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C) Each regular sequence [f1 , . . , fr ] ⊆ R+ , for r ≤ dim(R), can be extended to a homogeneous system of parameters. In particular, a regular sequence of length r = dim(R) is a homogeneous system of parameters. Proof. a) Let g ∈ annR (M/M f ), and let {m1 , . . , ms } generate M as an R-module. Hence by the Cayley-Hamilton Theorem there is h(T ) = T s + s−1 i i=0 hi T ∈ R[T ] monic such that hi ∈ f R and h(g) ∈ annR (M ). Hence we 35 have g s ∈ annR (M ) + f R, thus annR (M/M f ) ⊆ annR (M ) + f R.

24) Exercise: Prime avoidance. a) Let R be a commutative ring, let P1 , . . , Pn R be prime ideals, for n ∈ N, n and let I R be an ideal such that I ⊆ i=1 Pi . Then there is i ∈ {1, . . , n} such that I ⊆ Pi . b) Let R be Noetherian, let I R be an ideal and let M = {0} be a finitely generated R-module. Show that either I contains a non-zerodivisor on M , or there is 0 = m ∈ M such that I ⊆ annR (m). Proof. 3). 2]. 25) Exercise: Krull’s Principal Ideal Theorem. Let R be Noetherian and let P R be a prime ideal such that ht(P ) = r, for some r ∈ N.

2]. 36) Exercise: Hironaka decomposition. Let G = (1, 2)(3, 4), (1, 3)(2, 4) ≤ S4 be the Klein group of order 4, and let Q[X ] = Q[X1 , . . , X4 ]. 3 a) Show that the Hilbert series of Q[X ]G is given as HQ[X ]G = (1−T1+T )·(1−T 2 )3 . b) Find primary invariants {f1 , . . , f4 } ⊆ Q[X ]G such that deg(f1 ) = 1 and deg(f2 ) = deg(f3 ) = deg(f4 ) = 2, and secondary invariants {g1 , g2 } ⊆ Q[X ]G such that deg(g1 ) = 0 and deg(g2 ) = 3, yielding the Hironaka decomposition 2 Q[X ]G = i=1 (gi · Q[f1 , .

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More Invariant Theory of Finite Groups by Jürgen Müller


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