Proposition (S_1,…,S_r) from commutative algebra. Let k be a field, and I a proper ideal of k[X_1,…,X_n].

cimithe4c

cimithe4c

Answered question

2022-10-22

Proposition ( S 1 , , S r ) from commutative algebra
Let k be a field, and I a proper ideal of k [ X 1 , , X n ]. Then there exist a polynomial sub-k-algebra k [ S 1 , , S n ] of k [ X 1 , , X n ] and an integer 0 r n such that:
(a) k [ X 1 , , X n ] is finite over k [ S 1 , , S n ];
(b) k [ S 1 , , S n ] I = ( S 1 , , S r ) (this is the zero ideal if r = 0);
(c) k [ S r + 1 , , S n ] k [ X 1 , , X n ] / I is finite injective.
I know that ( S 1 , , S r ) is the smallest ideal containing { S 1 , , S r } but is it the ideal of k [ X 1 , , X n ] or the ideal of k [ S 1 , , S n ]?

Answer & Explanation

namotanimfc

namotanimfc

Beginner2022-10-23Added 7 answers

Explanation:
Since the LHS is not an ideal in k [ X 1 , X n ] in general, the intention must be that ( S 1 , , S r ) is an ideal in k [ S 1 , , S n ].

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