## 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 - DOI: https://doi.org/10.1090/S0894-0347-05-00488-1
- Published electronically: May 25, 2005
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## Abstract:

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 $d$.## References

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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: lfm@math.upenn.edu
**Ezra Miller**- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- Email: ezra@math.umn.edu
**Uli Walther**- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- Email: walther@math.purdue.edu
- 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 - © 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: https://doi.org/10.1090/S0894-0347-05-00488-1
- MathSciNet review: 2163866

Dedicated: Uli Walther dedicates this paper to the memory of his father, Hansjoachim Walther.