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Sheaf cohomology and free resolutions over exterior algebras. (English) Zbl 1063.14021

In this paper, the authors give a very explicit version of the Bernstein-Gel’fand-Gel’fand correspondence between bounded complexes of coherent sheaves on projective space and free resolutions over the exterior algebra. More precisely, they study a pair R, L of adjoint functors between categories of complexes of graded modules over the symmetric algebra \(S=\text{Sym}(V^*)\) and the exterior algebra \(E=\bigwedge V\). This gives, for example, an equivalence between graded \(S\)-modules and complexes of free \(E\)-modules that are eventually linear (given by matrices whose entries are linear forms). Under this equivalence, the Castelnuovo-Mumford regularity of \(M\) is equal to the degree where the corresponding complex becomes linear.
The authors use this to study the Beilinson monad, which is a complex associated to a sheaf on projective space whose homology is the sheaf. The explicit nature of their results not only provides new information on the Beilinson monad, but also gives new and efficient methods for machine computation of the cohomology of sheaves on projective space. This last development has had a number of applications, some of which are described in the the paper. For example, A. Khetan [J. Symb. Comput. 36, No. 3–4, 425–442 (2003; Zbl 1068.14070)] has been using this to obtain new, exact formulas for sparse resultants. The reviewer recommends reading this paper, as its clear exposition is a nice introduction to this topic. Particularly appealing is the marriage of the abstract (derived categories of sheaves on projective space) to the concrete (computer algebra algorithms).

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
13D02 Syzygies, resolutions, complexes and commutative rings
14Q20 Effectivity, complexity and computational aspects of algebraic geometry
18E30 Derived categories, triangulated categories (MSC2010)

Citations:

Zbl 1068.14070

Software:

BGG; Macaulay2
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Full Text: DOI arXiv

References:

