We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field ${\mathbb K}$ of characteristic $\neq 2$ from scratch. We show that the formal model of a semi-infinite flag variety admits a unique nice (ind-)scheme structure, its projective coordinate ring has a $\mathbb {Z}$ -model and it admits a Frobenius splitting compatible with the boundaries and opposite cells in positive characteristic. This establishes the normality of the Schubert varieties of the quasi-map space with a fixed degree (instead of their limits proved in [K, Math. Ann. 371 no.2 (2018)]) when $\mathsf {char}\, {\mathbb K} =0$ or $\gg 0$ , and the higher-cohomology vanishing of their nef line bundles in arbitrary characteristic $\neq 2$ . Some particular cases of these results play crucial roles in our proof [47] of a conjecture by Lam, Li, Mihalcea and Shimozono [60] that describes an isomorphism between affine and quantum K-groups of a flag manifold.

中文翻译：

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半无限旗形流形舒伯特变体的 Frobenius 分裂

我们在代数闭域上展示了半无限标志簇及其舒伯特簇的形式模型的基本代数几何结果 ${\mathbb K}$ 有特色的 $\neq 2$ 从头开始。我们证明了一个半无限旗形变体的形式模型承认一个独特的好（ind-）方案结构，它的射影坐标环有一个 $\mathbb {Z}$ -模型，它承认 Frobenius 分裂与正特性中的边界和相对单元兼容。这建立了具有固定度的准映射空间的舒伯特变体的正态性（而不是在 [K, Math. Ann.371no.2 (2018)]) 当 $\mathsf {char}\, {\mathbb K} =0$ 要么 $\gg 0$ ，以及它们的 nef 线束在任意特征中的高上同调消失 $\neq 2$ . 这些结果的一些特殊情况在我们对 Lam、Li、Mihalcea 和 Shimozono [60] 的猜想的证明 [47] 中起着至关重要的作用，该猜想描述了仿射和量子之间的同构ķ- 标志流形的组。