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Melvin Hochster | |
---|---|

Born | |

Nationality | American |

Alma mater | Princeton University Harvard University |

Spouse(s) | Margie Morris |

Children | 5 |

Awards | Cole Prize (1980) Guggenheim Fellowship (1981) Putnam Fellow (1960) |

Scientific career | |

Fields | Mathematics |

Institutions | University of Michigan |

Thesis | Prime Ideal Structure in Commutative Rings |

Doctoral advisor | Goro Shimura |

Doctoral students | Piotr Blass Sankar Dutta 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,^{ [1] } 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.^{ [2] }

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.

In commutative algebra, a **Gorenstein local ring** is a commutative Noetherian local ring *R* with finite injective dimension as an *R*-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense.

In ring theory and homological algebra, the **global dimension** of a ring *A* denoted gl dim *A*, is a non-negative integer or infinity which is a homological invariant of the ring. It is defined to be the supremum of the set of projective dimensions of all *A*-modules. Global dimension is an important technical notion in the dimension theory of Noetherian rings. By a theorem of Jean-Pierre Serre, global dimension can be used to characterize within the class of commutative Noetherian local rings those rings which are regular. Their global dimension coincides with the Krull dimension, whose definition is module-theoretic.

**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

**Combinatorial commutative algebra** is a relatively new, rapidly developing mathematical discipline. As the name implies, it lies at the intersection of two more established fields, commutative algebra and combinatorics, and frequently uses methods of one to address problems arising in the other. Less obviously, polyhedral geometry plays a significant role.

In mathematics, a **Stanley–Reisner ring**, or **face ring**, is a quotient of a polynomial algebra over a field by a square-free monomial ideal. Such ideals are described more geometrically in terms of finite simplicial complexes. The Stanley–Reisner ring construction is a basic tool within algebraic combinatorics and combinatorial commutative algebra. Its properties were investigated by Richard Stanley, Melvin Hochster, and Gerald Reisner in the early 1970s.

**Maurice Auslander** was an American mathematician who worked on commutative algebra and homological algebra. He proved the Auslander–Buchsbaum theorem that regular local rings are factorial, the Auslander–Buchsbaum formula, and introduced Auslander–Reiten theory and Auslander algebras.

In algebra, the **Hochster–Roberts theorem**, introduced by Hochster and Roberts (1974), states that rings of invariants of linearly reductive groups acting on regular rings are Cohen–Macaulay.

**Hossein Zakeri**, Prof. Dr. is an Iranian mathematician. He, along with Prof. R. Y. Sharp, are the founders of generalized fractions, a branch in theory of commutative algebra which expands the concept of fractions in commutative rings by introducing the modules of generalized fractions. This topic later found applications in local cohomology, in the monomial conjecture, and other branches of commutative algebra.

**Craig Lee Huneke** is an American mathematician specializing in commutative algebra. He is a professor at the University of Virginia.

**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*.

In algebra and algebraic geometry, given a commutative Noetherian ring and an ideal in it, the ** n-th symbolic power** of is the ideal

Given positive integers and , the -th **Macaulay representation** of is an expression for as a sum of binomial coefficients:

**Yves André** is a French mathematician, specializing in arithmetic geometry.

- Hochster, Melvin (1975).
*Topics in the homological theory of modules over commutative rings*. Providence: American Mathematical Society. ISBN 0-8218-1674-8. - Hochster, Melvin; Huneke, Craig (1993).
*Phantom homology*. Providence: American Mathematical Society. ISBN 0-8218-2556-9. - Huneke, Craig (2008),
*Mathematical biography of Melvin Hochster*,**57**, Michigan Mathematical Journal

- ↑ "Stuyvesant Math Team, Spring 1960". Archived from the original on 2011-05-29. Retrieved 2007-10-31.
- ↑ Melvin Hochster at the Mathematics Genealogy Project

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