# 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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5 citations in Web of Science®
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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics connectivity of sheafs; asymptotic depth; topology of the blowing up English 1990 01 Jun 2010 15:02 05 Apr 2016 13:29 American Mathematical Society 0002-9939 First published in [ Proc. Amer. Math. Soc. 108 (1990), no. 3, 573--581], published by the American Mathematical Society https://doi.org/10.2307/2047773 http://www.jstor.org/stable/2047773 http://www.ams.org/mathscinet-getitem?mr=1031674http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0695.13012

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