with some established notions in the theory of descent obstruction to the local-global principle, ...
oleout's user avatar
  • 875
5 votes
0 answers
294 views

2-descent on elliptic curves, and units modulo squares of units

Setup: Let $p$ be a prime, let $f(x) \in \mathbb{Q}_p[x]$ be a separable monic cubic polynomial cutting out the maximal order $\mathcal{O}_{K_f}$ in the etale algebra $K_f := \mathbb{Q}_p[x]/(f(x))$, ...
Ashvin Swaminathan's user avatar
7 votes
0 answers
249 views

Does a field extension define an effective descent morphism for locally ringed spaces?

Let $K'/K$ be an extension of fields and set $X=\operatorname{Spec}(K)$ and $X'=\operatorname{Spec}(K')$. As the category of locally ringed spaces has fibre products (see arXiv:1103.2139 or here) we ...
Michael's user avatar
  • 71
3 votes
0 answers
574 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^{\...
user267839's user avatar
  • 6,064
6 votes
1 answer
498 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}\...
Stabilo's user avatar
  • 1,479
3 votes
0 answers
438 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 ...
user127776's user avatar
  • 5,851
10 votes
1 answer
1k 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 ...
Kim's user avatar
  • 4,134
23 votes
3 answers
4k 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. ...
Friedrich Knop's user avatar
2 votes
0 answers
247 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 $X_Z$. ...
user127776's user avatar
  • 5,851
1 vote
1 answer
586 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)...
Z Wu's user avatar
  • 340
2 votes
0 answers
47 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 $...
Andy Sanders's user avatar
  • 2,890
2 votes
0 answers
304 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? ...
user avatar
2 votes
1 answer
130 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 $L$ ...
Adittya Chaudhuri's user avatar
6 votes
1 answer
405 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 ...
happymath's user avatar
  • 167
3 votes
0 answers
114 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 ...
Tri's user avatar
  • 1,388
4 votes
2 answers
344 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 ...
user avatar
0 votes
1 answer
240 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 ...
user avatar
4 votes
1 answer
403 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) $\mathscr{F}$ ...
k.j.'s user avatar
  • 1,352
3 votes
1 answer
268 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{...
Leo D's user avatar
  • 451
1 vote
0 answers
121 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 ...
Carot's user avatar
  • 121
2 votes
1 answer
358 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 ...
FelixBB's user avatar
  • 65
2 votes
0 answers
330 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....
Praphulla Koushik's user avatar
2 votes
1 answer
50 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 \...
Peter's user avatar
  • 546
1 vote
0 answers
127 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}}...
Ron's user avatar
  • 2,126
5 votes
1 answer
417 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 ...
user avatar
4 votes
0 answers
245 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 ...
user237334's user avatar
4 votes
1 answer
178 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 ...
Gaussler's user avatar
  • 295
7 votes
3 answers
650 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 ...
Mikhail Borovoi's user avatar
3 votes
1 answer
328 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 ...
zzz's user avatar
  • 878
4 votes
1 answer
250 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 ...
user avatar
4 votes
0 answers
292 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,...
user avatar
5 votes
0 answers
272 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 ...
user avatar
3 votes
0 answers
444 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 $N$ ...
Shubhodip Mondal's user avatar
1 vote
0 answers
157 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 ...
Ricardo's user avatar
  • 19