论文标题
riemann-roch for $ \ overline {\ text {spec} \ mathbb z} $
Riemann-Roch for $\overline{\text{Spec}\mathbb Z}$
论文作者
论文摘要
我们证明了整个代数频谱的Arakelov紧凑型在Arakelov紧凑型上的分隔线的Riemann-Roch定理。该结果依赖于引入三个关键概念的引入：共同体（连接到除数），它们的整数维度和二元性。这些概念直接将其经典函数扩展到功能字段。 The Riemann-Roch formula equates the (integer valued) Euler characteristic of a divisor with a slight modification of the traditional expression in terms of the sum of the degree of the divisor and the logarithm of 2. Both the definitions of the cohomologies and of their dimensions rely on a universal arithmetic theory over the sphere spectrum that we had previously introduced using Segal's Gamma rings.通过采用这种新的观点，我们可以平行Weil对Riemann-Roch公式的Adelic证明，用于功能字段，包括使用Pontryagin二重性。
We prove a Riemann-Roch theorem of an entirely novel nature for divisors on the Arakelov compactification of the algebraic spectrum of the integers. This result relies on the introduction of three key concepts: the cohomologies (attached to a divisor), their integer dimension, and Serre duality. These notions directly extend their classical counterparts for function fields. The Riemann-Roch formula equates the (integer valued) Euler characteristic of a divisor with a slight modification of the traditional expression in terms of the sum of the degree of the divisor and the logarithm of 2. Both the definitions of the cohomologies and of their dimensions rely on a universal arithmetic theory over the sphere spectrum that we had previously introduced using Segal's Gamma rings. By adopting this new perspective we can parallel Weil's adelic proof of the Riemann-Roch formula for function fields including the use of Pontryagin duality.
