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Transactions of the American Mathematical Society

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Sheaf cohomology and free resolutions over exterior algebras


Authors: David Eisenbud, Gunnar Fløystad and Frank-Olaf Schreyer
Journal: Trans. Amer. Math. Soc. 355 (2003), 4397-4426
MSC (2000): Primary 14F05, 14Q20, 16E05
DOI: https://doi.org/10.1090/S0002-9947-03-03291-4
Published electronically: July 10, 2003
MathSciNet review: 1990756
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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.


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  • [ABW] K. Akin, D. A. Buchsbaum, and J. Weyman: Schur functors and Schur complexes. Adv. in Math. 44 (1982) 207-278. MR 84c:20021
  • [AO] V. Ancona and G. Ottaviani: An introduction to the derived categories and the theorem of Beilinson. Atti Accademia Peloritana dei Pericolanti, Classe I de Scienze Fis. Mat. et Nat. LXVII (1989) 99-110. MR 92g:14013
  • [AAH] A. Aramova, L. A. Avramov, J. Herzog: Resolutions of monomial ideals and cohomology over exterior algebras. Trans. Amer. Math. Soc. 352 (2000) 579-594. MR 2000c:13021
  • [Bar] W. Barth: Moduli of vector bundles on the projective plane. Invent. Math. 42 (1977) 63-91. MR 57:324
  • [Bei] A. Beilinson: Coherent sheaves on $\mathbf{P} ^{n}$ and problems of linear algebra. Funct. Anal. and its Appl. 12 (1978) 214-216. (Trans. from Funkz. Anal. i. Ego Priloz. 12 (1978) 68-69.) MR 80c:14010b
  • [BGG] I. N. Bernstein, I. M. Gel'fand and S. I. Gel'fand: Algebraic bundles on $\mathbf{P}^{n}$ and problems of linear algebra. Funct. Anal. and its Appl. 12 (1978) 212-214. (Trans. from Funkz. Anal. i. Ego Priloz 12 (1978) 66-67.) MR 80c:14010a
  • [BE] D. A. Buchsbaum and D. Eisenbud: Generic free resolutions and a family of generically perfect ideals. Adv. Math. 18 (1975) 245-301. MR 53:391
  • [Buc1] R.-O. Buchweitz: Appendix to Cohen-Macaulay modules on quadrics, by R.-O. Buchweitz, D. Eisenbud, and J. Herzog. In Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), Springer-Verlag Lecture Notes in Math. 1273 (1987) 96-116. MR 88h:14001
  • [Buc2] R.-O. Buchweitz: Maximal Cohen-Macaulay modules and Tate-Cohomology over Gorenstein ring Preprint (1985).
  • [DE] 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.
  • [DS1] W. Decker and F.-O. Schreyer: On the uniqueness of the Horrocks-Mumford bundle. Math. Ann. 273 (1986) 415-443. MR 87d:14009
  • [DS2] W. Decker and F.-O. Schreyer: Non-general type surfaces in $\mathbf{P}^{4}$: Some remarks on bounds and constructions. J. Symbolic Comp. 29 (2000), 545-582. MR 2002a:14064
  • [Dec] W. Decker: Stable rank 2 bundles with Chern-classes $c_{1}=-1, c_{2}=4$. Math. Ann. 275 (1986) 481-500. MR 87j:14017
  • [Eis] D. Eisenbud: Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag, New York, 1995. MR 97a:13001
  • [EG] D. Eisenbud and S. Goto: Linear free resolutions and minimal multiplicity. J. Algebra 88 (1984) 89-133. MR 85f:13023
  • [EPo] D. Eisenbud and S. Popescu: Gale Duality and Free Resolutions of Ideals of Points. Invent. Math. 136 (1999) 419-449. MR 2000i:13014
  • [EPSW] 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.
  • [EPY] D. Eisenbud, S. Popescu, and S. Yuzvinsky: Hyperplane arrangements and resolutions of monomial ideals over an exterior algebra. Trans. Amer. Math. Soc., this issue.
  • [ES1] D. Eisenbud and F.-O. Schreyer: Sheaf cohomology and free resolutions over the exterior algebras, http://arXiv.org/abs/math.AG/0005055 Preprint (2000).
  • [ES2] D. Eisenbud, F.-O. Schreyer, and Jerzy Weyman: Resultants and Chow forms via exterior syzygies. J. Amer. Math. Soc. 16 (2003) 537-579.
  • [EPe] G. Ellingsrud and C. Peskine: Sur le surfaces lisse de $\mathbf{P}^{4}$. Invent. Math. 95 (1989) 1-12. MR 89j:14023
  • [EW] D. Eisenbud and J. Weyman: Fitting's Lemma for ${\mathbb{Z}}/{2}$-graded modules. Trans. Amer. Math. Soc., this issue.