[1] Kaan Akin, David A. Buchsbaum, and Jerzy Weyman, Schur functors and Schur complexes, Adv. in Math. 44 (1982), no. 3, 207 – 278. · Zbl 0497.15020 · doi:10.1016/0001-8708(82)90039-1
[2] Vincenzo Ancona and Giorgio Ottaviani, An introduction to the derived categories and the theorem of Beilinson, Atti Accad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur. 67 (1989), 99 – 110 (1991). · Zbl 0747.14004
[3] Annetta Aramova, Luchezar L. Avramov, and Jürgen Herzog, Resolutions of monomial ideals and cohomology over exterior algebras, Trans. Amer. Math. Soc. 352 (2000), no. 2, 579 – 594. · Zbl 0930.13011
[4] W. Barth, Moduli of vector bundles on the projective plane, Invent. Math. 42 (1977), 63 – 91. · Zbl 0386.14005 · doi:10.1007/BF01389784
[5] I. N. Bernšteĭn, I. M. Gel\(^{\prime}\)fand, and S. I. Gel\(^{\prime}\)fand, Algebraic vector bundles on \?\(^{n}\) and problems of linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 66 – 67 (Russian). A. A. Beĭlinson, Coherent sheaves on \?\(^{n}\) and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68 – 69 (Russian).
[6] I. N. Bernšteĭn, I. M. Gel\(^{\prime}\)fand, and S. I. Gel\(^{\prime}\)fand, Algebraic vector bundles on \?\(^{n}\) and problems of linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 66 – 67 (Russian). A. A. Beĭlinson, Coherent sheaves on \?\(^{n}\) and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68 – 69 (Russian).
[7] David A. Buchsbaum and David Eisenbud, Generic free resolutions and a family of generically perfect ideals, Advances in Math. 18 (1975), no. 3, 245 – 301. · Zbl 0336.13007 · doi:10.1016/0001-8708(75)90046-8
[8] G.-M. Greuel and G. Trautmann , Singularities, representation of algebras, and vector bundles, Lecture Notes in Mathematics, vol. 1273, Springer-Verlag, Berlin, 1987. · Zbl 0619.00007
[9] R.-O. Buchweitz: Maximal Cohen-Macaulay modules and Tate-Cohomology over Gorenstein ring Preprint (1985).
[10] W. Decker and D. Eisenbud: Sheaf algorithms using the exterior algebra, in Computations in Algebraic Geometry with Macaulay2, ed. D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels. Springer-Verlag, New York, 2001.
[11] Wolfram Decker and Frank-Olaf Schreyer, On the uniqueness of the Horrocks-Mumford bundle, Math. Ann. 273 (1986), no. 3, 415 – 443. · Zbl 0598.14013 · doi:10.1007/BF01450731
[12] Wolfram Decker and Frank-Olaf Schreyer, Non-general type surfaces in \?\(^{4}\): some remarks on bounds and constructions, J. Symbolic Comput. 29 (2000), no. 4-5, 545 – 582. Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998). · Zbl 1012.14014 · doi:10.1006/jsco.1999.0323
[13] Wolfram Decker, Stable rank 2 vector bundles with Chern-classes \?\(_{1}\)=-1,\?\(_{2}\)=4, Math. Ann. 275 (1986), no. 3, 481 – 500. · Zbl 0584.55015 · doi:10.1007/BF01458618
[14] David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. · Zbl 0819.13001
[15] David Eisenbud and Shiro Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), no. 1, 89 – 133. · Zbl 0531.13015 · doi:10.1016/0021-8693(84)90092-9
[16] David Eisenbud and Sorin Popescu, Gale duality and free resolutions of ideals of points, Invent. Math. 136 (1999), no. 2, 419 – 449. · Zbl 0943.13011 · doi:10.1007/s002220050315
[17] D. Eisenbud, S. Popescu, F.-O. Schreyer and C. Walter: Exterior algebra methods for the Minimal Resolution Conjecture, Duke Math. J. 112 (2002), 379-395. · Zbl 1035.13008
[18] D. Eisenbud, S. Popescu, and S. Yuzvinsky: Hyperplane arrangements and resolutions of monomial ideals over an exterior algebra. Trans. Amer. Math. Soc., this issue. · Zbl 1034.52019
[19] D. Eisenbud and F.-O. Schreyer: Sheaf cohomology and free resolutions over the exterior algebras, http://arXiv.org/abs/math.AG/0005055 Preprint (2000).
[20] D. Eisenbud, F.-O. Schreyer, and Jerzy Weyman: Resultants and Chow forms via exterior syzygies. J. Amer. Math. Soc. 16 (2003) 537-579. · Zbl 1069.14019
[21] Geir Ellingsrud and Christian Peskine, Sur les surfaces lisses de \?\(_{4}\), Invent. Math. 95 (1989), no. 1, 1 – 11 (French). · Zbl 0676.14009 · doi:10.1007/BF01394141
[22] D. Eisenbud and J. Weyman: Fitting’s Lemma for \({\mathbb{Z}}/{2}\)-graded modules. Trans. Amer. Math. Soc., this issue. · Zbl 1068.13001
[23] G. Fløystad: Koszul duality and equivalences of categories. http://arXiv.org/ abs/math.RA/0012264 Preprint (2000a). · Zbl 1101.16020
[24] G. Fløystad: Describing coherent sheaves on projective spaces via Koszul duality. http://arXiv.org/abs/math.RA/0012263 Preprint (2000b). · Zbl 0977.14007
[25] Gunnar Fløystad, Monads on projective spaces, Comm. Algebra 28 (2000), no. 12, 5503 – 5516. Special issue in honor of Robin Hartshorne. · Zbl 0977.14007 · doi:10.1080/00927870008827171
[26] Векторные расслоения на комплексных проективных пространствах, Математика: Новое в Зарубежной Науке [Матхематицс: Рецент Публицатионс ин Фореигн Сциенце], вол. 36, ”Мир”, Мосцощ, 1984 (Руссиан). Транслатед фром тхе Енглиш бы В. Я. Лин; Щитх ан аппендиш бы С. И. Гел\(^{\приме}\)фанд; Транслатион едитед анд щитх а префаце бы Ю. И. Манин.
[27] Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra, Springer-Verlag, Berlin, 1996. Translated from the 1988 Russian original.
[28] D. Grayson and M. Stillman: Macaulay2. http://www.math.uiuc.edu/Macaulay2/.
[29] Mark L. Green, The Eisenbud-Koh-Stillman conjecture on linear syzygies, Invent. Math. 136 (1999), no. 2, 411 – 418. · Zbl 0979.13013 · doi:10.1007/s002220050314
[30] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). · Zbl 0135.39701
[31] Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988. · Zbl 0635.16017
[32] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001
[33] J. Herzog and T. Römer: Resolutions of modules over the exterior algebra, working notes, 1999.
[34] G. Horrocks, Projective modules over an extension of a local ring, Proc. London Math. Soc. (3) 14 (1964), 714 – 718. · Zbl 0132.28103 · doi:10.1112/plms/s3-14.4.714
[35] G. Horrocks and D. Mumford, A rank 2 vector bundle on \?\(^{4}\) with 15,000 symmetries, Topology 12 (1973), 63 – 81. · Zbl 0255.14017 · doi:10.1016/0040-9383(73)90022-0
[36] M. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), no. 3, 479 – 508. · Zbl 0651.18008 · doi:10.1007/BF01393744
[37] M. M. Kapranov, On the derived category and \?-functor of coherent sheaves on intersections of quadrics, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 1, 186 – 199 (Russian); English transl., Math. USSR-Izv. 32 (1989), no. 1, 191 – 204.
[38] Mireille Martin-Deschamps and Daniel Perrin, Sur la classification des courbes gauches, Astérisque 184-185 (1990), 208 (French). · Zbl 0717.14017
[39] Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980. · Zbl 0438.32016
[40] D. O. Orlov, Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 4, 852 – 862 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 41 (1993), no. 1, 133 – 141. · Zbl 0798.14007 · doi:10.1070/IM1993v041n01ABEH002182
[41] Stewart B. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39 – 60. · Zbl 0261.18016
[42] Prabhakar Rao, Liaison equivalence classes, Math. Ann. 258 (1981/82), no. 2, 169 – 173. · Zbl 0493.14009 · doi:10.1007/BF01450532
[43] Frank-Olaf Schreyer, Small fields in constructive algebraic geometry, Moduli of vector bundles (Sanda, 1994; Kyoto, 1994) Lecture Notes in Pure and Appl. Math., vol. 179, Dekker, New York, 1996, pp. 221 – 228. · Zbl 0876.14040
[44] Richard G. Swan, \?-theory of quadric hypersurfaces, Ann. of Math. (2) 122 (1985), no. 1, 113 – 153. · Zbl 0601.14009 · doi:10.2307/1971371
[45] C. Walter: Algebraic cohomology methods for the normal bundle of algebraic space curves. Preprint (1990).
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