Abelian hereditary fractionally Calabi-Yau categories (Q2895803)

A \(k\)-linear triangulated category with finite-dimensional morphism spaces is called fractionally Calabi-Yau if it admits a Serre functor \(\mathbb{S}\) and there is an isomorphism of exact functors \(\mathbb{S}^n\cong [m]\) for some \(n>0\). As a matter of terminology, the fractional Calabi-Yau dimension is interpreted as a rational number, that is, one simplifies the fraction \(\frac{m}{n}\). Note that a fractionally Calabi-Yau category of dimension 1 is not necessarily a 1-Calabi-Yau category (in the latter case: \(\mathbb{S}\cong [1]\)). An abelian category \(\mathcal{A}\) is fractionally Calabi-Yau if its bounded derived category is. Recall that \(\mathcal{A}\) has Serre duality if and only if it has Auslander-Reiten (AR) triangles. Denoting the Auslander-Reiten shift by \(\tau\), one has \(\mathbb{S}\cong \tau[1]\). Further, \(\mathcal{A}\) is indecomposable if it is not the coproduct of two nonzero categories and hereditary if \(\text{Ext}^i(A_1,A_2)=0\) for all \(A_j\in \mathcal{A}\) and for all \(i>1\). The main result of the paper under review is the classifation of indecomposable hereditary abelian categories over an algebraically closed field which are fractionally Calabi-Yau. Namely, if \(\mathcal{A}\) is such a category, then it is derived equivalent to (1) the category of finite-dimensional representations over a Dynkin quiver or (2) the category of finite-dimensional nilpotent representations of an oriented cycle with \(n+1\) (\(n\geq 0\)) arrows or (3) the category of coherent sheaves over \(X\), where \(X\) is either an elliptic curve or a weighted projective line of tubular type.NEWLINENEWLINEThe proof of the result is a detailed analysis of the possible cases. Firstly, the case where \(\mathcal{A}\) is 1-Calabi-Yau was considered by the author [Int. Math. Res. Not. 2008, Article ID rnn003 (2008; Zbl 1144.18008)]. It is shown in Prop.\ 5.5 that the fractional Calabi-Yau dimension is always between 0 and 1. The classification thus falls into three cases. If the dimension is strictly smaller than 1, then it follows from an earlier result of the author [Trans. Am. Math. Soc. 360, No. 5, 2467--2503 (2008; Zbl 1155.16018)] that one has (1) above. The remaining two cases are where the fractional Calabi-Yau dimension is equal to 1 and all objects either all lie in one tube or in several. Recall that tubes are stable components of the Auslander-Reiten quiver of \(\mathcal{A}\) of the form \(\mathbb{Z}A_\infty/\langle\tau^n\rangle\).NEWLINENEWLINEIf the dimension is 1 and there is only 1 tube, then we are in the case (2) above. If the dimension is 1 and there is more than one tube, then the author proves that \(D^b(\mathcal{A})\) admits a tilting object and hence, by a result of \textit{D. Happel} [Invent.\ Math.\ 144, No.\ 2, 381--398 (2001; Zbl 1015.18006)], \(\mathcal{A}\) is derived equivalent to either the category of finitely generated modules over a finite-dimensional hereditary algebra or to the category of coherent sheaves on a weighted projective line. The statement that \(X\) is of tubular type then follows from the fact that every AR-component is a standard tube.