Tilting on non-commutative rational projective curves

Abstract

In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show that the dimension of the bounded derived category of coherent sheaves on a rational projective curve with only nodal or cuspidal singularities is at most two. In the case of the Kodaira cycles of projective lines, the corresponding tilted algebras belong to a well-known class of gentle algebras. We work out in details the tilting equivalence in the case of the Weierstrass nodal curve zy 2 = x 3 + x 2 z.

Singular Curves and Quasi-hereditary Algebras

“…In fact, gl.dim Coh(X) ≤ 2n, where n is a certain (purely commutative) invariant of X called level. If the original curve X has only nodes and cusps as singularities, the sheaf A coincides with Auslander's order O O I O introduced in [5], where I is the ideal sheaf of the singular locus of X.…”

“…Proof. The result follows from the corresponding local statements in Proposition 2.4 and Theorem 2.6 and the fact that For any 1 ≤ i ≤ n + 1, let e i ∈ Γ(X, A) be the i-th standard idempotent with respect to the matrix presentation (5). As in the affine case, we use the following notation.…”

“…Remark 7.3. In the case X is rational with only simple nodes or cusps as singularities, the bound (13) has been obtained in [5,Theorem 10]. Note that n = 1 and d = 0 is this case.…”

Spherical subcategories in representation theory

“…Burban & Drozd [11, §7] have shown that Λ is derived equivalent to a non-commutative nodal cubic curve X := (X, A), where X ⊂ P 2 is a nodal cubic curve and A = End X (O X ⊕ J ) is a sheaf of O X -algebras; here J ⊂ O X is the ideal sheaf of the singularity. This equivalence T −1 : D b (Λ) → D b (X) sends the Λ-module E to the exceptional simple A-module S γ supported on the singular point of X, see [11,Prop. 12].…”

“…which corresponds to the other exceptional simple A-module S α supported on the singular point of X, see [11,Prop. 12].…”