Serre duality for non-commutative ℙ¹-bundles

Nyman, Adam

doi:10.1090/S0002-9947-04-03523-8

Serre duality for non-commutative ${\mathbb {P}}^{1}$-bundles
HTML articles powered by AMS MathViewer

by Adam Nyman PDF

Trans. Amer. Math. Soc. 357 (2005), 1349-1416 Request permission

Abstract:

Let $X$ be a smooth scheme of finite type over a field $K$, let $\mathcal {E}$ be a locally free $\mathcal {O}_{X}$-bimodule of rank $n$, and let $\mathcal {A}$ be the non-commutative symmetric algebra generated by $\mathcal {E}$. We construct an internal $\operatorname {Hom}$ functor, ${\underline {{\mathcal {H}}\textit {om}}_{\mathsf {Gr} \mathcal {A}}} (-,-)$, on the category of graded right $\mathcal {A}$-modules. When $\mathcal {E}$ has rank 2, we prove that $\mathcal {A}$ is Gorenstein by computing the right derived functors of ${\underline {{\mathcal {H}}\textit {om}}_{\mathsf {Gr} \mathcal {A}}} (\mathcal {O}_{X},-)$. When $X$ is a smooth projective variety, we prove a version of Serre Duality for ${\mathsf {Proj}} \mathcal {A}$ using the right derived functors of $\underset {n \to \infty }{\lim } \underline {\mathcal {H}\textit {om}}_{\mathsf {Gr} \mathcal {A}} (\mathcal {A}/\mathcal {A}_{\geq n}, -)$.

References

M. Artin, J. Tate, and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 33–85. MR 1086882
Marcel Bökstedt and Amnon Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993), no. 2, 209–234. MR 1214458
Nicolas Bourbaki, Elements of mathematics. Commutative algebra, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1972. Translated from the French. MR 0360549
Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 232821
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Robin Hartshorne, Cohomological dimension of algebraic varieties, Ann. of Math. (2) 88 (1968), 403–450. MR 232780, DOI 10.2307/1970720
Peter Jørgensen, Intersection theory on non-commutative surfaces, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5817–5854. MR 1695026, DOI 10.1090/S0002-9947-00-02565-4
P. Jörgensen, Non-commutative projective geometry, unpub. notes, 1996.
Peter Jørgensen, Serre-duality for $\textrm {Tails}(A)$, Proc. Amer. Math. Soc. 125 (1997), no. 3, 709–716. MR 1363171, DOI 10.1090/S0002-9939-97-03670-8
Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
I. Mori, Riemann-Roch like theorem for triangulated categories, J. Pure Appl. Agebra, to appear.
I. Mori and S.P. Smith, The Grothendieck group of a quantum projective space bundle, submitted.
Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205–236. MR 1308405, DOI 10.1090/S0894-0347-96-00174-9
A. Nyman, Points on quantum projectivizations, Mem. Amer. Math. Soc., 167 (2004).
A. Nyman, Serre finiteness and Serre vanishing for non-commutative ${\mathbb {P}}^{1}$-bundles, J. Algebra, to appear.
N. Popescu, Abelian categories with applications to rings and modules, London Mathematical Society Monographs, No. 3, Academic Press, London-New York, 1973. MR 0340375
S.P. Smith, Non-commutative algebraic geometry, unpub. notes, 1999.
Michel Van den Bergh, A translation principle for the four-dimensional Sklyanin algebras, J. Algebra 184 (1996), no. 2, 435–490. MR 1409223, DOI 10.1006/jabr.1996.0269
Michel Van den Bergh, Blowing up of non-commutative smooth surfaces, Mem. Amer. Math. Soc. 154 (2001), no. 734, x+140. MR 1846352, DOI 10.1090/memo/0734
M. Van den Bergh, Non-commutative ${\mathbb {P}}^{1}$-bundles over commutative schemes, to appear.
M. Van den Bergh, Non-commutative quadrics, in preparation.
Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136