bio | website | |
---|---|---|
location | Barcelona | |
age | 27 | |
visits | member for | 1 year, 9 months |
seen | Apr 11 at 9:22 | |
stats | profile views | 1,112 |
I'm a PhD student at the Universitat Autònoma de Barcelona.
Feb 19 |
comment |
(Short) Exact sequences with no commutative diagram between them
I think $C_2$ should be $A_2$, I'm not allowed to edit just one symbol. Nice answer! |
Oct 31 |
awarded | Benefactor |
Oct 31 |
accepted | Do constructible sets have Krull dimension? |
Oct 31 |
comment |
Do constructible sets have Krull dimension?
Cormulier: Are you aware of the following paper: tandfonline.com/doi/pdf/10.1081/AGB-120004483 it seems to be strongly connected to what you are mentioning. Do you know further references on the topic? |
Oct 27 |
comment |
(Co)localization of the derived category
Yes, you are right again, thanks! It really seems that there is a decomposition of the category in a direct sum of the category in "connected components" (each "connected component" of the category is given by objects whose injective envelope is the clique of a given indecomposable injective... ). I'm formalizing a proof. This seems to solve completely my question |
Oct 27 |
comment |
(Co)localization of the derived category
let me also say that it would be safe to assume that the category is also locally Noetherian (otherwise this notion of link is not sufficient...) |
Oct 27 |
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(Co)localization of the derived category
yes you are right, it's not just what I have written. Again in a semi-artinian Grothendieck category, you have a link between two indecomposable injectives $E$ and $E′$ if $Hom(E,E′)\neq 0$ or $Hom(E′,E)\neq 0$. The Gabriel spectrum becomes a (undirected) graph where the edges are the links I have just defined. The clique of E is the connected component of E in this graph. Sorry for the mistake before... (this is not exactly the usual terminology of, say, Jategaonkar but I tried to give some natural notion of links and cliques in more "categorical" terms) |
Oct 27 |
accepted | (Co)localization of the derived category |
Oct 27 |
comment |
(Co)localization of the derived category
yes I agree, thanks for the nice "minimal" example. This happens essentially because there is a link between the indecomposable injective object which cogenerates $\tau$ and the other indecomposable injective. Now, suppose that you are in a semi-artinian Grothendieck category (just to make it simpler) and that, instead of being cogenerated by a single indecomposable injective, $\tau$ is cogenerated by an entire clique of indecomposable injectives (all the injectives that have non-trivial morphisms to or from a given indecomposable injective). Any thought in this more restricted setting? |
Oct 27 |
revised |
(Co)localization of the derived category
added 2 characters in body |
Oct 25 |
comment |
Do constructible sets have Krull dimension?
I'm sorry to bother you asking for more detail but I need to fully understand the answer. So, after your explanation its all clear excluding your last sentence: "Do an induction, using the fact that taking a finite product of lattices has Krull dimension equal to the max of the lattice". I would really appreciate if you edit your answer being more explicit. |
Oct 25 |
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Do constructible sets have Krull dimension?
@YvesCornulier: Yes, it would be very nice! |
Oct 25 |
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Do constructible sets have Krull dimension?
I guess it depends on the order you take on your lattices (ordinary or reverse inclusion). I prefer that my lattice of closed subsets has the same dimension of the classical Krull dimension of the associated $\mathbb K$-algebra |
Oct 25 |
comment |
Do constructible sets have Krull dimension?
I'm not doubting about the correctness of your answer, it is exactly what I want. Anyway I would like to understand it better. First of all, suppose $X\subseteq K^n$ is an affine variety and let $A\subset B\subseteq X$ be constructible subsets. Are you saying that $A-B$ is a finite union of closed sets? (any reference?) I do not understand the sentence "So choose $\bar n$ to be the first point where the usual dimension of the constructible set attains its maximum value." By "point", do you mean "point of the variety" or "point in the lattice $\mathcal C(X)$". What is the "usual dimension"? |
Oct 23 |
asked | Do constructible sets have Krull dimension? |
Jul 16 |
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Rejects and injectives
Well... it is certainly true that there are asymmetries in the category of modules, the dual of a module category is never a module category. Anyway, there should be something that can be said in this case, eventually in the setting of Grothendieck categories, forgetting the language of ring theory. I'll think about that. |
Jul 14 |
awarded | Citizen Patrol |
Jul 9 |
comment |
Rejects and injectives
Try to look in the first part of this book: "Equivalence and Duality for Module Categories with Tilting and Cotilting for Rings" |
Jul 9 |
revised |
Deligne-Mumford Stacks and exactness of products
edited body |
Jul 9 |
revised |
Deligne-Mumford Stacks and exactness of products
added 4 characters in body |