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 Post subject: Homology and right exact functorsPosted: Thu, 12 Jan 2012 09:21:02 UTC
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Joined: Fri, 3 Sep 2010 09:29:45 UTC
Posts: 140
Let be a covariant functor of abelian categories and a chain complex in

I want to show that if is right exact there is a natural map

There is some useful hints for the problem. First start with the exact sequence

Apply to get

and then it is not hard to see

We then use the exact sequence

Apply F

Now

Assuming this is correct thus far, how does this give an epimorphism ?

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 Post subject: Re: Homology and right exact functorsPosted: Thu, 12 Jan 2012 22:35:54 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12101
Location: Austin, TX
qwirk wrote:
Let be a covariant functor of abelian categories and a chain complex in

I want to show that if is right exact there is a natural map

There is some useful hints for the problem. First start with the exact sequence

Apply to get

and then it is not hard to see

We then use the exact sequence

Apply F

Now

Assuming this is correct thus far, how does this give an epimorphism ?

Do you mean homology or cohomology, your topic says one and your notation indicates the other.

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 Post subject: Re: Homology and right exact functorsPosted: Thu, 12 Jan 2012 22:57:22 UTC
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Joined: Fri, 3 Sep 2010 09:29:45 UTC
Posts: 140
Cohomology - but what exactly is looking like homology?

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 Post subject: Re: Homology and right exact functorsPosted: Fri, 13 Jan 2012 06:12:20 UTC
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qwirk wrote:
Cohomology - but what exactly is looking like homology?

Reread what you posted as the topic.

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 Post subject: Re: Homology and right exact functorsPosted: Fri, 13 Jan 2012 13:23:55 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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qwirk wrote:
Let be a covariant functor of abelian categories and a chain complex in

I want to show that if is right exact there is a natural map

There is some useful hints for the problem. First start with the exact sequence

Apply to get

and then it is not hard to see

We then use the exact sequence

Apply F

Now

Assuming this is correct thus far, how does this give an epimorphism ?

Recall, for any morphism in an abelian category, you have the 4 associated objects and a canonical isomorphism which gives the commutative diagram

with exact columns.

Now, is the kernel of , so by the universal property of kernel, the map factors through , and that is your morphism . And it is not hard to show it is an epimorphism (use together with the isomorphisms)

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 Post subject: Re: Homology and right exact functorsPosted: Tue, 17 Jan 2012 04:46:00 UTC
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Joined: Fri, 3 Sep 2010 09:29:45 UTC
Posts: 140
Very nice!

Thank you for that

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