## The Tensor Algebra and Exterior Algebra (Pt. III)

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

*Exterior Algebra*

Now that we know that is a graded -algebra we can start modding out by homogeneous ideals and get new graded -algebras.The one we are going to consider is the ideal . This ideal is definitively homogenous and so we can consider the quotient which we denote and call the *exterior algebra *of . We denote the degree homogeneous part of as .

With rings such that one can think about as being the tensor algebra where we have forced “antisymmetry” in the sense that we can now commute simple tensors with any permutation as long as we are willing to concede to multiply the new tensor by . Indeed, let us denote the coset as , we’ll call these *simple wedges, *then the equality

and noting that for all we see that

so that

We see then that we can change into with so many transpositions, at each swap picking up a factor of , and thus after all the total swaps picking up .

So, why the caveat that ? Well, in general I think it should be easy to see that really what modding out by does is give us a new where we have forced to be zero if –in other words, adjacent repetitions are zero. Of course, it’s not hard to see that this always implies the antisymmetry relation as mentioned above, but the two are equivalent if and only if we can divide by –i.e. that is invertible. Really, if and only if doesn’t annihilate anything nonzero, but saying is sufficient for our purposes.

Now, by definition we can think of , called the *exterior product of ,* as being where is the submodule generated by all elements with two adjacent entries being equal (i.e. just –the degree homogeneous part of ). That said, it’s easy to see from the anticommutativity relations described above that really is just the ideal generated by all of the elements that look simple tensors where any two elements of the tensor (not necessarily adjacent!) are equal.

What we’d first like to claim is that has a nice universal property. Indeed, we know that allows us to trade -linear maps in for linear maps . What we’d like to claim is that allows us to trade in -linear *alternating maps * in for linear maps .

Recall that an alternating map is a -linear map such that if any . The classic example of an alternating map is the -linear map given by the determinant (where we are obviously identifying an -tuple of elements with their associated -matrix).

Now, our claim is that for satisfies the following universal mapping property: given any -module and an -linear alternating map there exists a unique -linear map such that .

Indeed, since the set of all simple wedges generates as an -module it’s clear that any such map is unique, and so we must merely prove existence. To do this we note that since is -linear we get an -map and so it suffices to show that kernel of this map contains . But, this is clear for if then maps to . Thus, we see the that the map descends to an -linear map with the property that

as desired. We summarize this as follows:

**Theorem: ***Let be an -module (for a commutative unital ring) and . Then, for every alternating -linear map (where is some other module) there exists a unique -linear map with *

It’s obvious that since this is a universal mapping property that we could have defined to be any module satisfying this property, since any two such -modules are necessarily canonically isomorphic.

Note that this also gives us a way to define -linear maps out of exterior products. Indeed, just like the tensor product, while the simple tensors span we know that they aren’t, in general, linear dependent so that we can’t define an -map on the simple tensors and “extend by linearity”. Indeed, to construct an -map we first define a -linear -map and then use the universal property of the tensor product to produce an -linear map that acts on simple tensors “how we want”. Similarly, to define maps we cannot merely define the map on the simple wedges and extend by linearity (for the exact same reason we can’t do this for tensor products). Instead we need to construct an alternating -linear -map and then use the universal property to produce an -linear map with the desired action on simple wedges.

Note that thinking about exterior products in this way gives us an interesting way to view the elements of . Indeed, note that an element is nonzero in if and only if there exists some -module and some alternating map such that is not sent to zero (why?). In particular, this gives us an interesting view of what it should mean for to be the zero module. It should mean that the only alternating maps out of to any module is the zero map.

For example, we can now prove easily that if is an integral domain then . Indeed, suppose that is any left -module and is alternating. It’s easy to see that (the numerators come out for free, and the denominators come out by multiplying on the outside by, say, and bringing the numerator into the second coordinate). But, since was alternating it follows that and since were arbitrary it follows that . Thus, there exists non nonzero -alternating maps out of from where the above tells us that as desired. This tells us, for example, that and .

**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.

[6] Conrad, Keith. “Exterior Powers.” *Www.math.uconn.edu/~kconrad*. Web. <http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/extmod.pdf>.

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