Non-commutative curves and tilting Igor Burban and Yuriy Drozd

Non-commutative curves and tilting Igor Burban and Yuriy Drozd

E-mail address:burban@math.uni-bonn.de, drozd@imath.kiev.ua

Anon-commutative curveis a pair (X,A_X), whereX is an algebraic curve over a fieldkandA_X is a sheaf ofO_X-algebras, coherent as a sheaf ofO_X-modules. We always suppose that this curve isreduced, i.e. AX has no nilpotent ideals. An Auslander curve of such a (non-commutative, reduced) curve is a curve (X,End_AX(M)), where M is a coherent torsion free sheaf of AX-modules such that, for each pointx∈ X, the stalkMx is an additive generator of the category of torsion free (finitely generated) AX, x-modules. Then gl.dimAX ≤ 2 (exactly 2 if the curve is not smooth, i.e not all stalks AX, x are hereditary).

Suppose thatX is a rational (commutative) curve, which only has simple nodes and cusps as singu- larities,π: ˜X→Xis its normalization,O=OX and ˜O=π∗(OX˜). SetM=O ⊕O˜andA=EndO(M).

Then (X,AX) is an Auslander curve of (X,OX). We construct a tilting complex T in the derived cat- egory D^b(Coh(AX)) and calculate its endomorphism algebra ΛX. Namely, let {X1, X2, . . . , Xs} be the irreducible components of ˜X (all of them are identified with the projective line), {x1, x2, . . . , xm} be the nodes of X and {y1, y₂, . . . , y_n} be its cusps. Let also π⁻¹(x_j) = {x⁰_j, x⁰⁰_j}, where x⁰_j ∈ X_i0(j) and x⁰⁰_j ∈ X_i⁰⁰_(j) (possibly i⁰⁰(j) = i⁰(j)), while π⁻¹(yk)∈ X_i(k). Then ΛX =kQ/I, whereQ is the quiver with the set of vertices

{vi, v_i⁰, xj, yk|1≤i≤s,1≤j ≤m, 1≤k≤n}

and the set of arrows

{a⁰_j:xj →vi⁰(j), a⁰⁰_j :xj→vi⁰⁰(j), b⁰_k, b⁰⁰_k :yk→vi(k), ci, di:vi→v_i⁰}.

The idealI is generated by the relations

(η⁰_jc_i0(j)−ξ_j⁰d_i0(j))a⁰_j ifx⁰_j= (ξ_j⁰ :η_j⁰), (η⁰⁰_jc_i⁰⁰_(j)−ξ⁰⁰_jd_i⁰⁰_(j))a⁰⁰_j ifx⁰⁰_j = (ξ_j⁰⁰:η⁰⁰_j),

(η_kc_i(k)−ξ_kd_i(k))b⁰_k and (η_kc_i(k)−ξ_kd_i(k))b⁰⁰_k−b⁰_k ifπ⁻¹(y_k) = (ξ_k :η_k).

In particular, gl.dim ΛX= 2 andD^b(Coh(AX))' D^b(ΛX-mod), while the categoryD⁻(Coh(X)) embeds intoD⁻(ΛX-mod) as a full triangulated subcategory.

Note that the algebra ΛX is gentle ifX is either a chain or a cycle of projective lines. Since gentle algebras are known to be derived-tame, this gives a new proof of tameness of the category of coherent sheaves Coh(X) and its perfect derived category. Other way arround, such a geometric description of the obtained gentle algebras leads to interesting corollaries about their Serre functor and the full subcategory ofband complexes.

This result can be generalized to non-commutative curves such that all their singularities are nodal algebras.

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