Serre duality for non-commutative \({\mathbb{P}}^{1}\)-bundles. (English) Zbl 1067.14001
Applying results of P. Jørgensen [Proc. Am. Math. Soc. 125, No. 3, 709–716 (1997; Zbl 0860.14002)] one has here a version of Serre duality for non-commutative \(\mathbb{P}^1\)-bundles over smooth projective varieties of dimension \(d\) over a field \(K\).
Let \(X\) be a smooth scheme of finite type over \(K\), \(E\) be a locally free \({\mathcal O}_X\)-bimodule of rank \(n\), \(A\) be the non-commutative symmetric algebra generated by \(E\) and \(\text{Gr\,}A\) be the category of graded right \(A\)-modules. Internal Hom and tensor functors are constructed on \(\text{Gr\,}A\).
When \(E\) has rank 2, computing derived functors of \(\operatorname{Hom}_{\text{Gr\,}A}({\mathcal O}_X,-)\) one has precisions on \(A\). When \(X\) is a smooth projective variety, using the right derived of \(\lim_{n\to\infty}\operatorname{Hom}_{\text{Gr\,}A}(A/A_{\geq n},-)\) is proved the Serre duality for \(\text{Proj\,}A= \text{Gr\,}A/\text{Tors\,}A\) where \(\text{Tors\,}A\) is the full subcategory of direct limits of right bounded modules.
Some compatibilities between the duality functor and the tensor product are also proved.
Let \(X\) be a smooth scheme of finite type over \(K\), \(E\) be a locally free \({\mathcal O}_X\)-bimodule of rank \(n\), \(A\) be the non-commutative symmetric algebra generated by \(E\) and \(\text{Gr\,}A\) be the category of graded right \(A\)-modules. Internal Hom and tensor functors are constructed on \(\text{Gr\,}A\).
When \(E\) has rank 2, computing derived functors of \(\operatorname{Hom}_{\text{Gr\,}A}({\mathcal O}_X,-)\) one has precisions on \(A\). When \(X\) is a smooth projective variety, using the right derived of \(\lim_{n\to\infty}\operatorname{Hom}_{\text{Gr\,}A}(A/A_{\geq n},-)\) is proved the Serre duality for \(\text{Proj\,}A= \text{Gr\,}A/\text{Tors\,}A\) where \(\text{Tors\,}A\) is the full subcategory of direct limits of right bounded modules.
Some compatibilities between the duality functor and the tensor product are also proved.
Reviewer: Georges Hoff (Villetaneuse)
MSC:
| 14A22 | Noncommutative algebraic geometry |
| 16S99 | Associative rings and algebras arising under various constructions |
| 18G50 | Nonabelian homological algebra (category-theoretic aspects) |
| 18G10 | Resolutions; derived functors (category-theoretic aspects) |
Keywords:
Serre duality; non-commutative projective bundle; smooth scheme and variety; internal Hom and tensor functorsCitations:
Zbl 0860.14002References:
| [1] | 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. · Zbl 0744.14024 |
| [2] | Marcel Bökstedt and Amnon Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993), no. 2, 209 – 234. · Zbl 0802.18008 |
| [3] | Nicolas Bourbaki, Elements of mathematics. Commutative algebra, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1972. Translated from the French. · Zbl 0279.13001 |
| [4] | Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323 – 448 (French). · Zbl 0201.35602 |
| [5] | Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001 |
| [6] | Robin Hartshorne, Cohomological dimension of algebraic varieties, Ann. of Math. (2) 88 (1968), 403 – 450. · Zbl 0169.23302 · doi:10.2307/1970720 |
| [7] | Peter Jørgensen, Intersection theory on non-commutative surfaces, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5817 – 5854. · Zbl 0965.14003 |
| [8] | P. Jörgensen, Non-commutative projective geometry, unpub. notes, 1996. |
| [10] | Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. · Zbl 0193.34701 |
| [11] | Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. · Zbl 0906.18001 |
| [12] | I. Mori, Riemann-Roch like theorem for triangulated categories, J. Pure Appl. Agebra, to appear. · Zbl 1075.14006 |
| [13] | I. Mori and S.P. Smith, The Grothendieck group of a quantum projective space bundle, submitted. · Zbl 1114.14003 |
| [14] | Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205 – 236. · Zbl 0864.14008 |
| [15] | A. Nyman, Points on quantum projectivizations, Mem. Amer. Math. Soc., 167 (2004). · Zbl 1050.14003 |
| [16] | A. Nyman, Serre finiteness and Serre vanishing for non-commutative \({\mathbb{P}}^{1}\)-bundles, J. Algebra, to appear. · Zbl 1057.14003 |
| [17] | N. Popescu, Abelian categories with applications to rings and modules, Academic Press, London-New York, 1973. London Mathematical Society Monographs, No. 3. · Zbl 0271.18006 |
| [18] | S.P. Smith, Non-commutative algebraic geometry, unpub. notes, 1999. |
| [19] | Michel Van den Bergh, A translation principle for the four-dimensional Sklyanin algebras, J. Algebra 184 (1996), no. 2, 435 – 490. · Zbl 0876.17011 · doi:10.1006/jabr.1996.0269 |
| [20] | Michel Van den Bergh, Blowing up of non-commutative smooth surfaces, Mem. Amer. Math. Soc. 154 (2001), no. 734, x+140. · Zbl 0998.14002 · doi:10.1090/memo/0734 |
| [21] | M. Van den Bergh, Non-commutative \({\mathbb{P}}^{1}\)-bundles over commutative schemes, to appear. · Zbl 1082.14005 |
| [22] | M. Van den Bergh, Non-commutative quadrics, in preparation. · Zbl 1082.14005 |
| [23] | Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. · Zbl 0797.18001 |
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