Journal of Algebraic Geometry

Author(s): Max Lieblich
Journal: J. Algebraic Geom. 15 (2006), 175-206.
Posted: September 7, 2005
Retrieve article in: PDF DVI PostScript

Abstract: Given a proper morphism $X\to S$ , we show that a large class of objects in the derived category of

naturally form an Artin stack locally of finite presentation over

. This class includes

-flat coherent sheaves and, more generally, contains the collection of all

-flat objects which can appear in the heart of a reasonable sheaf of

-structures on

. In this sense, this is the Mother of all Moduli Spaces (of sheaves). The proof proceeds by studying the finite presentation properties, deformation theory, and Grothendieck existence theorem for objects in the derived category, and then applying Artin's representability theorem.

1.: Théorie des intersections et théorème de Riemann-Roch.
Springer-Verlag, Berlin, 1971.
Séminaire de Géométrie Algébrique du Bois-Marie 1966-1967 (SGA 6), Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre, Lecture Notes in Mathematics, Vol. 225. MR 0354655 (50:7133)
2.: Dan Abramovich, Alessio Corti, and Angelo Vistoli.
Twisted bundles and admissible covers.
Comm. Algebra, 31(8):3547-3618, 2003.
Special issue in honor of Steven L. Kleiman. MR 2007376 (2005b:14049)
3.: Dan Abramovich and Alexander Polishchuk.
Sheaves of t-structures and valuative criteria for stable complexes.
math. AG/0309435
4.: Dan Abramovich and Angelo Vistoli.
Compactifying the space of stable maps.
J. Amer. Math. Soc., 15(1):27-75 (electronic), 2002. MR 1862797 (2002i:14030)
5.: M. Artin.
Versal deformations and algebraic stacks.
Invent. Math., 27:165-189, 1974. MR 0399094 (53:2945)
6.: A. A. Be ${\u{\i}}\kern.15em$ linson, J. Bernstein, and P. Deligne.
Faisceaux pervers.
In Analysis and topology on singular spaces, I (Luminy, 1981), volume 100 of Astérisque, pages 5-171. Soc. Math. France, Paris, 1982. MR 0751966 (86g:32015)
7.: Marcel Bökstedt and Amnon Neeman.
Homotopy limits in triangulated categories.
Compositio Math., 86(2):209-234, 1993. MR 1214458 (94f:18008)
8.: Alexei Bondal and Dmitri Orlov.
Reconstruction of a variety from the derived category and groups of autoequivalences.
Compositio Math., 125(3):327-344, 2001. MR 1818984 (2001m:18014)
9.: Tom Bridgeland.
Flops and derived categories.
Invent. Math., 147(3):613-632, 2002. MR 1893007 (2003h:14027)
10.: Jiun-Cheng Chen.
Flops and equivalences of derived categories for threefolds with only terminal Gorenstein singularities.
J. Differential Geom., 61(2):227-261, 2002. MR 1972146 (2004d:14012)
11.: A. Grothendieck.
Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I.
Inst. Hautes Études Sci. Publ. Math., (11):167, 1961.
12.: A. Grothendieck.
Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III.
Inst. Hautes Études Sci. Publ. Math., (28):255, 1966. MR 0217086 (36:178)
13.: Robin Hartshorne.
Algebraic geometry.
Springer-Verlag, New York, 1977.
Graduate Texts in Mathematics, No. 52. MR 0463157 (57:3116)
14.: Michi-aki Inaba.
Toward a definition of moduli of complexes of coherent sheaves on a projective scheme.
J. Math. Kyoto Univ., 42(2):317-329, 2002. MR 1966840 (2004e:14022)
15.: Gérard Laumon and Laurent Moret-Bailly.
Champs algébriques, volume 39 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics].
Springer-Verlag, Berlin, 2000. MR 1771927 (2001f:14006)
16.: Hideyuki Matsumura.
Commutative ring theory, volume 8 of Cambridge Studies in Advanced Mathematics.
Cambridge University Press, Cambridge, second edition, 1989.
Translated from the Japanese by M. Reid. MR 1011461 (90i:13001)
17.: David Mumford.
Abelian varieties.
Tata Institute of Fundamental Research Studies in Mathematics, No. 5. Published for the Tata Institute of Fundamental Research, Bombay, 1970. MR 0282985 (44:219)
18.: Michael Schlessinger.
Functors of Artin rings.
Trans. Amer. Math. Soc., 130:208-222, 1968. MR 0217093 (36:184)
19.: N. Spaltenstein.
Resolutions of unbounded complexes.
Compositio Math., 65(2):121-154, 1988. MR 0932640 (89m:18013)
20.: Bertrand Toën and Michel Vaquié.
Moduli of objects in dg-categories. math. AG/0503269

Max Lieblich
Affiliation: Department of Mathematics, Brown University, Box 1917, Providence, Rhode Island 02912
Address at time of publication: Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544-1000
Email: lieblich@math.princeton.edu

PII: S 1056-3911(05)00418-2
Received by editor(s): February 21, 2005
Received by editor(s) in revised form: June 5, 2005
Posted: September 7, 2005
Additional Notes: Work on this paper was supported by a National Science Foundation Postdoctoral Fellowship.

Previous issue \| Table of contents \| Next issue Recently posted articles \| Most recent issue \| All issues