  • [Flo1] G. Fløystad: Koszul duality and equivalences of categories. http://arXiv.org/ abs/math.RA/0012264 Preprint (2000a).
  • [Flo2] G. Fløystad: Describing coherent sheaves on projective spaces via Koszul duality. http://arXiv.org/abs/math.RA/0012263 Preprint (2000b).
  • [Flo3] G. Fløystad: Monads on projective spaces. Communications in Algebra. 28 (2000c) 5503-5516. MR 2002e:14070
  • [Gel] S. I. Gel'fand: Sheaves on $\mathbf{P}^{n}$ and problems in linear algebra. Appendix to the Russian edition of [OSS], Mir, Moscow, 1984. MR 86i:14005
  • [GM] S. I. Gel'fand and Yu. I. Manin: Methods of Homological Algebra. Springer-Verlag, New York, 1996. MR 97j:18001
  • [GS] D. Grayson and M. Stillman: Macaulay2. http://www.math.uiuc.edu/Macaulay2/.
  • [Gre] M. Green: The Eisenbud-Koh-Stillman Conjecture on Linear Syzygies. Invent. Math. 136 (1999) 411-418. MR 2000j:13024
  • [GD] A. Grothendieck and J. Dieudonné: Éléments de la Géometrie Algébrique IV: Étude locale des schémas et de morphismes de schémas (deuxième partie). Inst. Hautes Etudes Sci. Publ. Math. 24 (1965). MR 33:7330
  • [Hap] D. Happel: Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras. London Math. Soc. Lecture Notes Ser. 119, 1988. MR 89e:16035
  • [Har] R. Hartshorne: Algebraic geometry , Springer Verlag, 1977. MR 57:3116
  • [HR] J. Herzog and T. Römer: Resolutions of modules over the exterior algebra, working notes, 1999.
  • [Hor] G. Horrocks: Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc. (3) 14 (1964), 714-718. MR 30:121
  • [HM] G. Horrocks and D. Mumford: A rank $2$ vector bundle on $\mathbf{P}^{4}$ with 15,000 symmetries. Topology 12 (1973) 63-81. MR 52:3164
  • [Kap1] M. M. Kapranov: On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math. 92 (1988) 479-508. MR 89g:18018
  • [Kap2] 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) 186-199; English transl., Math. USSR Izv. 32 (1989), 191-204. MR 89f:14014
  • [MP] M. Martin-Deschamps and D. Perrin: Sur la classification des courbes gauches. Astérisque 184-185 (1990). MR 91h:14039
  • [OSS] C. Okonek, M. Schneider, and H. Spindler: Vector Bundles on Complex Projective Spaces. Birkhäuser, Boston 1980. MR 81b:14001
  • [Orl] D. O. Orlov: Projective bundles, monoidal transformations, and derived categories of coherent sheaves. Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992) 852-862; translation in Russian Acad. Sci. Izv. Math. 41 (1993) 133-141. MR 94e:14024
  • [Pri] S. B. Priddy: Koszul resolutions. Trans. Amer. Math. Soc. 152 (1970) 39-60. MR 42:346
  • [Rao] P. Rao: Liaison equivalence classes, Math. Ann. 258 (1981) 169-173. MR 83j:14045
  • [Sch] F.-O. Schreyer: Small fields in constructive algebraic geometry. In S. Maruyama ed.: Moduli of Vector Bundles, Sanda 1994. New York, Dekker 1996, 221-228. MR 97h:14050
  • [Swa] R. G. Swan: $K$-theory of quadric hypersurfaces. Ann. of Math. 122 (1985) 113-153. MR 87g:14006
  • [Wal] C. Walter: Algebraic cohomology methods for the normal bundle of algebraic space curves. Preprint (1990).

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Additional Information

David Eisenbud
Affiliation: Department of Mathematics, University of California Berkeley, Berkeley, California 94720
Email: eisenbud@math.berkeley.edu

Gunnar Fløystad
Affiliation: Mathematisk Institutt, Johs. Brunsgt. 12, N-5008 Bergen, Norway
Email: gunnar@mi.uib.no

Frank-Olaf Schreyer
Affiliation: FB Mathematik, Universität Bayreuth D-95440 Bayreuth, Germany
Email: schreyer@btm8x5.mat.uni-bayreuth.de

DOI: https://doi.org/10.1090/S0002-9947-03-03291-4
Received by editor(s): December 1, 2001
Published electronically: July 10, 2003
Additional Notes: The first and third authors are grateful to the NSF for partial support during the preparation of this paper. The third author wishes to thank MSRI for its hospitality.
Article copyright: © Copyright 2003 American Mathematical Society

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