Supplementary Problem Set 3 - Commutative Algebra | MATH 614 | Assignments Mathematics

Math 614, Fall 2008 Supplementary Problem Set #3

1. Let Kbe a field and S=K[x, y] be the polynomial ring in two variables over K. For

every real number β≥0 let Rβ=K[xmyn:n > βm]⊆S. Show that Rβis not Noetherian

and, therefore, not finitely generated as a K-algebra. This gives an uncountable chain of

such subrings inside K[x, y], for if β < γ, then Rβ⊃Rγ.

2. Let R=C[xu, xv, yu, yv]⊆C[x, y, u, v], where x, y, u, v are indeterminates. Rep-

resent Rexplicitly as a finite module over a polynomial subring Aover C. Give the

algebraically independent generators of Aas a C-algebra, and give a finite set of genera-

tors for Ras an A-module. (You may want to begin by finding an algebraic relation on

xu, xv, yu, yv.)

3. Let Kbe an infinite field and let f6= 0 be in R=K[x1, . . . , xn]. Show that for suitable

λi∈K, λn6= 0, there is a K-linear automorphism φof Rthat sends xito xi+λixn,

1≤i<n, and xnto λnxnsuch that φ(f) is a nonzero scalar multiple of a polynomial that

is monic in xnwith coefficients in K[x1, . . . , xn−1]. (This leads to an alternative proof of

Noether normalization for infinite Kin which the algebraically independent elements are

K-linear combinations of the original generators.)

4. (a) Consider a chain of ideals in a ring Reach of which is not finitely generated. Show

that the union Iof the chain is also not finitely generated.

(b) It follows from Zorn’s lemma that if Ris not Noetherian, there is an ideal Pthat is

maximal with respect to not being finitely generated. Prove that such an ideal Pis prime.

(Thus: Ris Noetherian iff every prime ideal is finitely generated (Cohen).)

5. Let Kbe an algebraically closed field and Lan extension field.

(a) Given that a certain finite system of polynomial equations in finitely many variables

with coefficients in Khas a solution in L, prove that it has a solution in K.

(b) Consider a polynomial f∈K[x1, . . . , xn]− {0}. Show that if fis the product

of two polynomials of strictly smaller degree in L[x1, . . . , xn], then this is also true in

K[x1, . . . , xn]. (Perhaps part (a) will be helpful.)

6. (E. Noether) Let A⊆Rbe rings with ANoetherian and Rfinitely generated over A.

Let G={g1, . . . , gn}be a finite group acting on Rby A-algebra automorphisms. Show

that the fixed ring RG={r∈R: for all g∈G, g(r) = r}is finitely generated over A.

(Let R=A[r1, . . . , rk]. For each j, show that the elementary symmetric functions eij of

g1(rj), . . . , gn(rj) are in RG. Let R0=A[eij : 1 ≤i≤n, 1≤j≤k]. Show that Ris

integral and finitely generated over R0, and use this and that R0⊆RG⊆R.)

EXTRA CREDIT Let R⊆Sbe rings and sin Sbe such that for every prime Qof S,

the image of sin S/Q is integral over R/(Q∩R). Show that Sis integral over R. (This

is an alternative way of doing problem 5. of Problem Set #2.)

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Math 614, Fall 2008 Supplementary Problem Set #

Let K be a field and S = K[x, y] be the polynomial ring in two variables over K. For every real number β ≥ 0 let Rβ = K[xmyn^ : n > βm] ⊆ S. Show that Rβ is not Noetherian and, therefore, not finitely generated as a K-algebra. This gives an uncountable chain of such subrings inside K[x, y], for if β < γ, then Rβ ⊃ Rγ.
Let R = C[xu, xv, yu, yv] ⊆ C[x, y, u, v], where x, y, u, v are indeterminates. Rep- resent R explicitly as a finite module over a polynomial subring A over C. Give the algebraically independent generators of A as a C-algebra, and give a finite set of genera- tors for R as an A-module. (You may want to begin by finding an algebraic relation on xu, xv, yu, yv.)
Let K be an infinite field and let f 6 = 0 be in R = K[x 1 ,... , xn]. Show that for suitable λi ∈ K, λn 6 = 0, there is a K-linear automorphism φ of R that sends xi to xi + λixn, 1 ≤ i < n, and xn to λnxn such that φ(f ) is a nonzero scalar multiple of a polynomial that is monic in xn with coefficients in K[x 1 ,... , xn− 1 ]. (This leads to an alternative proof of Noether normalization for infinite K in which the algebraically independent elements are K-linear combinations of the original generators.)
(a) Consider a chain of ideals in a ring R each of which is not finitely generated. Show that the union I of the chain is also not finitely generated.

(b) It follows from Zorn’s lemma that if R is not Noetherian, there is an ideal P that is maximal with respect to not being finitely generated. Prove that such an ideal P is prime. (Thus: R is Noetherian iff every prime ideal is finitely generated (Cohen).)

Let K be an algebraically closed field and L an extension field.

(a) Given that a certain finite system of polynomial equations in finitely many variables with coefficients in K has a solution in L, prove that it has a solution in K.

(b) Consider a polynomial f ∈ K[x 1 ,... , xn] − { 0 }. Show that if f is the product of two polynomials of strictly smaller degree in L[x 1 ,... , xn], then this is also true in K[x 1 ,... , xn]. (Perhaps part (a) will be helpful.)

(E. Noether) Let A ⊆ R be rings with A Noetherian and R finitely generated over A. Let G = {g 1 ,... , gn} be a finite group acting on R by A-algebra automorphisms. Show that the fixed ring RG^ = {r ∈ R : for all g ∈ G, g(r) = r} is finitely generated over A. (Let R = A[r 1 ,... , rk]. For each j, show that the elementary symmetric functions eij of g 1 (rj ),... , gn(rj ) are in RG. Let R 0 = A[eij : 1 ≤ i ≤ n, 1 ≤ j ≤ k]. Show that R is integral and finitely generated over R 0 , and use this and that R 0 ⊆ RG^ ⊆ R.)

EXTRA CREDIT Let R ⊆ S be rings and s in S be such that for every prime Q of S, the image of s in S/Q is integral over R/(Q ∩ R). Show that S is integral over R. (This is an alternative way of doing problem 5. of Problem Set #2.)

Supplementary Problem Set 3 - Commutative Algebra | MATH 614, Assignments of Mathematics

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