Vanishing, singularities and effective bounds via prime characteristic local algebra. (English) Zbl 0913.13004

Kollár, János (ed.) et al., Algebraic geometry. Proceedings of the Summer Research Institute, Santa Cruz, CA, USA, July 9–29, 1995. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 62(pt.1), 289-325 (1997).
In this article, Smith brings together research, some known and some new, from commutative algebra and algebraic geometry, most of which use the technique of reduction to prime characteristic \(p\). This technique has been used since at least 1857 when Dedekind proved the irreducibility of cyclotomic polynomials. More recent examples of using reduction to characteristic \(p\) are: the theorem of Mori on the existence of rational curves on smooth projective variety [S. Mori, Ann. Math., II. Ser. 110, 593-606 (1979; Zbl 0423.14006)], proofs of several homological conjectures by C. Peskine and L. Szpiro [Publ. Math., Inst. Hautes Etud. Sci. 42, 47-119 (1972; Zbl 0268.13008)], and by M. Hochster [“Topics in the homological theory of modules over commutative rings”, Regional Conference Series in Mathematics, No. 24, (1975; Zbl 0302.13003)], the proof of the Cohen-Macaulayness of rings of invariants by M. Hochster and J.L. Roberts [Adv. Math. 13, 115-175 (1974; Zbl 0289.14010)], and Kodaira vanishing theorems of C. Huneke and K.E. Smith [J. Reine Angew. Math. 484, 127-152 (1997; see the preceding review)] and of N. Hara [Am. J. Math. 120, No. 5, 981-996 (1998)] to mention a few. Much of the recent activity in the reduction of the characteristic \(p\) methods have been inspired by Hochster and Huneke’s theory of tight closure in which Smith has made many contributions. In this beautifully written article Smith brings together some previously not-brought-together researches of various groups: tight closure, Kodaira vanishing, various singularities, and Metha-Ramanathan’s work on algebraic groups. Moreover, Smith gives new insight, new proofs and new connections. This article makes for a delightful reading!
For the entire collection see [Zbl 0882.00032].


13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
14F17 Vanishing theorems in algebraic geometry
14B15 Local cohomology and algebraic geometry
13D45 Local cohomology and commutative rings
14B05 Singularities in algebraic geometry
13B22 Integral closure of commutative rings and ideals