Talk:Ext functor

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According to Osborne it is Left derived functors that are defined using projective resolutions and right derived functors that use injective resolutions. Doesn't this make Ext left derive.

IE start with a covariant functor Hom(X,-) and some object A, take a projective resolution of X, P_n --> ... --> P_0 --> X, then apply Hom, cut the end off: Hom(P_n,A) --> ... --> Hom(P_0,A), then take cohomology of the complex?

Fixed that. And a few things more. Michiexile 15:18, 9 February 2007 (UTC)Reply

Isn't that Hom(P_n,A) <-- ... <-- Hom(P_0,A) ? Charles Matthews 15:53, 9 February 2007 (UTC)Reply

It is. But in my rewrite, things are tweaked the right way around, regardless of what's in the discussion. Michiexile 21:51, 10 February 2007 (UTC)Reply

Hom(A,-) is [covariant] left-exact, but contrary to what is in the article, Hom(-,B) is also left-exact. (See exact functor.) This is why both have right-derived functors. (Changed.) Tesseran 20:08, 11 February 2007 (UTC)Reply

In answer to the original question: left-exact covariant functors, as well as right-exact contravariant functors, have their derived functors (right-derived and left-derived respectively) defined using injective resolutions. Right-exact covariant and left-exact contravariant functors have their derived functors (left-derived and right-derived resp.) defined using projective resolutions. Your statement is only true if all your original functors are covariant. Tesseran 20:12, 11 February 2007 (UTC)Reply

Hom(-,B) is contravariant though. Which, upon a reread, meshes perfectly with what you wrote. Thanks for the correction. Michiexile 16:29, 12 February 2007 (UTC)Reply

The examples have dangling references. Should R and M be the same? --MarSch 13:53, 2 May 2007 (UTC)Reply

No, the first paragraph of the article introduces ${\displaystyle {\mathcal {C}}}$ as the category of modules over some ring ${\displaystyle R}$. So, the statements in the example sequence just fix the category in which the example is exhibited, and ${\displaystyle M}$ is the module of coefficients for the functor.

I'm not certain how to make this distinction clearer - suggestions are very welcome. Michiexile 22:33, 3 May 2007 (UTC)Reply

Mmmh, at least this solves where the R dependency should go, although it currently makes no sense. The solution is to be explicit about when C is some particular category and when it is merely _a_ category (which is the usual meaning) and especially not suppress R-dependence. Any idea what M is? --MarSch 10:38, 5 May 2007 (UTC)Reply

Ayup, M is some R-module. Any R-module.

The interesting examples subsection reads fine as English (I can't follow the math at this level). I made a couple of minor edits in earlier subsections for clarity; for instance Ring Structure... subsection above. I didn't change any math/markup/symbols/ , so someone might like to check the argument remains sound.Newbyguesses 23:13, 3 May 2007 (UTC)Reply

The claims of this chapter should be made more explicit. Quote:

Let ${\displaystyle R}$ be a ring and let ${\displaystyle \mathrm {Mod} _{R}}$ be the category of modules over R. Let ${\displaystyle B}$ be in ${\displaystyle \mathrm {Mod} _{R}}$ and set ${\displaystyle T(B)=\operatorname {Hom} _{\mathrm {Mod} _{R}}(A,B)}$, for fixed ${\displaystyle A}$ in ${\displaystyle \mathrm {Mod} _{R}}$. This is a left exact functor and thus has right derived functors ${\displaystyle R^{n}T}$. Define

${\displaystyle \operatorname {Ext} _{R}^{n}(A,B)=(R^{n}T)(B),}$

i.e., take an injective resolution

${\displaystyle J(B)\leftarrow B\leftarrow 0,}$

compute

${\displaystyle \operatorname {Hom} _{\mathrm {Mod} _{R}}(A,J(B))\leftarrow \operatorname {Hom} _{\mathrm {Mod} _{R}}(A,B)\leftarrow 0,}$

and take the cohomology of this complex.

So is Extⁿ supposed to be the n-th cohomology group of this complex? And why is the right derived functor the same as this cohomology? This should be explained. --Roentgenium111 (talk) 03:25, 5 July 2009 (UTC)Reply

Wouldn't such explanations belong in the Derived functor article? I would think so. And they are there. RobHar (talk) 05:15, 5 July 2009 (UTC)Reply

By this logic the whole "i.e." part should be deleted. But since it's there, it should also be stated that R^n is the n-th co(?)homology group of the complex, if this is the case (the Derived functor article says homology). --Roentgenium111 (talk) 14:32, 5 July 2009 (UTC)Reply

This needs some correction or clarification:

"For Fp the finite field on p elements, we also have that H*(G,M) = Ext* Fp[G](Fp, M), and it turns out that the group cohomology doesn't depend on the base ring chosen." — Preceding unsigned comment added by 80.42.253.109 (talk) 17:59, 7 May 2013 (UTC)Reply

I just deleted this claim, because it is completely false. Tesseran (talk) 17:51, 15 October 2017 (UTC)Reply

As the article currently looks, it is not very good at explaining why this concept should be called an extension of ${\displaystyle A}$ and ${\displaystyle B}$, or indeed why this would be a natural thing to consider. It is not until one gets to the Yoneda definition in the context of abelian categories we get to a definition that looks natural to a reader who does not compulsively extend every object to a resolution: we consider exact sequences with ${\displaystyle n}$ steps between ${\displaystyle B}$ and ${\displaystyle A}$, and want to classify these up to isomorphism — that's a reasonably natural thing to do, even for a reader who does not yet understand why the sequences should be exact, or why we go from ${\displaystyle B}$ to ${\displaystyle A}$ rather than the other way around, or indeed why this may be considered "an extension" (although the Ext and extensions section gives some clues to the diligent reader). To instead throw up two different definitions, one of which singles out ${\displaystyle A}$ and the other ${\displaystyle B}$, simply cries out that there must be a more unified perspective!

The framing of the Yoneda definition, with the reservation that it is equivalent to the first definition for modules only if there are enough projectives or injectives, also strikes me as misleading, because isn't it for modules always possible to find an epimorphism from a free module? I can however believe equivalence of the characterisations using right/left derived functors breaks down if there aren't necessarily enough projectives/injectives to construct the resolutions that machinery makes use of. 130.243.94.123 (talk) 11:59, 11 May 2023 (UTC)Reply