Commutative Algebra - summer term 2017

Commutative Algebra - summer term 2017

Assignment sheet 11

Prof. Dr. Mohamed Barakat, M.Sc. Kamal Saleh

Exercise 1. (Tensor product of finitely presented modules)

LetRbe a commutative unitary computable ring. LetAbe anm×n matrix andB ap×q matrix with entries in R. The Kronecker product A⊗B is the mp×nq block matrix:

A⊗B :=







a₁₁B . . . a_1nB ... . .. ... a_m1B . . . a_mnB





.

For any two finitely generated R-modules M = hm₁, . . . , m_ri_R, N = hn₁, . . . , n_si_R, the set {m₁ ⊗n₁, m₁ ⊗n₂, . . . , m_r ⊗n_s} generates M ⊗_RN. We call this set the product generating set of M ⊗_RN.

1. Let f :R^{m A}−→Rⁿ, g :R^{p B}−→R^q be twoR-linear maps defined by the matrices Aand B, respectively. Show that the matrix of f ⊗g : R^m ⊗_RR^p → Rⁿ⊗_RR^q w.r.t. the product bases of R^m⊗_RR^p and Rⁿ⊗_RR^q isA⊗B.

2. Let N ∈ R-fpmod with presentation matrix N ∈R^u×s and R^p be the free R-module of rank p. Use 1. and the fact that R^p⊗ − is right exact to compute a presentation matrix for R^p⊗_RN.

3. Let h:R^{p H}−→R^q be an R-linear map inR-fpmod. Use 2. to compute a morphism in R-fpres corresponding to h⊗idN :R^p⊗_RN →R^q⊗_RN.

4. Let M ∈ R-fpmod with presentation matrix M ∈ R^p×q. Use 3. and the fact that

− ⊗N is right exact to compute a presentation matrix forM ⊗_RN. 5. (a) Let R=Z. Apply the above to compute Z/4Z⊗Z/6Z.

(b) LetR =Q[x, y, z],I :=hx²−y², xz−y,−x⁵+yzi_R,M := coker((^{x y}z y)). Compute Ann_R(R/I ⊗_RM).

Exercise 2. (Being a flat R-module is local property) LetR be a ring,M anR-module. The following are equivalent:

1. M is a flatR-module.

2. M_p is a flat R_p-module for all p∈SpecR.

3. M_m is a flatR_m-module for all m∈MaxR.

Hand in until Juli 25th 12:15 in the class or in the Box in ENC, 2nd floor, at the entrance of the building part D.