# Math 551, Homework 8

Problems in bold are to be handed in

Unless noted otherwise, problems are from Dummit and Foote.

• Chapter 11.4 / 2, 3

• Chapter 11.5 / 5, 6, 9

• Chapter 12.1 / 1, 2, 3, 11, 12

In the following problems, if R is a ring, and M is an R-module, we say that M is graded if we have a direct sum decomposition M = ⊕i ≥ 0Mi for R-modules Mi. If M =  ⊕ Mi, N =  ⊕ Ni are graded R-modules, a graded R-module homomorphism f : M → N, is an R-module homomorphism such that f(Mi) ⊂ Ni for all i.

We say that a graded R-module M is concentrated in degree j, and write M = Mj when M =  ⊕ Mi is a graded R-module with Mi = 0 for i ≠ j.

Optional problem

Let R be a commutative ring, and let V be an R-module. Show that the tensor algebra T(V) has the following universal property: it is an algebra containing V, as an R-submodule, and, considering V = V1 as a graded R-module concentrated in degree 1, if A is a graded R-algebra, then every graded homomorphism of R-modules V → A with V, can be uniquely extended to a homomorphism of graded R-algebras T(V) → A.

Optional problem

Let R be a commutative ring, and let V be an R-module. Show that the symmetric algebra S(V) has the following universal property: it is a commutative algebra containing V as an R-submodule, and, considering V = V1 as a graded R-module concentrated in degree 1, if A is a graded R-algebra which is commutative, then every graded homomorphism of R-modules V → A with V, can be uniquely extended to a homomorphism of graded R-algebras S(V) → A.

If A is a graded R-algebra, we say that A is graded-commutative if for a ∈ Ai, b ∈ Aj, we have ab = ( − 1)ijba. In other words, ab = ba whenever either a or b has even degree, and ab =  − ba whenever a and b both have odd degree.

Required problem

Let R be a commutative ring, and let V be an R-module. Show that the exterior algebra Λ(V) has the following universal property: it is a graded-commutative algebra containing V as a graded R-submodule, and, considering V = V1 as a graded R-module concentrated in degree 1, if A is a graded R-algebra which is graded-commutative, then every graded homomorphism of R-modules V → A with V, can be uniquely extended to a homomorphism of graded R-algebras Λ(V) → A.