**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

**Additional Problems**

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 ≥ 0}*M*_{i} for *R*-modules *M*_{i}. If *M* = ⊕ *M*_{i}, *N* = ⊕ *N*_{i} are graded *R*-modules, a graded *R*-module homomorphism *f* : *M* → *N*, is an *R*-module homomorphism such that *f*(*M*_{i}) ⊂ *N*_{i} for all *i*.

We say that a graded *R*-module *M* is concentrated in degree *j*, and write *M* = *M*_{j} when *M* = ⊕ *M*_{i} is a graded *R*-module with *M*_{i} = 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* = *V*_{1} 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* = *V*_{1} 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* ∈ *A*_{i}, *b* ∈ *A*_{j}, we have *a**b* = ( − 1)^{ij}*b**a*. In other words, *a**b* = *b**a* whenever either *a* or *b* has even degree, and *a**b* = − *b**a* 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 = V_{1} 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.**