Tilting objects in the stable category of vector bundles on a weighted projective line of type \((2,2,2,2;\lambda )\)
The paper under review explores the category of vector bundles on a weighted projective line \(\mathbb{X}\) of type \((2,2,2,2;\lambda)\), constructing a tilting object for the category.NEWLINENEWLINEA weighted projective line can be seen as a projective line endowed with finitely many weights. Its category of coherent sheaves is proven to be derived equivalent to the category of finite-dimensional modules over certain algebra and its category of vector bundles is equivalent, as an additive category, to the category of graded Cohen-Macaulay modules over its corresponding graded ring.NEWLINENEWLINEThe stable category of vector bundles (obtained by factoring out all line bundles) turns out to be a triangulated category equivalent to \(D^{b}(coh \mathbb{X})\). However, this equivalence is not explicit hence, although the stable category of vector bundles has tilting objects, it is dificult to find one and this is the main purpose of this article.NEWLINENEWLINEThe paper shows the basic definitions and concepts in section \(2\) and explores projective covers in section \(3\). In section \(4\) a formula to compute slopes of vector bundles under shifting is given, and all exceptional objects of the category are described in section \(5\). Theorem 6.2 in section \(6\) presents the main result by constructing an explicit tilting object consisting of a direct sum of five rank two bundles and one rank three bundle, whose endomorphism algebra is a canonical algebra of type \((2,2,2,2)\), remarking that there are no simpler tilting objects for this category.
scientific article; zbMATH DE number 6340902