摘要翻译:
在拟相干束的单形范畴的基础上,引入了交换代数几何的一个非交换对应体,我们证明了非交换几何中的各种构造(如Morita等价、Hopf-Galois扩张)在交换情形下都可以被赋予几何意义,从而扩展了它们的几何解释。另一方面,我们证明了交换几何中的某些构造(如忠实平坦下降理论、主纤维、等变几何和无穷小几何)可以解释为应用于交换对象的非交换几何构造。对于这样的广义几何,我们定义了构造循环对象的全局不变量,由此我们以标准的方式导出了Hochschild、循环和周期循环同调(带系数)。
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英文标题:
《Noncommutative geometry through monoidal categories I》
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作者:
Tomasz Maszczyk
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最新提交年份:
2007
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分类信息:
一级分类:Mathematics 数学
二级分类:Algebraic Geometry 代数几何
分类描述:Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇,叠,束,格式,模空间,复几何,量子上同调
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一级分类:Mathematics 数学
二级分类:K-Theory and Homology K-理论与同调
分类描述:Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras
代数和拓扑K-理论,与拓扑的关系,交换代数和算子代数
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英文摘要:
After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasi-coherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, Hopf-Galois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. For such generalized geometry we define global invariants constructing cyclic objects from which we derive Hochschild, cyclic and periodic cyclic homology (with coefficients) in the standard way.
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PDF链接:
https://arxiv.org/pdf/0707.1542