On derived equivalences and homological dimensions
On derived equivalences and homological dimensions
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关于派生等价和同调维数
DOI：
10.1515/crelle-2020-0006
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发表时间：
2020-04
期刊：
影响因子：
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通讯作者：
M. Fang;Weiqun Hu;S. Koenig
中科院分区：
文献类型：
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作者：
M. Fang;Weiqun Hu;S. Koenig
与Hochschild（上）同调和K-理论不同，代数的整体维数和支配维数在一般的导出等价下远不是不变的。然而，我们发现，全球范围内的尺寸和优势的尺寸时，限制到一类代数与反自同构保持单导出不变。这样的反自同构存在于所有的细胞代数，特别是在代数李理论中出现的许多有限维代数。这两个维度都可以在某些衍生的范畴内内在地被表征。在此过程中，证明并使用了一个限制定理，即具有正ν-支配维数的代数之间的导出等价总是限制于它们相关联的自内射代数之间的导出等价，在此假设下，这些导出等价确实存在。
Abstract Unlike Hochschild (co)homology and K-theory, global and dominant dimensions of algebras are far from being invariant under derived equivalences in general. We show that, however, global dimension and dominant dimension are derived invariant when restricting to a class of algebras with anti-automorphisms preserving simples. Such anti-automorphisms exist for all cellular algebras and in particular for many finite-dimensional algebras arising in algebraic Lie theory. Both dimensions then can be characterised intrinsically inside certain derived categories. On the way, a restriction theorem is proved, and used, which says that derived equivalences between algebras with positive ν-dominant dimension always restrict to derived equivalences between their associated self-injective algebras, which under this assumption do exist.