bio | website | |
---|---|---|
location | Barcelona | |
age | 27 | |
visits | member for | 2 years |
seen | yesterday | |
stats | profile views | 1,142 |
I'm a PhD student at the Universitat Autònoma de Barcelona.
Jul 4 |
awarded | Yearling |
Jul 2 |
awarded | Curious |
Jun 6 |
comment |
Krull dimension of dense extensions
I do not see why... the atomic Boolean algebras seem to have Gabriel dimension 1. |
Jun 6 |
comment |
Krull dimension of dense extensions
I've added the two definitions |
Jun 6 |
revised |
Krull dimension of dense extensions
added 1774 characters in body |
Jun 6 |
comment |
Krull dimension of dense extensions
very interesting, nice proof! Do you have any guess on the Gabriel dimension? |
Jun 6 |
revised |
Krull dimension of dense extensions
edited body |
Jun 5 |
asked | Krull dimension of dense extensions |
Apr 28 |
awarded | Revival |
Apr 28 |
answered | Analogy between Lagrange's Theorem and Rank-Nullity Theorem? |
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 |
comment |
(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 |