Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-13T05:48:58.741Z Has data issue: false hasContentIssue false
  • English
  • Français

Grothendieck–Neeman duality and the Wirthmüller isomorphism

Part of: (Co)homology theory Abelian categories Applied homological algebra and category theory

Published online by Cambridge University Press:  23 May 2016

Paul Balmer ,
Ivo Dell’Ambrogio  and
Beren Sanders
Paul Balmer
Affiliation:
Mathematics Department, UCLA, Los Angeles, CA 90095-1555, USA email balmer@math.ucla.edu
Ivo Dell’Ambrogio
Affiliation:
Laboratoire de Mathématiques Paul Painlevé, Université de Lille 1, Cité Scientifique – Bât. M2, 59665 Villeneuve-d’Ascq Cedex, France email ivo.dellambrogio@math.univ-lille1.fr
Beren Sanders
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark email sanders@math.ku.dk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We clarify the relationship between Grothendieck duality à la Neeman and the Wirthmüller isomorphism à la Fausk–Hu–May. We exhibit an interesting pattern of symmetry in the existence of adjoint functors between compactly generated tensor-triangulated categories, which leads to a surprising trichotomy: there exist either exactly three adjoints, exactly five, or infinitely many. We highlight the importance of so-called relative dualizing objects and explain how they give rise to dualities on canonical subcategories. This yields a duality theory rich enough to capture the main features of Grothendieck duality in algebraic geometry, of generalized Pontryagin–Matlis duality à la Dwyer–Greenless–Iyengar in the theory of ring spectra, and of Brown–Comenetz duality à la Neeman in stable homotopy theory.

Keywords

Grothendieck dualityur-Wirthmüllerdualizing objectadjointscompactly generated triangulated categorySerre functor

MSC classification

Primary: 18E30: Derived categories, triangulated categories
Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions 55U35: Abstract and axiomatic homotopy theory
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 8 , August 2016 , pp. 1740 - 1776
Copyright
© The Authors 2016 

