By Andreas Gathmann

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Let N be a submodule of a finitely generated R-module M. 5 (c) you have seen that N need not be finitely generated in this case. However, prove now that N is finitely generated if it is the kernel of a surjective R-module homomorphism ϕ : M → Rn for some n ∈ N. ) 5. 5. Tensor Products 43 Tensor Products In the last two chapters we have developed powerful methods to work with modules and linear maps between them. However, in practice bilinear (or more generally multilinear) maps are often needed as well, so let us have a look at them now.

One application of tensor products is to extend the ring of scalars for a given module. For vector spaces, this is a process that you know very well: suppose that we have e. g. a real vector space V with dimR V = n < ∞ and want to study the eigenvalues and eigenvectors of a linear map ϕ : V → V . We then usually set up the matrix A ∈ Mat(n × n, R) corresponding to ϕ in some chosen basis, and compute its characteristic polynomial. Often it happens that this polynomial does not split into linear factors over R, and that we therefore want to pass from the real to the complex numbers.

A) A sequence 0 −→ M −→ N is exact if and only if ker ϕ = 0, i. e. if and only if ϕ is injective. ϕ A sequence M −→ N −→ 0 is exact if and only if im ϕ = N, i. e. if and only if ϕ is surjective. (b) The sequence 0 −→ M −→ 0 is exact if and only if M = 0. ϕ By (a), the sequence 0 −→ M −→ N −→ 0 is exact if and only if ϕ is injective and surjective, i. e. if and only if ϕ is an isomorphism. 3 (Short exact sequences). 2, the first interesting case of an exact sequence occurs if it has at least three non-zero terms.

### Commutative Algebra by Andreas Gathmann

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