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Asymptotic depth and connectedness in projective schemes. (English) Zbl 0695.13012
For a local Noetherian ring $$(R,{\mathfrak m})$$ and $$I\subset R$$ an ideal of height $$>0$$ the asymptotic depth $$\bar t$$ of the higher conormal modules $$I^ n/I^{n+1}$$ is related to some topology of the blowing up $$\pi$$ of Spec(R) at I. One of the main results says that for large values of $$\bar t$$ and $$\text{grad}e(I,R)>1$$ the complement $$C:=\pi^{-1}(V(I))-\pi^{- 1}({\mathfrak m})$$ of the special fiber in the exceptional fiber is highly connected (i.e. the connectedness subdimension of C is large, see proposition (4.8) for more details). The grade-condition in proposition (4.8) implies that the exceptional fiber is connected. Then the statement of (4.8) follows from proposition (3.7) where instead of the blowing-up morphism $$\pi$$ one works with an arbitrary projective morphism induced by a homogeneous R-algebra S. In this more general context the author considers a Noetherian graded S-module $$M=\oplus M_ n$$ and relates the depth of the R-modules $$M_ n$$ to the connectivity of the sheaf $${\mathcal F}$$ induced by M on Proj(S), see (3.3), (3.7) and (3.8).
Reviewer: M.Herrmann

##### MSC:
 13C15 Dimension theory, depth, related commutative rings (catenary, etc.) 13H99 Local rings and semilocal rings 14F45 Topological properties in algebraic geometry 14A15 Schemes and morphisms
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##### References:
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