Abstract Nonsense

Crushing one theorem at a time

Functorial Properties of the Tensor Product (Pt. III)


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

\text{ }

The important thing to note about the previous two theorems is that they really only require (for ingredients) left or right R-modules and R-maps, and the only parts of the proof, in fact the entire notion of exactness, really only has to do with the underlying abelian group structure. From this we can glean the following two theorems:

\text{ }

Theorem: If M,N,L are right (S,R)-bimodules and we have an exact sequence

\text{ }

M\xrightarrow{f}N\xrightarrow{g}L\to0

\text{ }

is an exact sequence of (S,R)-maps, and if A is a left (R,T)-module, then

\text{ }

M\otimes_R A\xrightarrow{f\otimes 1_A}N\otimes_R A\xrightarrow{g\otimes 1_A}L\otimes_R A\to 0

\text{ }

is an exact sequence of (S,T)-modules and (S,T)-maps.

\text{ }

and

\text{ }

Theorem: If M,N,L are right (R,T)-bimodules and we have an exact sequence

\text{ }

M\xrightarrow{f}N\xrightarrow{g}L\to0

\text{ }

is an exact sequence of (R,T)-maps, and if A is a left (S,R)-bimodule, then

\text{ }

A\otimes_R M\xrightarrow{1_A\otimes f}A\otimes_R N\xrightarrow{1_A\otimes g}A\otimes_R L\to 0

\text{ }

is an exact sequence of (S,T)-bimodules and (S,T)-maps.

\text{ }

\text{ }

References:

[1] Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. Advanced Modern Algebra. Providence, RI: American Mathematical Society, 2010. Print.

[3] Blyth, T. S. Module Theory. Clarendon, 1990. Print.

[4] Lang, Serge. Algebra. Reading, MA: Addison-Wesley Pub., 1965. Print.

[5] Grillet, Pierre A. Abstract Algebra. New York: Springer, 2007. Print.

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January 4, 2012 - Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , ,

4 Comments »

  1. […] in terms of an exact sequence of the form then we know that is the cokernel of the initial, right exactness tells us then that is exact and so will be the cokernel of the first map. After showing some […]

    Pingback by Using Partial Exactness to Compute Things (Pt. I) « Abstract Nonsense | January 20, 2012 | Reply

  2. […] have proven that the tensor functor is ‘right exact’, and moreover we have seen that this partial […]

    Pingback by The Hom Functor is Left Exact « Abstract Nonsense | January 26, 2012 | Reply

  3. […] example, we have shown that is a right exact functor for commutative rings and we have also shown that is a left […]

    Pingback by Left Exact, Right Exact, and Exact Functors « Abstract Nonsense | April 24, 2012 | Reply

  4. […] fact, considering the right exactness of the tensor product it’s not hard to show […]

    Pingback by The Tensor Algebra and Exterior Algebra (Pt. V) « Abstract Nonsense | May 10, 2012 | Reply


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