an:07184421
Zbl 1453.12007
Wang, Jie
Toric P-difference varieties
EN
Sci. China, Math. 63, No. 4, 643-670 (2020).
1674-7283 1869-1862
2020
j
12H10 14M25
\(\mathbb{Z} [x]\)-lattice; affine \(P[x]\)-semimodule; P-difference variety; toric P-difference variety; difference torus
Summary: In this paper, we introduce the concept of P-difference varieties and study the properties of toric P-difference varieties. Toric P-difference varieties are analogues of toric varieties in difference algebraic geometry. The category of affine toric P-difference varieties with toric morphisms is shown to be antiequivalent to the category of affine \(P[x]\)-semimodules with \(P[x]\)-semimodule morphisms. Moreover, there is a one-to-one correspondence between the irreducible invariant P-difference subvarieties of an afne toric P-difference variety and the faces of the corresponding affine \(P[x]\)-semimodule. We also define abstract toric P-difference varieties by gluing affine toric P-difference varieties. The irreducible invariant P-difference subvariety-face correspondence is generalized to abstract toric P-difference varieties. By virtue of this correspondence, a divisor theory for abstract toric P-difference varieties is developed.