Proceedings of the 44th Symposium on Ring Theory and ...

Pro<strong>of</strong> <strong>of</strong> Claim 7. Take y ∈ [(a) : b]∩I n . Then, since by ∈ (a), we see bt·yt n = byt n+1 = 0
in G(I). Hence yt n ∈ [H 0 M (G(I))] n, because bt is a homogeneous parameter for <strong>the</strong> graded
ring G(I). Recall now that n ≥ 2, whence [H 0 M (G(I))] n = (0), so that yt n = 0. Thus
y ∈ (a) + I n+1 , whence y ∈ (a) + [((a) : b) ∩ I n+1 ], as claimed.
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Since x ∈ (a) + [((a) : b) ∩ I 2 ], thanks to Claim 7, we get x ∈ (a) + I n+1 for all n ≥ 1,
whence x ∈ (a), so that <strong>the</strong> map ρ is injective. Thus
as A-modules.
[H 1 M(G)] 0
∼ =
[(a) : b] ∩ I
(a)
Theorem 8. Suppose that d = 2 and depth A > 0. Let Q = (x, y) be a parameter ideal
in A and assume that x is superficial with respect to Q. Then
−h 1 (A) ≤ e 2 Q(A) ≤ 0
and <strong>the</strong> following three conditions are equivalent.
(1) e 2 Q (A) = 0.
(2) x, y forms a d-sequence in A.
(3) x l , y l forms a d-sequence in A for all integers l ≥ 1.
Pro<strong>of</strong>. By Lemma 5 we have
e 2 Q(A) = −l A
( [(x l ) : y l ] ∩ (x, y) l
(x l )
)
≤ 0
for all integers l ≫ 0. To show that −h 1 (A) ≤ e 2 Q (A), we may assume that H1 m(A) is
finitely generated. Take <strong>the</strong> integer l ≫ 0 so that <strong>the</strong> system a = x l , b = y l <strong>of</strong> parameters
<strong>of</strong> A is standard. Then since
[(a) : b] ∩ Q l
(a)
⊆ (a) : b
(a)
we get −h 1 (A) ≤ e 2 Q (A).
Let us consider <strong>the</strong> second assertion.
(1) ⇒ (3). Take an integer N ≥ 1 so that
for all l ≥ N (cf. Lemma 5); hence
∼ = H
0
m (A/(a)) ∼ = H 1 m(A),
e 2 Q(A) = −l A
( [(x l ) : y l ] ∩ (x, y) l
(x l )
[(x l ) : y l ] ∩ (x, y) l = (x l ).
Claim 9. [(x l ) : y l ] ∩ (x, y) l = (x l ) for all l ≥ 1.
Pro<strong>of</strong> <strong>of</strong> Claim 9. We may assume that 1 ≤ l < N. Take τ ∈ [(x l ) : y l ] ∩ (x, y) l . Then,
since y N (x N−l τ) = y N−l x N−l (y l τ) ∈ (x N ), we have x N−l τ ∈ [(x N ) : y N ] ∩ (x, y) N = (x N ).
Thus τ ∈ (x l ), because x is A-regular (recall that depth A > 0 and x is superficial with
respect to Q).
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