A quick proof of the Hartshorne-Lichtenbaum vanishing theorem. (English) Zbl 0816.13014

Brodmann, Markus; Huneke, Craig

Bajaj, Chandrajit L. (ed.), Algebraic geometry and its applications. Collections of papers from Shreeram S. Abhyankar’s 60th birthday conference held at Purdue University, West Lafayette, IN, USA, June 1-4, 1990. New York: Springer-Verlag. 305-308 (1994).

Let \((R,m)\) be a local Noetherian ring and let \(I\) be an ideal of \(R\). Set \(d = \dim (R)\). Then the following statements are equivalent:
(i) \(H^ d_ I (R)=0\);
(ii) For all minimal primes \(P\) of \(\widehat R\) with \(\dim (\widehat R/P) = \dim(R)\), \(\dim (\widehat R/(I \widehat R + R)) > 0\).
Here \(H^ j_ I (M)\) denotes the \(j\)-th local cohomology of \(M\) with respect to \(I\). \(\widehat R\) is the completion of \(R\) with respect to the powers of its maximal ideal. – The authors give an elementary proof of this equivalence which is known as Hartshorne-Lichtenbaum vanishing theorem.
For the entire collection see [Zbl 0788.00050].

Reviewer: V.K.Vedernikov (Moskva)

MSC:

13D45	Local cohomology and commutative rings
14B15	Local cohomology and algebraic geometry
14F17	Vanishing theorems in algebraic geometry

Keywords:

local cohomology; Hartshorne-Lichtenbaum vanishing theorem