Sheaf cohomology and free resolutions over exterior algebras

Eisenbud, David; Fløystad, Gunnar; Schreyer, Frank-Olaf

doi:10.1090/S0002-9947-03-03291-4

Sheaf cohomology and free resolutions over exterior algebras
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by David Eisenbud, Gunnar Fløystad and Frank-Olaf Schreyer PDF

Trans. Amer. Math. Soc. 355 (2003), 4397-4426 Request permission

Abstract:

We derive an explicit version of the Bernstein-Gel’fand-Gel’fand (BGG) correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free resolutions over its “Koszul dual” exterior algebra. Among the facts about the BGG correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the “linear part” of a resolution over the exterior algebra. We explore the structure of free resolutions over an exterior algebra. For example, we show that such resolutions are eventually dominated by their “linear parts" in the sense that erasing all terms of degree $>1$ in the complex yields a new complex which is eventually exact. As applications we give a construction of the Beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms. The explicitness of our version allows us to prove two conjectures about the morphisms in the monad, and we get an efficient method for machine computation of the cohomology of sheaves. We also construct all the monads for a sheaf that can be built from sums of line bundles, and show that they are often characterized by numerical data.

References

Kaan Akin, David A. Buchsbaum, and Jerzy Weyman, Schur functors and Schur complexes, Adv. in Math. 44 (1982), no. 3, 207–278. MR 658729, DOI 10.1016/0001-8708(82)90039-1
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). MR 1113077
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. MR 1603874, DOI 10.1090/S0002-9947-99-02298-9
W. Barth, Moduli of vector bundles on the projective plane, Invent. Math. 42 (1977), 63–91. MR 460330, DOI 10.1007/BF01389784
I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Algebraic vector bundles on $\textbf {P}^{n}$ and problems of linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 66–67 (Russian). MR 509387
I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Algebraic vector bundles on $\textbf {P}^{n}$ and problems of linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 66–67 (Russian). MR 509387
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. MR 396528, DOI 10.1016/0001-8708(75)90046-8
G.-M. Greuel and G. Trautmann (eds.), Singularities, representation of algebras, and vector bundles, Lecture Notes in Mathematics, vol. 1273, Springer-Verlag, Berlin, 1987. MR 915165, DOI 10.1007/BFb0078834
R.-O. Buchweitz: Maximal Cohen-Macaulay modules and Tate-Cohomology over Gorenstein ring Preprint (1985).
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.
Wolfram Decker and Frank-Olaf Schreyer, On the uniqueness of the Horrocks-Mumford bundle, Math. Ann. 273 (1986), no. 3, 415–443. MR 824431, DOI 10.1007/BF01450731
Wolfram Decker and Frank-Olaf Schreyer, Non-general type surfaces in $\textbf {P}^4$: 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). MR 1769655, DOI 10.1006/jsco.1999.0323
Wolfram Decker, Stable rank $2$ vector bundles with Chern-classes $c_1=-1,\;c_2=4$, Math. Ann. 275 (1986), no. 3, 481–500. MR 858291, DOI 10.1007/BF01458618
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
David Eisenbud and Shiro Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), no. 1, 89–133. MR 741934, DOI 10.1016/0021-8693(84)90092-9
David Eisenbud and Sorin Popescu, Gale duality and free resolutions of ideals of points, Invent. Math. 136 (1999), no. 2, 419–449. MR 1688433, DOI 10.1007/s002220050315
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.
D. Eisenbud, S. Popescu, and S. Yuzvinsky: Hyperplane arrangements and resolutions of monomial ideals over an exterior algebra. Trans. Amer. Math. Soc., this issue.
D. Eisenbud and F.-O. Schreyer: Sheaf cohomology and free resolutions over the exterior algebras, http://arXiv.org/abs/math.AG/0005055 Preprint (2000).
D. Eisenbud, F.-O. Schreyer, and Jerzy Weyman: Resultants and Chow forms via exterior syzygies. J. Amer. Math. Soc. 16 (2003) 537–579.
Geir Ellingsrud and Christian Peskine, Sur les surfaces lisses de $\textbf {P}_4$, Invent. Math. 95 (1989), no. 1, 1–11 (French). MR 969410, DOI 10.1007/BF01394141
D. Eisenbud and J. Weyman: Fitting’s Lemma for ${\mathbb {Z}}/{2}$-graded modules. Trans. Amer. Math. Soc., this issue.
G. Fløystad: Koszul duality and equivalences of categories. http://arXiv.org/abs/math.RA/0012264 Preprint (2000a).
G. Fløystad: Describing coherent sheaves on projective spaces via Koszul duality. http://arXiv.org/abs/math.RA/0012263 Preprint (2000b).
Gunnar Fløystad, Monads on projective spaces, Comm. Algebra 28 (2000), no. 12, 5503–5516. Special issue in honor of Robin Hartshorne. MR 1808585, DOI 10.1080/00927870008827171
K. Okonek, M. Shneĭder, and Kh. Shpindler, Vektornye rassloeniya na kompleksnykh proektivnykh prostranstvakh, Matematika: Novoe v Zarubezhnoĭ Nauke [Mathematics: Recent Publications in Foreign Science], vol. 36, “Mir”, Moscow, 1984 (Russian). Translated from the English by V. Ya. Lin; With an appendix by S. I. Gel′fand; Translation edited and with a preface by Yu. I. Manin. MR 778380
Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra, Springer-Verlag, Berlin, 1996. Translated from the 1988 Russian original. MR 1438306, DOI 10.1007/978-3-662-03220-6
D. Grayson and M. Stillman: Macaulay2. http://www.math.uiuc.edu/Macaulay2/.
Mark L. Green, The Eisenbud-Koh-Stillman conjecture on linear syzygies, Invent. Math. 136 (1999), no. 2, 411–418. MR 1688437, DOI 10.1007/s002220050314
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). MR 199181
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. MR 935124, DOI 10.1017/CBO9780511629228
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
J. Herzog and T. Römer: Resolutions of modules over the exterior algebra, working notes, 1999.
G. Horrocks, Projective modules over an extension of a local ring, Proc. London Math. Soc. (3) 14 (1964), 714–718. MR 169878, DOI 10.1112/plms/s3-14.4.714
G. Horrocks and D. Mumford, A rank $2$ vector bundle on $\textbf {P}^{4}$ with $15,000$ symmetries, Topology 12 (1973), 63–81. MR 382279, DOI 10.1016/0040-9383(73)90022-0
M. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), no. 3, 479–508. MR 939472, DOI 10.1007/BF01393744
M. M. Kapranov, On the derived category and $K$-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. MR 936529, DOI 10.1070/IM1989v032n01ABEH000752
Mireille Martin-Deschamps and Daniel Perrin, Sur la classification des courbes gauches, Astérisque 184-185 (1990), 208 (French). MR 1073438
Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980. MR 561910
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. MR 1208153, DOI 10.1070/IM1993v041n01ABEH002182
Stewart B. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39–60. MR 265437, DOI 10.1090/S0002-9947-1970-0265437-8
Prabhakar Rao, Liaison equivalence classes, Math. Ann. 258 (1981/82), no. 2, 169–173. MR 641822, DOI 10.1007/BF01450532
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. MR 1397991
Richard G. Swan, $K$-theory of quadric hypersurfaces, Ann. of Math. (2) 122 (1985), no. 1, 113–153. MR 799254, DOI 10.2307/1971371
C. Walter: Algebraic cohomology methods for the normal bundle of algebraic space curves. Preprint (1990).