Noncommutative linear systems and noncommutative elliptic curves
HTML articles powered by AMS MathViewer
- by Daniel Chan and Adam Nyman;
- Trans. Amer. Math. Soc. 377 (2024), 1957-1987
- DOI: https://doi.org/10.1090/tran/9051
- Published electronically: January 10, 2024
- HTML | PDF | Request permission
Abstract:
In this paper we introduce a noncommutative analogue of the notion of linear system, which we call a helix $\underline {\mathcal {L}} ≔(\mathcal {L}_{i})_{i \in \mathbb {Z}}$ in an abelian category $\mathsf {C}$ over a quadratic $\mathbb {Z}$-indexed algebra $A$. We show that, under natural hypotheses, a helix induces a morphism of noncommutative spaces from Proj $End(\underline {\mathcal {L}})$ to Proj $A$. We construct examples of helices of vector bundles on elliptic curves generalizing the elliptic helices of line bundles constructed by Bondal-Polishchuk, where $A$ is the quadratic part of $B≔End(\underline {\mathcal {L}})$. In this case, we identify $B$ as the quotient of the Koszul algebra $A$ by a normal family of regular elements of degree 3, and show that Proj $B$ is a noncommutative elliptic curve in the sense of Polishchuk [J. Geom. Phys. 50 (2004), pp. 162–187]. One interprets this as embedding the noncommutative elliptic curve as a cubic divisor in some noncommutative projective plane, hence generalizing some well-known results of Artin-Tate-Van den Bergh.References
- M. F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414–452. MR 131423, DOI 10.1112/plms/s3-7.1.414
- 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
- M. Artin and J. J. Zhang, Abstract Hilbert schemes, Algebr. Represent. Theory 4 (2001), no. 4, 305–394. MR 1863391, DOI 10.1023/A:1012006112261
- Martin Brandenburg and Alexandru Chirvasitu, Tensor functors between categories of quasi-coherent sheaves, J. Algebra 399 (2014), 675–692. MR 3144607, DOI 10.1016/j.jalgebra.2013.09.050
- A. I. Bondal and A. E. Polishchuk, Homological properties of associative algebras: the method of helices, Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), no. 2, 3–50 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 42 (1994), no. 2, 219–260. MR 1230966, DOI 10.1070/IM1994v042n02ABEH001536
- Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527. MR 1322847, DOI 10.1090/S0894-0347-96-00192-0
- D. Chan and A. Nyman, Species and non-commutative $\Bbb {P}^1$’s over non-algebraic bimodules, J. Algebra 460 (2016), 143–180. MR 3510397, DOI 10.1016/j.jalgebra.2016.03.048
- D. Chan and A. Nyman, Morphisms to noncommutative projective lines, Proc. Amer. Math. Soc. 149 (2021), no. 7, 2789–2803. MR 4257794, DOI 10.1090/proc/15386
- S. A. Kuleshov, Construction of bundles on an elliptic curve, Helices and vector bundles, London Math. Soc. Lecture Note Ser., vol. 148, Cambridge Univ. Press, Cambridge, 1990, pp. 7–22. MR 1074778, DOI 10.1017/CBO9780511721526.002
- I. Mori and A. Nyman, Local duality for connected $\Bbb {Z}$-algebras, J. Pure Appl. Algebra 225 (2021), no. 9, Paper No. 106676, 22. MR 4202590, DOI 10.1016/j.jpaa.2021.106676
- Adam Nyman, A structure theorem for $\Bbb {P}^1$—spec $k$-bimodules, Algebr. Represent. Theory 16 (2013), no. 3, 659–671. MR 3049664, DOI 10.1007/s10468-011-9324-0
- A. Nyman, Witt’s theorem for noncommutative conic curves, Appl. Categ. Structures 24 (2016), no. 4, 407–420. MR 3516079, DOI 10.1007/s10485-015-9402-2
- S. Okawa and K. Ueda, $\operatorname {AS}$-regular algebras from acyclic spherical helices, arXiv:2007.07620.
- Jorge Plazas, Arithmetic structures on noncommutative tori with real multiplication, Int. Math. Res. Not. IMRN 2 (2008), Art. ID rnm147, 41. MR 2418858, DOI 10.1093/imrn/rnm147
- A. Polishchuk, Noncommutative two-tori with real multiplication as noncommutative projective varieties, J. Geom. Phys. 50 (2004), no. 1-4, 162–187. MR 2078224, DOI 10.1016/j.geomphys.2003.11.007
- A. Polishchuk, Noncommutative proj and coherent algebras, Math. Res. Lett. 12 (2005), no. 1, 63–74. MR 2122731, DOI 10.4310/MRL.2005.v12.n1.a7
- Stewart B. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39–60. MR 265437, DOI 10.1090/S0002-9947-1970-0265437-8
- Susan J. Sierra, $G$-algebras, twistings, and equivalences of graded categories, Algebr. Represent. Theory 14 (2011), no. 2, 377–390. MR 2776790, DOI 10.1007/s10468-009-9193-y
- S. Paul Smith, Corrigendum to “Maps between non-commutative spaces”[ MR2052602], Trans. Amer. Math. Soc. 368 (2016), no. 11, 8295–8302. MR 3546801, DOI 10.1090/tran/6908
- Loring W. Tu, Semistable bundles over an elliptic curve, Adv. Math. 98 (1993), no. 1, 1–26. MR 1212625, DOI 10.1006/aima.1993.1011
- Michel Van den Bergh, Noncommutative quadrics, Int. Math. Res. Not. IMRN 17 (2011), 3983–4026. MR 2836401, DOI 10.1093/imrn/rnq234
Bibliographic Information
- Daniel Chan
- Affiliation: School of Mathematics and Statistics, UNSW Sydney, NSW 2052, Australia
- MR Author ID: 658386
- ORCID: 0000-0002-5546-4828
- Email: danielc@unsw.edu.au
- Adam Nyman
- Affiliation: Western Washington University, Bellingham, Washington
- MR Author ID: 687479
- Email: nymana@wwu.edu
- Received by editor(s): February 1, 2023
- Received by editor(s) in revised form: May 4, 2023, and July 18, 2023
- Published electronically: January 10, 2024
- © Copyright 2024 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 377 (2024), 1957-1987
- MSC (2020): Primary 14A22; Secondary 16S38
- DOI: https://doi.org/10.1090/tran/9051
- MathSciNet review: 4744746