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|Alma mater|| Princeton University |
|Awards|| Cole Prize (1980)|
Guggenheim Fellowship (1981)
Putnam Fellow (1960)
|Institutions||University of Michigan|
|Thesis||Prime Ideal Structure in Commutative Rings|
|Doctoral advisor||Goro Shimura|
|Doctoral students|| Piotr Blass |
Karen E. Smith
Melvin Hochster (born August 2, 1943) is an American mathematician working in commutative algebra. He is currently the Jack E. McLaughlin Distinguished University Professor of Mathematics at the University of Michigan.
Hochster attended Stuyvesant High School,where he was captain of the Math Team, and received a B.A. from Harvard University. While at Harvard, he was a Putnam Fellow in 1960. He earned his Ph.D. in 1967 from Princeton University, where he wrote a dissertation under Goro Shimura characterizing the prime spectra of commutative rings.
He held positions at the University of Minnesota and Purdue University before joining the faculty at Michigan in 1977.
Hochster's work is primarily in commutative algebra, especially the study of modules over local rings. He has established classic theorems concerning Cohen–Macaulay rings, invariant theory and homological algebra. For example, the Hochster–Roberts theorem states that the invariant ring of a linearly reductive group acting on a regular ring is Cohen–Macaulay. His best-known work is on the homological conjectures, many of which he established for local rings containing a field, thanks to his proof of the existence of big Cohen–Macaulay modules and his technique of reduction to prime characteristic. His most recent work on tight closure, introduced in 1986 with Craig Huneke, has found unexpected applications throughout commutative algebra and algebraic geometry.
He has had more than 40 doctoral students, and the Association for Women in Mathematics has pointed out his outstanding role in mentoring women students pursuing a career in mathematics. He served as the chair of the department of Mathematics at the University of Michigan from 2008 to 2017.
Hochster shared the 1980 Cole Prize with Michael Aschbacher, received a Guggenheim Fellowship in 1981, and has been a member of both the National Academy of Sciences and the American Academy of Arts and Sciences since 1992. In 2008, on the occasion of his 65th birthday, he was honored with a conference in Ann Arbor and with a special volume of the Michigan Mathematical Journal.
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not required to be commutative.
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings, as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.
In mathematics, homological conjectures have been a focus of research activity in commutative algebra since the early 1960s. They concern a number of interrelated conjectures relating various homological properties of a commutative ring to its internal ring structure, particularly its Krull dimension and depth.
In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well understood in many ways.
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David Eisenbud is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and was Director of the Mathematical Sciences Research Institute (MSRI) from 1997 to 2007. He was reappointed to this office in 2013, and his term has been extended until July 31, 2022.
In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic. It was introduced by Melvin Hochster and Craig Huneke.
In commutative and homological algebra, depth is an important invariant of rings and modules. Although depth can be defined more generally, the most common case considered is the case of modules over a commutative Noetherian local ring. In this case, the depth of a module is related with its projective dimension by the Auslander–Buchsbaum formula. A more elementary property of depth is the inequality
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Karen Ellen Smith is an American mathematician, specializing in commutative algebra and algebraic geometry. She completed her bachelor's degree in mathematics at Princeton University before earning her PhD in mathematics at the University of Michigan in 1993. Currently she is the Keeler Professor of Mathematics at the University of Michigan. In addition to being a researcher in algebraic geometry and commutative algebra, Smith with others wrote the textbook An Invitation to Algebraic Geometry.
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