Math 515 homework #17

Due date: 19980415

D. J. Bernstein

19980408

Graded modules. Let A be a ring. In the following exercises, M is an A[x_{1}, x_{2}, …, x_{d}]-module with a grading: a sequence of A-submodules M_{0}, M_{1}, … of M such that (1) every element of M is expressible uniquely in the form m_{0} + m_{1} + ⋯, with m_{n} ∈ M_{n} and almost all m_{n} equal to zero; (2) x_{k}M_{n} ⊆ M_{n+1} for all k and n.

Exercise 48. Let φ be an A[x_{1}, x_{2}, …, x_{d}]-module homomorphism from M to M such that φ(M_{n}) ⊆ M_{n+1}. Let K be the kernel of φ. Prove that the sequence of A-modules K ∩ M_{0}, K ∩ M_{1}, … is a grading of K.

Exercise 49. Let φ be an A[x_{1}, x_{2}, …, x_{d}]-module homomorphism from M to M such that φ(M_{n}) ⊆ M_{n+1}. Let L be the cokernel of φ, i.e., M/φ(M); let π be the natural map from M to L. Prove that the A-module π(M_{n}) is isomorphic to M_{n+1}/φ(M_{n}); and prove that π(M_{0}), π(M_{1}), … is a grading of L.