Symplectic geometry of homological algebra
Symplectic geometry of homological algebra
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同调代数的辛几何
DOI：
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发表时间：
2009
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通讯作者：
M. Kontsevich
中科院分区：
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作者：
M. Kontsevich
对于地面场k上的任一方案X，我们可以联系到完美复形的k-线性三角范畴Perf(X)，即X上由局部(在Zariski拓扑中)准同构于有限阶自由复形的有限复形的对象组成的拟凝聚层无界派生范畴的满子范畴。范畴Perf(X)本质上是小的，允许对微分分次(简称dg)范畴进行自然增强直到同伦等价，并且是Karoubi闭(如幂等闭)。导出的非交换代数几何的主要思想是将任何Karoubi闭的小dg范畴看作是“空间”上的完全复形范畴。根据A.Bondal和M.Van den Bergh的一个基本结果，任何有限类型的分离格式在导出意义上都是仿射的，即Perf(X)是由一个对象生成的。等同地，
With any scheme X over ground field k we can associate a k-linear triangulated category Perf(X) of perfect complexes, i.e. the full subcategory of the unbounded derived category of quasi-coherent sheaves on X, consisting of objects which are locally (in Zariski topology) quasi-isomorphic to finite complexes of free sheaves of finite rank. The category Perf(X) is essentially small, admits a natural enhancement to a differential graded (dg in short) category up to a homotopy equivalence, and is Karoubi (e.g. idempotent) closed. The main idea of derived noncommutative algebraic geometry is to treat any Karoubi closed small dg category as the category of perfect complexes on a “space”. By a foundamental result of A. Bondal and M. Van den Bergh, any separated scheme of finite type is affine in the derived sense, i.e. Perf(X) is generated by just one object. Equivalently,