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

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].

MSC:

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