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摘要翻译：
对于任意真DG代数A（即具有有限维全上同调的DG代数），我们引入了A的Hochschild同调上的配对，并给出了任意完美A-模的Chern型特征的显式公式（Chern特征取A的Hochschild同调中的值）。Hirzebruch-Riemann-Roch公式根据两个完全A-模之间Chern特征的配对，表达了它们之间Hom-复形的Euler特征。我们给出了两个适当的DG代数的例子和它们的HRR公式。第一个例子是带关系的颤振的Ringel公式。第二个例子与V/G形式的轨道奇点有关，其中V是复向量空间，G是SL(V)的有限子群。进一步，我们证明了当DG代数光滑时Hochschild同调上的上述配对是非退化的。我们还提出了对于Calabi-Yau DG代数a的配对与拓扑场论中的配对一致的猜想，并在Frobenius代数中进行了验证。
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英文标题：
《Hirzebruch-Riemann-Roch theorem for DG algebras》
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作者：
D. Shklyarov
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最新提交年份：
2008
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分类信息：
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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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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英文摘要：
For an arbitrary proper DG algebra A (i.e. DG algebra with finite dimensional total cohomology) we introduce a pairing on the Hochschild homology of A and present an explicit formula for a Chern-type character of an arbitrary perfect A-module (the Chern characters take values in the Hochschild homology of A). The Hirzebruch-Riemann-Roch formula in this context expresses the Euler characteristic of the Hom-complex between two perfect A-modules in terms of the pairing of their Chern characters. We mention two examples of proper DG algebras and the HRR formulas for them. The first example is Ringel's formula for quivers with relations. The second example is related to orbifold singularities of the form V/G where V is a complex vector space and G is a finite subgroup of SL(V). Furthermore, we prove that the above pairing on the Hochschild homology is non-degenerate when the DG algebra is smooth. We also formulate the conjecture that for a Calabi-Yau DG algebra A the pairing coincides with the one coming from the Topological Field Theory associated with A and verify it in the case of Frobenius algebras.
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PDF链接：
https://arxiv.org/pdf/0710.1937
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