代做MATH4312 COMMUTATIVE ALGEBRA: ASSIGMENT 2代写留学生Matlab程序

MATH4312 COMMUTATIVE ALGEBRA:

ASSIGMENT 2

Solutions should be rigorous and clearly presented.

Ambiguous arguments will not be given marks.

Any result from the lectures may be used without proof unless you are asked to prove the result.

If you use results from any other sources, provide proofs for them.

All rings are assumed to be commutative with identity.

Problem 1. Let M = Kn 2 . Arrange its coordinates into an n × n matrix, so that a point A ∈ M can be represented by A = (aij ) where aij ∈ K with i, j = 1, 2, . . . , n.

(1) Describe G = {A ∈ M | det A ≠ 0} as an affine variety.

(2) Give an explicit description of the coordinate ring of G.

Problem 2. Let I = (x 2 − y 3 , y2 − z 3 ) ≤ K[x, y, z], and V = V (I) ⊆ K3 , where K is an algebraically closed field.

(1) Find a one to one and onto polynomial map K−→V .

(2) Determine whether V is isomorphic to K as affine algebraic set.

Problem 3. Let R = K[x1, . . . , xn], where K is an algebraically closed field. Let I ≤ R be an ideal which is radical in the sense that I = √I.

(1) Show that I can be expressed as

(1.1)                I = P1 ∩ P2 ∩ · · · ∩ Pr

for finitely many prime ideals such that Pi 6⊆ Pj for all i ≠ j, and this decomposition is unique up to permutations of the Pi ’s.

(2) Assume that the radical ideal I is proper. Show that the prime ideals Pi in the decomposition (1.1) of I are precisely the elements of the set {proper prime ideal (I : f) | f ∈ R}.

Problem 4. Let R = K[x0, x1, . . . , xn], where K is an algebraically closed field. Denote by P the set of straight lines in Kn+1 passing through the origin 0 = (0, 0 . . . , 0). [Note: a straight line passing through 0 is {(µv0, µv1, . . . , µvn) | ∀µ ∈ K} for a given v = (v0, v1, . . . , vn) ∈ Kn+1 such that v ≠ 0.]

(1) Let e ∈ P, and let f = P i fi ∈ R where fi is the degree i homogeneous component. Show that if there exists a ∈ e such that fi(a) = 0, then fi(b) = 0, ∀b ∈ e. Further show that f(b) = 0, ∀b ∈ e , if and only if fi(b) = 0, ∀b ∈ e , for all i. [In this case, write f(e) = 0.]

(2) Given a homogeneous ideal J ≤ R, let

                      Vp(J) = {e ∈ P | f(e) = 0 ∀f ∈ J}.

Show that Vp(J) = ∅ if √J ⊇ (x0, x1, . . . , xn).

(3) Given any subset X ⊆ P, show that

                      Ip(X) = {f ∈ R | f(e) = 0 ∀e ∈ X}

is a homogeneous ideal in R.

(4) For any homogeneous ideal J ∈ R such that Vp(J) ≠ ∅, show that Ip(Vp(J)) = √J.