Constructing derived moduli stacks

Pridham, Jonathan P¹

¹ Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK

Geometry & topology, Tome 17 (2013) no. 3, pp. 1417-1495.

Résumé

We introduce frameworks for constructing global derived moduli stacks associated to a broad range of problems, bridging the gap between the concrete and abstract conceptions of derived moduli. Our three approaches are via differential graded Lie algebras, via cosimplicial groups, and via quasicomonoids, each more general than the last. Explicit examples of derived moduli problems addressed here are finite schemes, polarised projective schemes, torsors, coherent sheaves and finite group schemes.

Détail

DOI : 10.2140/gt.2013.17.1417

Classification : 14A20, 14D23, 14J10
Keywords: derived algebraic geometry, stacks, derived moduli, DGLAs

Bibliographie

[1] M Anel, Grothendieck topologies from unique factorisation systems (2009)

[2] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, 304, Springer (1972)

[3] A M Cegarra, J Remedios, The relationship between the diagonal and the bar constructions on a bisimplicial set, Topology Appl. 153 (2005) 21

[4] A M Cegarra, J Remedios, Diagonal fibrations are pointwise fibrations, J. Homotopy Relat. Struct. 2 (2007) 81

[5] I Ciocan-Fontanine, M Kapranov, Derived Quot schemes, Ann. Sci. École Norm. Sup. 34 (2001) 403

[6] I Ciocan-Fontanine, M M Kapranov, Derived Hilbert schemes, J. Amer. Math. Soc. 15 (2002) 787

[7] F R Cohen, T J Lada, J P May, The homology of iterated loop spaces, 533, Springer (1976)

[8] W G Dwyer, D M Kan, Homotopy theory and simplicial groupoids, Nederl. Akad. Wetensch. Indag. Math. 46 (1984) 379

[9] P G Goerss, J F Jardine, Simplicial homotopy theory, 174, Birkhäuser (1999)

[10] A Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique, IV : Les schémas de Hilbert, from: "Séminaire Bourbaki (1960/1961) (Exposé 221)", Benjamin (1960)

[11] A Grothendieck, Éléments de géométrie algébrique, III : Étude cohomologique des faisceaux cohérents, I, Inst. Hautes Études Sci. Publ. Math. 1961 (1961) 349

[12] M Haiman, B Sturmfels, Multigraded Hilbert schemes, J. Algebraic Geom. 13 (2004) 725

[13] R Hartshorne, Residues and duality, 20, Springer (1966)

[14] R Hartshorne, Algebraic geometry, 52, Springer (1977)

[15] L Illusie, Complexe cotangent et déformations, I, 239, Springer (1971)

[16] L Illusie, Cotangent complex and deformations of torsors and group schemes, from: "Toposes, algebraic geometry and logic" (editor F W Lawvere), Lecture Notes in Math. 274, Springer (1972) 159

[17] G Laumon, L Moret-Bailly, Champs algébriques, 39, Springer (2000)

[18] J Lurie, Derived algebraic geometry, PhD thesis, Massachusetts Institute of Technology (2004)

[19] D Mumford, Lectures on curves on an algebraic surface, 59, Princeton Univ. Press (1966)

[20] J P Pridham, Deformations of schemes and other bialgebraic structures, Trans. Amer. Math. Soc. 360 (2008) 1601

[21] J P Pridham, Pro-algebraic homotopy types, Proc. Lond. Math. Soc. 97 (2008) 273

[22] J P Pridham, Derived deformations of Artin stacks (2009)

[23] J P Pridham, The homotopy theory of strong homotopy algebras and bialgebras, Homology, Homotopy Appl. 12 (2010) 39

[24] J P Pridham, Derived deformations of schemes, Comm. Anal. Geom. 20 (2012) 529

[25] J P Pridham, Derived moduli of schemes and sheaves, J. K–Theory 10 (2012) 41

[26] J P Pridham, Representability of derived stacks, J. K–Theory 10 (2012) 413

[27] J P Pridham, Presenting higher stacks as simplicial schemes, Adv. Math. 238 (2013) 184

[28] D Quillen, Rational homotopy theory, Ann. of Math. 90 (1969) 205

[29] D Quillen, On the (co-) homology of commutative rings, from: "Applications of categorical algebra" (editor A Heller), Amer. Math. Soc. (1970) 65

[30] J P Serre, Faisceaux algébriques cohérents, Ann. of Math. 61 (1955) 197

[31] B Toën, Descente fidèlement plate pour les n–champs d’Artin, Compos. Math. 147 (2011) 1382

[32] B Toën, G Vezzosi, Homotopical algebraic geometry, II : Geometric stacks and applications, 902, Amer. Math. Soc. (2008)

[33] C A Weibel, An introduction to homological algebra, 38, Cambridge Univ. Press (1994)