References

Avramov, L. L., Iyengar, S. B. and Lipman, J., Reflexivity and rigidity for complexes. I. Commutative rings , Algebra Number Theory 4 (2010), 47–86.Google Scholar
Avramov, L., Iyengar, S. B. and Lipman, J., Reflexivity and rigidity for complexes, II: Schemes , Algebra Number Theory 5 (2011), 379–429.Google Scholar
Balmer, P., Separability and triangulated categories , Adv. Math. 226 (2011), 4352–4372.CrossRefGoogle Scholar
Beĭlinson, A. A., The derived category of coherent sheaves on P n , Selecta Math. Soviet. 3 (1983/84), 233–237; selected translations.Google Scholar
Benson, D. and Greenlees, J., Stratifying the derived category of cochains on BG for G a compact Lie group , J. Pure Appl. Algebra 218 (2014), 642–650.Google Scholar
Bökstedt, M. and Neeman, A., Homotopy limits in triangulated categories , Compositio Math. 86 (1993), 209–234.Google Scholar
Bondal, A. I. and Kapranov, M. M., Representable functors, Serre functors, and reconstructions , Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 1183–1205, 1337.Google Scholar
Bondal, A. and Orlov, D., Reconstruction of a variety from the derived category and groups of autoequivalences , Compositio Math. 125 (2001), 327–344.Google Scholar
Bondal, A. and van den Bergh, M., Generators and representability of functors in commutative and noncommutative geometry , Mosc. Math. J. 3 (2003), 1–36, 258.Google Scholar
Bourbaki, N., Éléments de mathématique. Algèbre commutative. Chapitre 10 (Springer, Berlin, 2007); reprint of the 1998 original.Google Scholar
Boyarchenko, M. and Drinfeld, V., A duality formalism in the spirit of Grothendieck and Verdier , Quantum Topol. 4 (2013), 447–489.CrossRefGoogle Scholar
Brown, E. H. Jr and Comenetz, M., Pontrjagin duality for generalized homology and cohomology theories , Amer. J. Math. 98 (1976), 1–27.CrossRefGoogle Scholar
Calmès, B. and Hornbostel, J., Tensor-triangulated categories and dualities , Theory Appl. Categ. 22 (2009), 136–200.Google Scholar
Dwyer, W. G., Greenlees, J. P. C. and Iyengar, S., Duality in algebra and topology , Adv. Math. 200 (2006), 357–402.Google Scholar
Dwyer, W. G., Greenlees, J. P. C. and Iyengar, S. B., Gross–Hopkins duality and the Gorenstein condition , J. K-Theory 8 (2011), 107–133.Google Scholar
Dwyer, W. G. and Wilkerson, C. W., The elementary geometric structure of compact Lie groups , Bull. Lond. Math. Soc. 30 (1998), 337–364.Google Scholar
Grothendieck, A., Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. II , Publ. Math. Inst. Hautes Études Sci. 17 (1963), 5–91.Google Scholar
Elmendorf, A. D., Kriz, I., Mandell, M. A. and May, J. P., Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47 (American Mathematical Society, Providence, RI, 1997); with an appendix by M. Cole.Google Scholar
Fausk, H., Hu, P. and May, J. P., Isomorphisms between left and right adjoints , Theory Appl. Categ. 11 (2003), 107–131.Google Scholar
Greenlees, J. P. C., Algebraic models of change of groups in rational stable equivariant homotopy theory, Preprint (2015), arXiv:1501.06167 [math.AT].Google Scholar
Greenlees, J. P. C. and Shipley, B., An algebraic model for free rational G-spectra , Bull. Lond. Math. Soc. 46 (2014), 133–142.CrossRefGoogle Scholar
Hartshorne, R., Residues and duality: Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64, Lecture Notes in Mathematics, vol. 20 (Springer, Berlin, 1966); with an appendix by P. Deligne.CrossRefGoogle Scholar
Heard, D. and Stojanoska, V., K-theory, reality, and duality , J. K-Theory 14 (2014), 526–555.Google Scholar
Hovey, M., Palmieri, J. H. and Strickland, N. P., Axiomatic stable homotopy theory , Mem. Amer. Math. Soc. 128 (1997).Google Scholar
Hu, P., Base change functors in the A1 -stable homotopy category , Homology, Homotopy Appl. 3 (2001), 417–451.Google Scholar
Jantzen, J. C., Representations of algebraic groups, Pure and Applied Mathematics, vol. 131 (Academic Press, Boston, 1987).Google Scholar
Kelly, G. M., Basic concepts of enriched category theory , Repr. Theory Appl. Categ. (2005), vi+137; reprint of the 1982 original [Cambridge University Press].Google Scholar
Krause, H., A Brown representability theorem via coherent functors , Topology 41 (2002), 853–861.Google Scholar
Lewis, L. G. Jr., May, J. P., Steinberger, M. and McClure, J. E., Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213 (Springer, Berlin, 1986).Google Scholar
Lipman, J., Notes on derived functors and Grothendieck duality , in Foundations of Grothendieck duality for diagrams of schemes, Lecture Notes in Mathematics, vol. 1960 (Springer, Berlin, 2009), 1–259.Google Scholar
Lipman, J. and Neeman, A., Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor , Illinois J. Math. 51 (2007), 209–236.CrossRefGoogle Scholar
Mac Lane, S., Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5 (Springer, New York, 1998).Google Scholar
Matsumura, H., Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8 (Cambridge University Press, Cambridge, 1986).Google Scholar
May, J. P., The Wirthmüller isomorphism revisited , Theory Appl. Categ. 11 (2003), 132–142.Google Scholar
McCleary, J., A user’s guide to spectral sequences, Cambridge Studies in Advanced Mathematics, vol. 58, second edition (Cambridge University Press, Cambridge, 2001).Google Scholar
McGibbon, C. A. and Neisendorfer, J. A., On the homotopy groups of a finite-dimensional space , Comment. Math. Helv. 59 (1984), 253–257.Google Scholar
Morita, K., Adjoint pairs of functors and Frobenius extensions , Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 9 (1965), 40–71.Google Scholar
Neeman, A., Stable homotopy as a triangulated functor , Invent. Math. 109 (1992), 17–40.Google Scholar
Neeman, A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability , J. Amer. Math. Soc. 9 (1996), 205–236.CrossRefGoogle Scholar
Neeman, A., Triangulated categories, Annals of Mathematics Studies, vol. 148 (Princeton University Press, Princeton, NJ, 2001).CrossRefGoogle Scholar
Neeman, A., Derived categories and Grothendieck duality , in Triangulated categories, London Mathematical Socieyt Lecture Note Series, vol. 375 (Cambridge University Press, Cambridge, 2010), 290–350.Google Scholar
Porta, M., Shaul, L. and Yekutieli, A., On the homology of completion and torsion , Algebr. Represent. Theory 17 (2014), 31–67.Google Scholar
Riou, J., Dualité de Spanier–Whitehead en géométrie algébrique , C. R. Math. Acad. Sci. Paris 340 (2005), 431–436.Google Scholar
Rouquier, R., Derived categories and algebraic geometry , in Triangulated categories, London Mathematical Society, Lecture Note Series, vol. 375 (Cambridge University Press, Cambridge, 2010), 351–370.Google Scholar
Schwede, S. and Shipley, B., Stable model categories are categories of modules , Topology 42 (2003), 103–153.Google Scholar
Shipley, B., Hℤ-algebra spectra are differential graded algebras , Amer. J. Math. 129 (2007), 351–379.Google Scholar
Wirthmüller, K., Equivariant homology and duality , Manuscripta Math. 11 (1974), 373–390.Google Scholar
Yekutieli, A., Rigid dualizing complexes via differential graded algebras (survey) , in Triangulated categories, London Mathematical Society Lecture Note Series, vol. 375 (Cambridge University Press, Cambridge, 2010), 452–463.CrossRefGoogle Scholar