×

zbMATH — the first resource for mathematics

Hypergeometric D-modules and twisted Gauß-Manin systems. (English) Zbl 1181.13023
Summary: The Euler-Koszul complex is the fundamental tool in the homological study of \(A\)-hypergeometric differential systems and functions. We compare Euler-Koszul homology with D-module direct images from the torus to the base space through orbits in the corresponding toric variety. Our approach generalizes a result by I. M. Gelfand, M. M. Kapranav, and A. Y. Zelevinskij [Adv. Math. 84, No. 2, 255–271 (1990; Zbl 0741.33011), Thm. 4.6] and yields a simpler, more algebraic proof.
In the process we extend the Euler-Koszul functor to a category of infinite toric modules and describe multigraded localizations of Euler-Koszul homology.

MSC:
13N10 Commutative rings of differential operators and their modules
33C70 Other hypergeometric functions and integrals in several variables
16E05 Syzygies, resolutions, complexes in associative algebras
Software:
Macaulay2
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Adolphson, Alan, Hypergeometric functions and rings generated by monomials, Duke math. J., 73, 2, 269-290, (1994), MR 96c:33020 · Zbl 0804.33013
[2] Borel, A.; Grivel, P.-P.; Kaup, B.; Haefliger, A.; Malgrange, B.; Ehlers, F., Algebraic D-modules, Perspect. math., vol. 2, (1987), Academic Press Inc. Boston, MA, MR MR882000 (89g:32014)
[3] Gel’fand, I.M.; Graev, M.I.; Zelevinskiĭ, A.V., Holonomic systems of equations and series of hypergeometric type, Dokl. akad. nauk SSSR, 295, 1, 14-19, (1987), MR MR902936 (88j:58118)
[4] Gel’fand, I.M.; Kapranov, M.M.; Zelevinsky, A.V., Generalized Euler integrals and A-hypergeometric functions, Adv. math., 84, 2, 255-271, (1990), MR MR1080980 (92e:33015) · Zbl 0741.33011
[5] Gel’fand, I.M.; Zelevinskiĭ, A.V.; Kapranov, M.M., Hypergeometric functions and toric varieties, Funktsional. anal. i prilozhen., 23, 2, 12-26, (1989), MR 90m:22025 · Zbl 0721.33006
[6] Hochster, M., Rings of invariants of tori, cohen – macaulay rings generated by monomials, and polytopes, Ann. of math. (2), 96, 318-337, (1972), MR MR0304376 (46#3511) · Zbl 0233.14010
[7] Grayson, Daniel R.; Stillman, Michael E., Macaulay 2, a software system for research in algebraic geometry, available at
[8] Felicia Matusevich, Laura; Miller, Ezra; Walther, Uli, Homological methods for hypergeometric families, J. amer. math. soc., 18, 4, 919-941, (2005), (electronic), MR MR2163866 · Zbl 1095.13033
[9] Okuyama, Go, Local cohomology modules of A-hypergeometric systems of cohen – macaulay type, Tohoku math. J. (2), 58, 2, 259-275, (2006), MR MR2248433 · Zbl 1102.13019
[10] Pham, Frédéric, Singularités des systèmes différentiels de gauss – manin, Progr. math., vol. 2, (1979), Birkhäuser Boston Boston, MA, With contributions by Lo Kam Chan, Philippe Maisonobe and Jean-Étienne Rombald, MR MR553954 (81h:32015) · Zbl 0524.32015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.