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location  Barcelona  
age  28  
visits  member for  2 years, 7 months 
seen  7 hours ago  
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I'm a PhD student at the Universitat Autònoma de Barcelona.
23h

awarded  Notable Question 
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 