Linear Modules over Sklyanin Algebras

Let A = A n ( E , τ) be the n ‐dimensional Sklyanin algebra associated with a smooth elliptic curve E and point τ ∈ E . This paper classifies linear modules over A . We show that d ‐linear modules are in bijection with those d ‐planes in P( A 1 * ) which are either secant to E or are singular loci of certain rank ( n — d — 1)‐quadrics containing E . Moreover, linear modules are Cohen‐Macaulay and critical, and non‐isomorphic linear modules give non‐isomorphic linear spaces in Proj A . We also construct all short exact sequences of linear modules.