6
votes
0answers
125 views

Coherent sheaves and Mitchell's embedding theorem

Let $S$ be a scheme. By Freyd-Mitchell's theorem the category of (quasi)coherent sheaves of $\mathcal{O}_S$-modules is equivalent to a full subcategory of the category of left $R$-modules for some ...
5
votes
2answers
248 views

When is/isn't the monoidal unit compact projective?

I am interested in developing intuition for when the monoidal unit in a closed monoidal abelian category is or isn't compact projective. As such, my question is not looking for a yes/no answer, but ...
3
votes
1answer
121 views

Infinite Dimensional Weil Restriction and Ind-scheme

I'm studying the Weil Restriction $\mbox{Res}_{k/\textbf{F}_p}$ where $\textbf{F}_p$ is the field with $p$ elements and $k$ is a perfect field over $\textbf{F}_p$, not necessarily finite. In this ...
0
votes
0answers
94 views

How to prove a Lefschetzesque principle for relative homology and absolute homology

Here is my conjecture: If H is some homology theory from some abelian category A to some abelian category B, E is the class of all epimorphism in A, E' is some closed class of epimorphisms in A and ...
1
vote
0answers
146 views

Coarse moduli spaces and rational points [closed]

Let $K$ be a field (not necessarily algraically closed). Let $\mathcal{F}$ be a contravariant functor from the category of schemes over $K$ to sets and $M$ be a coase moduli space for the functor. So, ...
3
votes
1answer
193 views

Postnikov towers in bounded t-structures

If $\mathcal{H}$ is the heart of a bounded t-structure in a triangulated category $\mathcal{T}$, then for every object $E$ in $\mathcal{T}$ there exists a finite sequence of integers ...
1
vote
0answers
96 views

Flat and injective quasi-coherent sheaves

Let $X$ be a quasi-compact semi-separated scheme and $$\varepsilon: 0\to A \to B \to C \to 0$$ be a short exact sequence of quasi-coherent sheaves. $\varepsilon$ is called a (categorical) pure exact ...
0
votes
3answers
263 views

Rank vanishing in tensor categories

Let $\mathcal{C}$ be a cocomplete $\mathbb{Q}$-linear symmetric monoidal category (if convenient, assume that it is locally finitely presentable and/or abelian). Recall that the rank (or dimension) of ...
2
votes
0answers
447 views

The link between the subfactors and the motives as enriched Galois theories?

On 29th March 2007, at the "École normale supérieure" of Paris, the mathematician Vaughan Jones, gave a conference (in French) entitled "Les sous-facteurs : une théorie de Galois enrichie" (see here), ...
9
votes
2answers
343 views

Is the category of schemes wellpowered? regularly wellpowered?

Wellpowered means that for every scheme $X$, the subobject lattice of monormophisms $Y \to X$ is essentially small; regularly wellpowered means that for every scheme $X$, the regular subobject lattice ...
6
votes
3answers
371 views

Do any Stone-like dualities have some self-dualities hidden inside them?

This question originated from the observation that in most cases when one has duality of structured sets induced by a dualizing set-with-two-structures $D$, both sides of the duality are substructures ...
11
votes
2answers
439 views

Microlocalizing Hochschild homology

A recent paper of Ben-Zvi and Nadler gives a general formalism for understanding "dimensions" in sheaf theories. Without getting too far into details, amongst other things, this formalism allows us ...
3
votes
0answers
214 views

Why does this setting imply that a category is Grothendieck?

I came across the following Lemma in Mitsuyasu Hashimoto's Equivariant Twisted Inverses; it is Lemma 11.2 on page 107 of this pdf. Let $\mathcal{A}$ be an abelian category which satisfies the (AB3) ...
12
votes
3answers
1k views

History and motivation for Tannaka, Krein, Grothendieck, Deligne et al. works on Tannaka-Krein theory?

I am trying to wrap my mind around Tannaka-Krein duality and it seems quite mysterious for me, as well, as its history. So let me ask: Question: What was the motivation and historical context for ...
8
votes
1answer
575 views

Analogy between topology and algebraic geometry

In topos theory, there are many generalizations of topological concepts. For example, open, closed, proper and etale morphisms between toposes. However, there are also such analogous concepts in ...
4
votes
1answer
280 views

How to characterize flasque sheaves in more functorial way?

