Melvin Hochster

Melvin Hochster

Melvin Hochster (born August 2, 1943 in Brooklyn , New York ) is an American mathematician who made outstanding contributions in the field of commutative algebra . He is currently the Jack E. McLaughlin Distinguished University Professor of Mathematics at the University of Michigan at Ann Arbor .

Hochster attended Stuyvesant High School, where he was the captain of the math team. He won the Putnam competition in 1960, received his BA from Harvard University in 1964 and his doctorate from Princeton University under Gorō Shimura in 1967 for a work that characterizes the spectrum of a Noetherian ring . From 1967 to 1973 he held positions at the University of Minnesota and then until 1977 a professorship at Purdue University . In 1976 he came to the University of Michigan as a visiting professor. Hochster received the Cole Prize in 1980 (alongside Michael Aschbacher ), a Guggenheim scholarship in 1982 and has been a member of the National Academy of Sciences and the American Academy of Arts and Sciences since 1992.

Hochster mainly works in the field of commutative algebra , in particular on modules over local rings . He proved important theorems about Cohen-Macaulay rings , invariant theory and homological algebra . The Hochster-Roberts Theorem states that the invariant ring of a linear reductive group is Cohen-Macaulay. Much of his work is based on homological conjectures, many of which he was able to confirm in the case that the rings contain a body , by proving the existence of so-called large Cohen-Macaulay modules. Another focus is the proof technique of reducing to positive characteristics . The theory of tight closure , which Hochster introduced in 1986 together with Craig Huneke , and which has a multitude of applications in homological algebra, commutative algebra and algebraic geometry , also goes in this direction .

Hochster supervised more than thirty PhD students and enjoys playing bridge in his spare time .

In 1978 he was invited speaker at the International Congress of Mathematicians in Helsinki ( Cohen-Macaulay rings and modules ).

Karen Smith is one of his PhD students .

Fonts

• Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Annals of Math. 96 (1972), 318-337
• Cohen-Macaulay modules, in Conference on Commutative Algebra (Lawrence, Kansas, 1972), Lecture Notes in Mathematics. 311, Springer-Verlag, Berlin, 1973, pp. 120–152.
• Cohen-Macaulay rings, combinatorics, and simplicial complexes, in Proceedings of the Second Oklahoma Ring Theory Conference (March 1976), Marcel-Dekker, New York, 1977, pp. 171-223.
• Topics in the homological theory of modules over commutative rings, CBMS Regional Conference, Lincoln (Nebraska) 1974, American Mathematical Society 1975
• Big and small Cohen-Macaulay modules, in Proceedings of the Special Session on Module Theory (Seattle, Aug. 1977), Lecture Notes in Mathematics 700, Springer 1979, 119-142
• with Craig Huneke: Tightly closed ideals, Bulletin AMS, 18, 1988, 45-48
• with Craig Huneke: Tight closure, in Commutative Algebra, Math. Sci. Research Inst. Publ. 15, Springer-Verlag, New York-Berlin-Heidelberg, 1989, 305-324.
• with Craig Huneke: Tight closure, invariant theory, and the Briançon – Skoda theorem, Journal of the American Mathematical Society 3 (1), 1990, pp. 31–116.
• with Craig Huneke: Infinite integral extensions and big Cohen-Macaulay algebras, Annals of Math. 135 (1992), 53-89.
• with Craig Huneke: Phantom Homology, Memoirs American Mathematical Society 103, 1993, 1-91
• Homological conjectures, old and new, Illinois J. Math. 51 (2007) 151-169.

Web links

• Mel Hochster , homepage at the University of Michigan