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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
Macaulay2
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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
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