Abelianness of the “missing part” from a sheaf category to a module category

Tilting bundles and the missing part on a weighted projective line of type (2,2,n)

“…Moreover, several examples there indicate that the missing part is not, in general, abelian if the weight type is different from (2, 2, n), or if the tilting sheaf contains a direct summand of finite length. In this paper, we extend the main result of [3] to a more general case. Namely, we show that for weight type (2, 2, n), the missing part induced by any tilting bundle carries the structure of an abelian category.…”

“…Firstly, we recall the definition of missing part from [3]. [3].) Let T be a tilting bundle in coh X with endomorphism algebra Λ.…”

“…In general, the factor category C is only an additive category. In [3], we focused on the weighted projective lines of type (2, 2, n) to show that for canonical tilting sheaf T can = 0≤ x≤ c O( x), the corresponding missing part is an abelian category which is equivalent to the module category mod(k A n−1 ), where k A n−1 denotes the path algebra of the equioriented quiver A n−1 . Moreover, several examples there indicate that the missing part is not, in general, abelian if the weight type is different from (2, 2, n), or if the tilting sheaf contains a direct summand of finite length.…”

Tilting bundles and the "missing part" on the weighted projective line of type $(2, 2, n)$

“…The factor category C is called the "missing part" from coh X to modΛ op in [2]. It is only an additive category in general.…”

“…It is only an additive category in general. Chen, Lin and Ruan [2] focused on the weighted projective line of weight type (2, 2, n), and showed that for the canonical tilting sheaf T can , the corresponding "missing part" C can is an abelian category and isomorphic to mod(k − → A n−1 ). Moreover, some examples there indicated that the abelianness is not true if the tilting sheaf contains a direct summand of finite length sheaf.…”