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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotic depth and connectedness in projective schemes


Author: M. Brodmann
Journal: Proc. Amer. Math. Soc. 108 (1990), 573-581
MSC: Primary 13C15; Secondary 13H99, 14A15
MathSciNet review: 1031674
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Abstract: Let $ I \subseteq \mathfrak{m}$ be an ideal of a local noetherian ring $ \left( {R,\mathfrak{m}} \right)$. Consider the exceptional fiber $ {\pi ^{ - 1}}\left( {V\left( 1 \right)} \right)$ of the blowing-up morphism

$\displaystyle \pi :\operatorname{Proj}\left( {{ \oplus _{n \geq 0}}{I^n}} \right) \to \operatorname{Spec}\left( R \right)$

and the special fiber $ {\pi ^{ - 1}}\left( \mathfrak{m} \right)$. We show that the complement set

$\displaystyle {\pi ^{ - 1}}\left( {V\left( I \right)} \right) - {\pi ^{ - 1}}\left( \mathfrak{m} \right)$

is highly connected if the asymptotic depth of the higher conormal modules $ {I^n}/{I^{n + 1}}$ is large.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1031674-6
PII: S 0002-9939(1990)1031674-6
Article copyright: © Copyright 1990 American Mathematical Society