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 Post subject: Some Tor and Ext calculationsPosted: Sun, 18 Mar 2012 22:55:55 UTC
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Joined: Fri, 3 Sep 2010 09:29:45 UTC
Posts: 140
Let be a field. I'd like to calculate and

A good projective (free) resolution of as a -module is given by

where is the augmentation and is multiplication by .

So to calculate Tor we get the sequence
which is just

My question is what is the map ? It comes from , but what happens when we use the isomoprhism ? (I think the map must be the zero map?)

Similar for . We end up with . Since a homomorphism is determined by where we send and it can go anywhere in , I think that , but then I'm not sure what the map should be.

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 Post subject: Re: Some Tor and Ext calculationsPosted: Mon, 19 Mar 2012 00:19:03 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12170
Location: Austin, TX
The tensor works by modding out by the action of x, so your multiplication by x is now multiplication by 0.

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 Post subject: Re: Some Tor and Ext calculationsPosted: Mon, 19 Mar 2012 02:22:37 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6066
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qwirk wrote:
Let be a field. I'd like to calculate and

A good projective (free) resolution of as a -module is given by

where is the augmentation and is multiplication by .

So to calculate Tor we get the sequence
which is just

My question is what is the map ? It comes from , but what happens when we use the isomoprhism ? (I think the map must be the zero map?)

Similar for . We end up with . Since a homomorphism is determined by where we send and it can go anywhere in , I think that , but then I'm not sure what the map should be.

Hmm... you haven't seen dual map and the induced map on tensor products when you were doing linear algebra?

Shadow has done the Tor, so I'll do the Ext.

Recall the map is defined as .

So the map is defined as taking the map "evaluate at " and mapping it to "multiply by then evaluate at ". Of course, the multiplication by is absorbed in the identification (j=0,1), so is the identity map.

Indeed, you wouldn't be asking the question had you written instead of for your (although you will then have to be careful when you do the Tor functor calculation not to fall into the obvious trap).

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 Post subject: Re: Some Tor and Ext calculationsPosted: Fri, 30 Mar 2012 09:43:34 UTC
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Joined: Fri, 3 Sep 2010 09:29:45 UTC
Posts: 140
Thanks a lot to both of you!

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