  We exhibit basic algebro-geometric results on the formal model of
semi-infinite flag varieties and its Schubert varieties over an algebraically
closed field $\bK$ 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 $\Z$-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. {\bf 371} no.2 (2018)]) when $\mathsf{char} \, \bK =0$
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 [K, arXiv:1805.01718] of a conjecture by
Lam-Li-Mihalcea-Shimozono [J. Algebra {\bf 513} (2018)] that describes an
isomorphism between affine and quantum $K$-groups of a flag manifold.
