Asymptotic depth and connectedness in projective schemes

Brodmann, M (1990). Asymptotic depth and connectedness in projective schemes. Proceedings of the American Mathematical Society, 108(3):573-581.

Abstract

Let $I \subseteq \mathfrak{m}$ be an ideal of a local noetherian ring (R, m). Consider the exceptional fiber π-1(V(I)) of the blowing-up morphism $\pi: \operatorname{Proj} \bigg(\bigoplus_{n \geq 0}I^n\bigg) \rightarrow \operatorname{Spec}(R)$ and the special fiber π-1(m). We show that the complement set π-1(V(I)) - π-1(m) is highly connected if the asymptotic depth of the higher conormal modules In/In + 1 is large.

Abstract

Let $I \subseteq \mathfrak{m}$ be an ideal of a local noetherian ring (R, m). Consider the exceptional fiber π-1(V(I)) of the blowing-up morphism $\pi: \operatorname{Proj} \bigg(\bigoplus_{n \geq 0}I^n\bigg) \rightarrow \operatorname{Spec}(R)$ and the special fiber π-1(m). We show that the complement set π-1(V(I)) - π-1(m) is highly connected if the asymptotic depth of the higher conormal modules In/In + 1 is large.

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