680 reputation
417
bio website
location Barcelona
age 27
visits member for 2 years, 5 months
seen 17 hours ago
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