仿射概形与层的可幂等化
代数几何
2024-12-30 v3 组合数学
摘要
在本文中，我们引入可幂等化过程，它在哲学和数学上与现代解析化（analytification）和热带化（tropicalization）有某些相似之处。可幂等化将任意仿射概形关联于一个关于固定由 distinguished 仿射开子概形覆盖的幂等版本。一旦该覆盖固定，我们就能函子性地将一个幂等半环层或幂等半模层关联到一个Zariski环层或模层。我们表明可幂等化独立于所选覆盖，且在Noetherian情形下，结构层的可幂等化恢复全局截面。我们形式体系的底层是有序理论对象子对象格的组合反射，这些格被视为来自交换代数的格。这对交换半环的减理想半环具有拓扑后果：一方面，对于粗下拓扑，它是的同余关系半环的拓扑收缩；另一方面，对于粗上拓扑，它是的理想半环的拓扑收缩。
引用
@article{arxiv.2304.04872,
title = {Idempotentization of Affine Schemes and Sheaves},
author = {Félix Baril Boudreau and Cristhian Garay},
journal= {arXiv preprint arXiv:2304.04872},
year = {2024}
}
备注
20 pages. A thorough revision of the paper was conducted to improve its content and exposition. Realizable semirings were removed. Reason: Proposition B.16 of version 2 does not hold if a realizable semiring has non-compact elements. Thanks go to Kalina Mincheva for this observation and to her and Jaiung Jun for providing detailed comments on the previous version and relevant literature