The motivation to ask this question is some proposition of flasque sheaves. Let's recall the definition of flasque sheaf:A sheaf $F$ on a topological space $X$ is flasque if for every inclusion ...
6
votes
1answer
466 views

Is “stackiness” transitive? (and a couple other basic questions about stacks)

Say, $B$ is a category fibered in groupoids over some category $C$, and $A$ is a category fibered in groupoids over $B$. Suppose $A$ is a stack (over whatever site) over $B$, and $B$ is a stack over ...
3
votes
0answers
207 views

Categorical proof for Chavelley theorem on affiness of scheme

The question is related to another question asked here a couple of minutes ago: Does vanishing of cohomology of locally free sheaves imply affiness of scheme In Hartshorne Exercise 4.2,we have the ...
4
votes
0answers
204 views

Reconstruction of noncommutative scheme

It is known that a quasi compact scheme(even quasi separated scheme)can be determined uniquely by the category of quasi coherent sheaves on it by Gabriel-Rosenberg reconstruction theorem The ...
1
vote
1answer
292 views

A $2$-torsion version of the motivic stable homotopy category?

For a field $k$ there exists the motivic stable homotopy $SH(k)$; it is compactly generated. My question: does there exist a 'reasonable' functor $p$ from $SH(k)$ to a certain triangulated category ...
4
votes
1answer
236 views

Is a group scheme determined by its category of representations?

More precisely, let $G$ be an affine group scheme over a field $k$, $Rep_k(G)$ be the category of finite dimensional representations of G, and $\omega_0$ be the forgetful functor from $Rep_k(G)$ to ...
6
votes
2answers
264 views

Properness of the category of modules over a spectrum (that represents algebraic cobordism or motivic cohomology)

The abstract form of the question: let $C$ be a closed proper stable model category, $R$ is a ring object in it. Which conditions ensure that the category $R-mod$ is also proper? Since weak ...
5
votes
0answers
145 views

Nice references for injective model structures and Quillen functors between motivic homotopy categories

It seems that for the injective model structures for the categories simplicial presheaves and simplicial Nisnevich sheaves (on the category of smooth varieties over a perfect field $k$) there exist ...
13
votes
5answers
728 views

is the category of coherent sheaves some kind of abelian envelope of the category of vector bundles?

This might be obvious to experts, but I'm not sure where to look for the answer. On a reasonably nice, at least noetherian, scheme (or variety, algebraic space, stack), can the category of coherent ...
2
votes
1answer
195 views

how do automorphisms act on the right in grothendieck's galois theory

So, I'm reading through some notes on the etale fundamental group (mostly Murre, but also some other notes I have), and I find it confusing how in a galois category $\mathcal{C}$ with fundamental ...
1
vote
2answers
229 views

2-sheaf definition in nlab

I'm looking at the definition of 2-sheaf in the nlab http://ncatlab.org/nlab/show/2-sheaf and I get stuck with the definition of 2-separated. Especially with the expression ...
12
votes
1answer
630 views

Janelidze's Galois theory

I am interested in learning about categorical Galois theory, as developed by Janelidze. I am a graduate student who has good familiarity with category theory, but not in the level of doing research on ...
3
votes
1answer
327 views

Can we define the tensor product in the derived category $D^b_{\text{coh}}(X)$ just from $D^b_{\text{coh}}(X)$ in certain cases?

This question arise from the comparision of the reconstruction theorems of Bondal-Orlov and Balmer and is inspired by Shizhuo Zhang's mathoverflow question: How to unify various reconstruction ...
3
votes
1answer
196 views

Is the filtered colimit of sheaves of abelian groups a sheaf?

This might be embarrassingly simple, but I want to be 100% sure I am not missing any subtleties. Let $F_i$, $i\in I$ be a filtered inductive system of sheaves of abelian groups on some site. Take the ...
0
votes
0answers
191 views

is this a simplicial model category?

A basic result in the theory of model categories is that simplicial sets form a simplicial model category. The same is true for simplicial $k$-algebras. I have two questions related to this. One ...
5
votes
1answer
187 views

Comparison of two traces

Suppose $X$ is a smooth quasi-projective variety over $\mathbb{C}$ and $Z$ a proper subscheme, there is a formal duality isomorphism (here we consider the Zariski topology) due to Hartshorne: $$ tr: ...
4
votes
2answers
298 views

Lingering foundational question about sheaves of abelian groups

Motivation for the question: I have a standard working knowledge of sheaves. Given a scheme, a coherent module over its structure sheaf and a few hours I can compute things. Despite this I have ...
1
vote
0answers
111 views

Question on existence and atomicity of a geometric morphism

I am curious to know where we can find a geometric morphism from the Zariski topos to the étale topos and more specifically when this is atomic. I would like to know, actually, in which instances is ...
4
votes
1answer
428 views

Is the category of quasi-coherent $\mathcal{O}_X$-algebras cocomplete?

