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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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