# Abstract Nonsense

## Using Partial Exactness to Compute Things (Pt. II)

Point of Post: This is a continuation of this post.

January 20, 2012

## Using Partial Exactness to Compute Things (Pt. I)

Point of Post: In this post we show how one can use partial exactness to actually compute, explicitly, the isomorphism type of certain tensor products.

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Motivation

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Up until this point I have committed a sort of embarrassing crime–I have yet to explicitly compute a tensor product! This is not good, because I strongly believe that if one doesn’t stop and compute some things, one can get a little lost in the abstraction. So, this post is devoted primarily to discussing a way in which we can use partial exactness to explicitly compute the isomorphism type of some actual examples. Roughly the idea is that if we can express a module in terms of an exact sequence of the form $A\to B\to C\to0$ then we know that $C$ is the cokernel of the initial, right exactness tells us then that $A\otimes D\to B\otimes D\to C\otimes D\to0$ is exact and so $C\otimes D$ will be the cokernel of the first map. After showing some examples where this is useful we shall describe a general technique for computing tensor products by using “free presentations” which ultimately amount to expressing a module in terms of generators and relations.

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January 20, 2012