Note

Semiorthogonal Decompositions of Perfect Complexes on Split Smooth Projective Toric Schemes over Z

Author: Wei Yang · Added: June 18, 2026

Article date: June 17, 2026.

Abstract

Let $X$ be a split smooth projective toric scheme over $S = \mathrm{Spec}\,\mathbb{Z}$ . We prove that $\mathrm{Perf}(X)$ admits a finite $S$ -linear semiorthogonal decomposition whose components are all equivalent to $\mathrm{Perf}(\mathbb{Z})$ , represented by relative perfect-complex blocks $M \mapsto \pi^*M \otimes E_i$ . As a consequence, for every split base change $X_R$ one has

K(X_R) \simeq \bigoplus_{i=1}^N K(R)

for nonconnective algebraic $K$ -theory. The proof is an integral Cox-VGIT argument using finite monomial descent and a directed path from an outside point, lying outside the effective cone and hence having empty semistable locus, to the target chamber; the auxiliary quotient stacks have finite split diagonalizable stabilizers, although their acting groups are usually tori.