Ladders and simplicity of derived module categories

Abstract

Abstract. Recollements of derived module categories are investigated, using a new technique, ladders of recollements, which are maximal mutation sequences. The position in the ladder is shown to control whether a recollement restricts from unbounded to another level of derived category. Ladders also turn out to control derived simplicity on all levels. An algebra is derived simple if its derived category cannot be deconstructed, that is, if it is not the middle term of a non-trivial recollement whose outer terms are again derived categories of algebras. Derived simplicity on each level is characterised in terms of heights of ladders.These results are complemented by providing new classes of examples of derived simple rings, in particular indecomposable commutative rings, as well as by a finite dimensionsal counterexample to the Jordan-Hölder problem for derived module categories. Moreover, recollements are used to compute homological and K-theoretic invariants.

Lifting of recollements and gluing of partial silting sets

“…For example, this holds in the situation of a recollement of homologically non-positive homologically finite length dg algebras , and . This allows us to generalize some of the results from [5] (see proposition 3.18).…”

“…For instance, given a recollement of (bounded) derived categories of finite dimensional algebras and over a field, Hochschild and cyclic homology and cohomology of and , as well as, -theory, are related by long exact sequences [20, 25, 31, 44, 56, 58, 60]. Furthermore, global dimension, Cartan determinants and finitistic dimensions of and are related [5, 17, 22, 30, 51].…”

“…This is the first goal of this paper. The motivation for this stems from [5], where the authors show that recollements of bounded derived categories of finite dimensional algebras lift up to equivalence to corresponding recollements of the unbounded derived categories, which, in addition, can be extended upwards and downwards to ladders of recollements of height three. We wanted to know to what extent this relies on the context of finite dimensional algebras and to what extent this is a general phenomenon occurring in triangulated categories.…”

$\boldsymbol{n}$-Recollements and upper recollements of derived module categories

“…(3) D + (modA) 有一个关于 D + (modB) 和 D + (modC) 的下黏合. 我们有 (1)⇔(2)⇒ (3). 特别地, 若 A 是域 k 上的有限维代数, 则以上叙述等价.…”

“…(2) K b (projA) 有一个关于 K b (projB) 和 K b (projC) 的上黏合; 本文的另一个方面是, 在黏合的框架下讨论三个代数的同调性质. 在黏合框架下, 已被研究的同 调性质包括整体维数的有限性 [3,11,12] 、有限维数的有限性 [10,13] 、同调光滑性 [2] 和 Gorenstein 性 [2,14] 等. 特别地, 设三个环 A、 B 和 C 的无界导出范畴 (DB、 DA、 DC) 构成一黏合, Angeleri Hügel 等在文 献 [3, 定理 I(b)] 中证明了, 若 A 为域 k 上的有限维代数, 则 B 和 C 也如此.…”