bio | website | |
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location | Barcelona | |
age | 27 | |
visits | member for | 2 years, 5 months |
seen | 17 hours ago | |
stats | profile views | 1,188 |
I'm a PhD student at the Universitat Autònoma de Barcelona.
Dec 13 |
comment |
Exactness of an additive left Kan extension
Many thanks for your edit! The Noetherian case (when you have Kernels) was clear to me but I did not know how to proceed in the general case... |
Dec 13 |
accepted | Exactness of an additive left Kan extension |
Dec 12 |
comment |
Exactness of an additive left Kan extension
So do you claim that also the left adjoint between the categories of contravariant functors on $fp(R)$ and $fp(S)$ is exact provided $\phi_!$ is exact? I'm trying to prove this but I do not see how to prove that the slice category is directed... could you add some details for this other case? (If you prefer I can ask "officially" a new question) |
Nov 25 |
comment |
Exactness of an additive left Kan extension
@FilippoAlbertoEdoardo I'm not defining $F_!$ as a restriction, I define $F$ as a restriction. $F_!$ is the additive left Kan extension to $F$, in particular, while $F$ is a functor $fp(R)\to fp(S)$, $F_!$ goes from the category of additive functors on $fp(R)$ to Abelian groups, to the same category of presheaves over $fp(S)$. |
Nov 24 |
comment |
Exactness of an additive left Kan extension
You are right, $\mathbb Z\to \mathbb Q$ is flat (an Abelian group is flat if and only if it is torsion free). So $f^*$ cannot be restricted to a functor $fp(\mathbb Q)\to fp(\mathbb Z)$, but I am not asking that. I am just restricting $f_!$ and using its exactness (in your example it is clear that the image of a finitely presented module is a finite dimensional rational vector space). |
Nov 24 |
comment |
Exactness of an additive left Kan extension
OK sorry! Now I see the problem, there was a missprint (now corrected). $f_!$ had to be $\phi_!$ |
Nov 24 |
revised |
Exactness of an additive left Kan extension
added 3 characters in body |
Nov 24 |
comment |
Exactness of an additive left Kan extension
Well both $(\phi_!,\phi^*)$ and $(F_!,F^*)$ are adjunctions (where $(-)_!$ is used here to denote left adjoints and $(-)^*$ indicates right adjoints). The relation with $\phi$ is explained in the question. |
Nov 24 |
revised |
Exactness of an additive left Kan extension
edited title |
Nov 24 |
asked | Exactness of an additive left Kan extension |
Oct 20 |
accepted | Cocomplete but not complete abelian category |
Sep 24 |
awarded | Autobiographer |
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 |