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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Homological methods for hypergeometric families
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by Laura Felicia Matusevich, Ezra Miller and Uli Walther
J. Amer. Math. Soc. 18 (2005), 919-941
Published electronically: May 25, 2005


We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for hypergeometric systems $H_A(\beta )$ arising from a $d \times n$ integer matrix $A$ and a parameter $\beta \in \mathbb {C}^d$. To do so we introduce an Euler–Koszul functor for hypergeometric families over $\mathbb {C}^d$, whose homology generalizes the notion of a hypergeometric system, and we prove a homology isomorphism with our general homological construction above. We show that a parameter $\beta \in \mathbb {C}^d$ is rank-jumping for $H_A(\beta )$ if and only if $\beta$ lies in the Zariski closure of the set of $\mathbb {C}^d$-graded degrees $\alpha$ where the local cohomology $\bigoplus _{i < d} H^i_\mathfrak m(\mathbb {C}[\mathbb {N} A])_\alpha$ of the semigroup ring $\mathbb {C}[\mathbb {N} A]$ supported at its maximal graded ideal $\mathfrak m$ is nonzero. Consequently, $H_A(\beta )$ has no rank-jumps over $\mathbb {C}^d$ if and only if $\mathbb {C}[\mathbb {N} A]$ is Cohen–Macaulay of dimension $d$.
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Bibliographic Information
  • Laura Felicia Matusevich
  • Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
  • Address at time of publication: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
  • MR Author ID: 632562
  • Email:
  • Ezra Miller
  • Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
  • Email:
  • Uli Walther
  • Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
  • Email:
  • Received by editor(s): June 22, 2004
  • Published electronically: May 25, 2005
  • Additional Notes: The first author was partially supported by a postdoctoral fellowship from MSRI and an NSF Postdoctoral Fellowship
    The second author was partially supported by NSF Grant DMS-0304789
    The third author was partially supported by the DfG, the Humboldt foundation, and NSF Grant DMS-0100509

  • Dedicated: Uli Walther dedicates this paper to the memory of his father, Hansjoachim Walther.
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 18 (2005), 919-941
  • MSC (2000): Primary 13N10, 13D45, 14D99, 13F99, 16E99; Secondary 32C38, 35A27, 14M25, 70F20, 33C70, 13C14, 13D07
  • DOI:
  • MathSciNet review: 2163866