The negative K-theory of normal surfaces
The negative K-theory of normal surfaces
复制标题
法向表面的负 K 理论
DOI：
--
复制
发表时间：
2001
期刊：
影响因子：
--
通讯作者：
C. Weibel
中科院分区：
文献类型：
--
作者：
C. Weibel
我们将法曲面的负K-理论与奇点分解联系起来。唯一非零的K-群是K-−2，它计数例外fi环中的圈，以及K−1，它与完备局部环在奇点的除数类群有关。我们还验证了斯里尼瓦斯关于曲面的K0-正则性和K−1的两个猜想。本文给出了法面X的负K-理论的几何解释。如果R是X在其fi个奇点的半局部环，则X和R的负K-理论由[W3，定理1.2]相同。由此，我们还刻画了任意2维正则半局部环R的负K-理论。正如我们在下一页所解释的那样，我们的结果还给出了投射R[T]-模的几乎完全的Classifi正则。我们的解释将群K−j(X)与曲面的奇点˜X→X的分解联系起来。对于初学者(见定理4.4)，我们证明了对所有j−3，K≥j(X)=0，即使X是奇异的。(这一点fi均方根在[W1，第180页]中是一个猜测。)如果X是正规的，且j=2，我们证明了K−2(X)∼=Zλ，其中λ表示例外fi中的“环”的个数；在Defi定理2.1中，曲线中的“环”的个数是精确的。
We relate the negative K -theory of a normal surface to a resolution of singularities. The only nonzero K -groups are K − 2 , which counts loops in the exceptional fiber, and K − 1 , which is related to the divisor class groups of the complete local rings at the singularities. We also verify two conjectures of Srinivas about K 0 -regularity and K − 1 of a surface. This paper gives a geometric interpretation for the negative K -theory of a normal surface X . If R is the semilocal ring of X at its finitely many singularities, then the negative K -theory of X and R are the same by [W3, Theorem 1.2]. Thus we also describe the negative K -theory of any excellent 2-dimensional normal semilocal ring R . As we explain on the next page, our results also give an almost complete classification of projective R [ T ] -modules. Our interpretation relates the groups K − j (X) to a resolution of singularities ˜ X → X of the surface. For starters (see Theorem 4.4), we show that K − j (X) = 0 for all j ≥ 3, even if X is singular. (This confirms a guess in [W1, p. 180].) If X is normal and j = 2, we prove that K − 2 (X) ∼= Z λ , where λ denotes the number of “loops” in the exceptional fiber E ; the number of “loops” in a curve is made precise in Definition 2.1.