论文标题
（复合）du val单数的K理论
The K-theory of (compound) Du Val singularities
论文作者
论文摘要
本论文给出了针对两个和三维奇点的大型家庭的Grothendieck小组和Divisor Class组的完整描述。整个过程中呈现的主要结果，并在定理8.1.1中进行了总结，给出了grothendieck组和kleinian奇点的班级组的明确描述，它们的变形和复合du val（CDV）奇异性在各种环境中。对于此类环R，主要结果断言，存在$ G_0（R）$和$ \ Mathbb {Z} \ oplus \ Mathrm {Cl}（r）$之间的同构和类别组。 More precisely, we establish these results for 2-dimensional deformations of global type A Kleinian singularities, 3-dimensional isolated complete local cDV singularities admitting a noncommutative crepant resolution, any 3-dimensional type A complete local cDV singularity, polyhedral quotient singularities (which are non-isolated), and any isolated cDV singularity admitting a minimal model with only type cAn cDV奇异性。我们还研究了在符号商奇点的设置中，为此，同构不结合，并且基于计算机证据表明，在对称组的情况下，降低了Grothendieck组的猜想。
This thesis gives a complete description of the Grothendieck group and divisor class group for large families of two and three dimensional singularities. The main results presented throughout, and summarised in Theorem 8.1.1, give an explicit description of the Grothendieck group and class group of Kleinian singularities, their deformations, and compound Du Val (cDV) singularities in a variety of settings. For such rings R, the main results assert that there exists an isomorphism between $G_0(R)$ and $\mathbb{Z} \oplus \mathrm{Cl}(R)$, and the class group is explicitly presented. More precisely, we establish these results for 2-dimensional deformations of global type A Kleinian singularities, 3-dimensional isolated complete local cDV singularities admitting a noncommutative crepant resolution, any 3-dimensional type A complete local cDV singularity, polyhedral quotient singularities (which are non-isolated), and any isolated cDV singularity admitting a minimal model with only type cAn cDV singularities. We also study various complex reflection groups in the setting of symplectic quotient singularities, for which this isomorphism does not hold, and conjecture based on computer evidence that the reduced Grothendieck group in the case of the symmetric group has size $n!$.