Let $X$ be a scheme. Is the category of quasi-coherent (commutative) $\mathcal{O}_X$-algebras cocomplete? Remark. The same question was asked in MSE last year. Since nobody has answered it, I post ...
1
vote
0answers
237 views

Categories of sheaves and Kan Extensions

This is quite a broad question regarding constructions of categories of sheaves in geometry. Let $\textbf{Sch}$ denote the category of schemes. Let $\textbf{SchAff}$ denote the full subcategory of ...
-1
votes
1answer
164 views

Pure Quotient and pure sub-object

Let $\mathcal{C}$ be the category of modules over a ring. Let also $\mathcal{F}$ be a class of objects in $\mathcal C$ closed under pure subobject (pure quotient) and direct limit. Is $\mathcal{F}$ ...
6
votes
1answer
527 views

Is every module the colimit of its finitely generated submodules? (for algebraic spaces or stacks)

For (quasi-compact and quasi-separated) schemes there is a categorical way to characterise quasi-coherent sheaves of finite type using purely the abelian category $\operatorname{QCoh}(X)$. In an ...
19
votes
1answer
770 views

What is $Aut(Ell)$?

Consider the stack $Ell$ (of groupoids) of elliptic curves. I'm interested in the autoequivalence 2-group of $Ell$, the objects of which consists of transformations $Ell \Rightarrow Ell: Ring \to Gpd$ ...
16
votes
3answers
1k views

Is there a scheme corresponding to the unit interval?

Can someone complete the following table? $\begin{array}{cc} \text{Topology over } \mathbb{R} & \text{Topology over } \mathbb{C} & \text{Algebraic Geometry} \\\\ \hline \mathbb{R} & ...
4
votes
2answers
366 views

Construction of the spectral sequence of Katz/Oda

In their famous paper "On the differentiation of De Rham cohomology classes..." Katz and Oda construct the spectral sequence for de Rham cohomology for the situation of a smooth morphism $\pi: X ...
5
votes
2answers
579 views

Can Inequivalent Topologies Have Same Sheaves/Cohomology?

Let $C$ be a fixed category, and let $T_1$ and $T_2$ be two Grothendieck (pre)topologies on $C$. We say $T_1$ is subordinate to $T_2$ if every covering in $T_1$ has a refinement in $T_2$. We say $T_1$ ...
13
votes
1answer
533 views

How much of a variety can be reconstructed from codimension-zero data?

This is an improved version of this question. Maybe it should be an edit -- I'm not sure what the MO convention is. I'm curious, more or less, how much information one can get out of the derived ...
1
vote
1answer
296 views

How much of a variety can be reconstructed from codimension-zero data?

Suppose $X$ is a nice (finite type over $\mathbb{C}$, smooth and proper if necessary) variety. Suppose we're given the $\mathbb{C}$-points of $X$ as a set and for any finite subset $S\subset X$ we ...
2
votes
1answer
357 views

Calculate $Hom$ in derived category

Suppose $X$ is a smooth variety over $\mathbb{C}$. Let $K^{b}(X)$ be the homotopy category of bounded complex of coherent sheaves, and $D^{b}(X))$ be the derived category of bounded complex of ...
4
votes
2answers
252 views

When is a substack closed?

For this question we will consider the Zariski site of affine schemes and a stack $\mathcal{M}$ over it. I don't know what a substack is, but I have a guess. The stack $\mathcal{M}$ has an underlying ...
2
votes
1answer
163 views

Twisting an object P by an H-Torsor I

I am reading Brylinski's Loop Spaces, Characteristic Classes, and Geometric Quantization. The Statement Let $C$ be a gerbe on a space $X$ with "abelian" band $H$, $f: Y \to X$ a local homeomorphism ...
5
votes
0answers
232 views

Exactness is often an open condition. How often?

Let $\mathcal C$ denote an abelian category. Consider a formal $\mathbb Z$-graded object $V_\bullet$ in $\mathcal C$, by which I simply mean a $\mathbb Z$-indexed list $n\mapsto V_n$ of objects. A ...
3
votes
2answers
483 views

Injective Modules over Group Rings

Given the group ring $\mathbb{Z}[G]$ of a finite group $G$ over $\mathbb{Z}$, is there a way to generalize the notion of the "frobenius algebra" in some cases? One can show that every group ring ...
8
votes
0answers
423 views

Conceptual proofs for the computation of the structure sheaf

The following lemma in commutative algebra is important for the foundations of algebraic geometry: If $A$ is a commutative ring, $U \subseteq A$ is a finite subset generating the unit ideal, then ...
2
votes
0answers
88 views

stratification by gerbes of a symmetric power of a gerbe

We work over $\mathbb{C}$. I'm trying to understand the following result (a lemma from the Stacks project), in some particular example. The Lemma says that for an algebraic stack $X$ for which the ...