代写MATH4312 COMMUTATIVE ALGEBRA: ASSIGMENT 1代做Java语言

MATH4312 COMMUTATIVE ALGEBRA:

ASSIGMENT 1

Problem 1. Let R[x] be the ring of polynomials in the indeterminate x over a Noetherian ring R.

(1) Given any sequence C of elements in R[x], denote by (C) the ideal of R[x] generated by the elements of C. Show that there exists a finite subsequence Cf of C such that (C) = (Cf ).

(2) Given any prime ideal P of R, let P[x] be its extension in R[x]. Show that P[x] and (P, x) are prime ideals in R[x], and P[x] ⊊ (P, x).

Problem 2. Let R be the polynomial ring of 2 indeterminates δ1, δ2 over C, that is, R = C[δ1, δ2]. Let M = C[x, y].

(1) (a) Show that M forms an R-module with the action defined, for any ele-ment g = P i≥0 gi(δ1)δi2 of R, where gi(δ1) ∈ C[δ1], by

(b) Show that M0 = C[x] + yC[x] and M′ = y 3C[x, y] are R submodules of M.

(2) Let j : M→M′ be the R-module map defined by j(f) = δ2.f for all f ∈ M. Show that the following sequence of R-modules is exact

where ι : M0→M is the inclusion map.

Problem 3. Let C3 = {1, ω, ω2} be the cyclc group of order 3, where ω 3 = 1. It acts on C[x, y] by

ω.x = exp(2πi/3)x,   ω.y = exp(4πi/3)y,

where i = √−1. Denote by R = C[x, y] C3 the subalgebra of invariants.

(1) Show that C[x, y] admits a grading such that x is of degree 1 and y is of degree 2, and describe the homogeneous components with respect to this grading.

(2) Show that R is finitely generated as a subalgebra of C[x, y], and give a finite set of generators.

Problem 4. Let R = C[z1, z2, z3, . . .] be the ring of polynomials in infinitely many variables z1, z2, z3, . . . over C, and let M be a maximal ideal of R. Denote by ιi : C[zi ]→R the natural inclusions and by π : R→R/M the natural surjection.

(1) Show that the ring homomorphism φi = π ◦ ιi : C[zi ]→R/M has a non-zero kernel for all i = 1, 2, . . . .

(2) Show that there exist a sequence (a1, a2, . . .) with ai ∈ C for all i = 1, 2, . . . , such that M = (z1 − a1, z2 − a2, . . .).