# zbMATH — the first resource for mathematics

Orlov’s equivalence and maximal Cohen-Macaulay modules over the cone of an elliptic curve. (English) Zbl 1329.14034
Let $$K$$ be a field and $$f\in K[x_0, \ldots, x_n]$$ be a homogeneous polynomial of degree $$n+1$$ and $$X=\mathrm{Proj}\;K[x_0, \ldots, x_n]/f$$. D. Orlov [Prog. Math. 270, 503–531 (2009; Zbl 1200.18007)] proved that there is an equivalence between the bounded derived category of coherent sheaves on $$X$$ and the homotopy category of graded matrix factorisations of $$f$$. It is described how to compute this equivalence in case of a smooth elliptic curve over an algebraically closed field. The indecomposable graded matrix factorisations of rank one are described. Every indecomposable maximal Cohen-Macaulay module over the completion of a smooth curve is gradable. This gives explicit descriptions of all indecomposable rank one matrix factorisations. It is explained how to compute all indecomposable matrix factorisations of higher rank using a computer algebra system as Singular.

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 16G50 Cohen-Macaulay modules in associative algebras 14H52 Elliptic curves
SINGULAR
Full Text:
##### References:
 [1] Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. 7 (3) pp 415– (1957) · Zbl 0084.17305 [2] Ballard, Orlov spectra: bounds and gaps, Invent. Math. 189 (2) pp 359– (2012) · Zbl 1266.14013 [3] Bondal, Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat. 53 pp 25– (1989) · Zbl 0692.18002 [4] Bondal, Representable functors, Serre functors, and reconstruction, Izv. Akad. Nauk SSSR Ser. Mat. 53 (6) pp 1183– (1989) [5] Brodmann, Local Cohomology: An Algebraic Introduction with Geometric Applications (1998) [6] Bruns, Cohen-Macaulay Rings (1993) [7] R.-O. Buchweitz Maximal Cohen-Macaulay modules and tate cohomology over Gorenstein rings https://tspace.library.utoronto.ca/handle/1807/16682 [8] Burban, Trends in Representation Theory of Algebras and Related Topics (2008) [9] Conrad, Grothendieck Duality and Base Change (2000) · Zbl 0992.14001 [10] W. Decker G.-M. Greuel G. Pfister H. Schnemann SINGULAR, a System for Polynomial Computations http://www.singular.uni-kl.de/ [11] Dolgachev, Lectures on Invariant Theory (2003) [12] R Fossum and H-B Foxby, The category of graded modules, Math. Scand. 35 pp 288– (1974) · Zbl 0298.13012 [13] Grothendieck, SGA2 cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (1968) [14] Hartshorne, Residues and Dualities (1966) · Zbl 0212.26101 [15] Hartshorne, Ample Subvarities of Algebraic Varieties (1970) · Zbl 0208.48901 [16] Herzog, Advanced Studies in Pure Mathematics (1987) [17] Kahn, Reflexive Moduln auf einfach-elliptischen Flächensingularitäten, Dissertation (1988) · Zbl 0674.14025 [18] Kahn, Reflexive modules on minimally elliptic singularities, Math. Ann. 285 pp 141– (1989) · Zbl 0662.14022 [19] Laza, Maximal Cohen-Macaulay modules over the cone of an elliptic curve, J. Algebra 253 pp 209– (2002) · Zbl 1056.14048 [20] Lenzing, Carelton-Ottawa Mathematical Lecture Note Series (1993) [21] Miyachi, Localisation of triangulated categories and derived categories, J. Algebra 141 pp 463– (1991) · Zbl 0739.18006 [22] Nastasescu, Methods of Graded Rings (2004) [23] Orlov, Progress in Mathematics (2009) [24] Seidel, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 pp 37– (2001) · Zbl 1092.14025 [25] Serre, Faisceaux algèbraique cohèrents, Annals of Math. 61 pp 197– (1955) · Zbl 0067.16201 [26] Yoshino, Cohen-Macaulay Modules over Cohen-Macaulay Rings (1990) [27] Yoshino, The fundamental module of a normal local domain of dimension 2, Trans. Amer. Math. Soc. 309 pp 425– (1988) · Zbl 0656.13027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.