Realisation functors in tilting theory
Realisation functors in tilting theory
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DOI：
10.1007/s00209-017-1923-y
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发表时间：
2015-11
影响因子：
0.8
通讯作者：
Chrysostomos Psaroudakis;J. Vitória
中科院分区：
文献类型：
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作者：
Chrysostomos Psaroudakis;J. Vitória
导出等价与t-结构密切相关。我们利用三角范畴中与t-结构相关的实现函子，建立了具有投射生成元或内射余生成元的交换范畴的衍生Morita理论。为此，我们发展了一种(非紧凑的，或大的)倾斜和共倾斜物体的理论，该理论推广了文献中的上述概念。在环的衍生森田理论的范围内，我们证明了，在某些假设下，实现函子是衍生张量积。这一事实允许我们探讨Rickard关于导出等价的形状的问题。最后，我们应用这一新的派生Morita理论的技巧来证明派生范畴的回忆是阿贝尔范畴的回忆的派生版本当且仅当存在倾斜或共倾斜的t-结构粘合到倾斜或共倾斜的结构上。作为进一步的应用，我们回答了Xi关于有限维遗传代数的导模范畴的标准形式的一个问题。
Derived equivalences and t-structures are closely related. We use realisation functors associated to t-structures in triangulated categories to establish a derived Morita theory for abelian categories with a projective generator or an injective cogenerator. For this purpose we develop a theory of (non-compact, or large) tilting and cotilting objects that generalises the preceding notions in the literature. Within the scope of derived Morita theory for rings we show that, under some assumptions, the realisation functor is a derived tensor product. This fact allows us to approach a problem by Rickard on the shape of derived equivalences. Finally, we apply the techniques of this new derived Morita theory to show that a recollement of derived categories is a derived version of a recollement of abelian categories if and only if there are tilting or cotilting t-structures glueing to a tilting or a cotilting t-structure. As a further application, we answer a question by Xi on a standard form for recollements of derived module categories for finite dimensional hereditary algebras.