Toric singularities revisited

Abstract

In [K. Kato, Toric singularities, Amer. J. Math. 116 (5) (1994) 1073-1099], Kato defined his notion of a log regular scheme and studied the local behavior of such schemes. A toric variety equipped with its canonical logarithmic structure is log regular. And, these schemes allow one to generalize toric geometry to a theory that does not require a base field. This paper will extend this theory by removing normality requirements. Conventions and notationAll monoids considered in this paper are commutative and cancellative. All rings considered in this paper are commutative and unital. See Kato [2] for an introduction to log schemes. There Kato defines pre-log structures and log structures on the étale site of X. However, we will use the Zariski topology throughout this paper. P * the unit group of the monoid P . P the sharp image of the monoid P , P = P /P * is the orbit space under the natural action of P * on P . P + P + = P \ P * is the maximal ideal of the monoid P .P gp the group generated by P , that is, image of P under the left adjoint of the inclusion functor from Abelian groups to monoids. P sat the saturation of P , that is, {p ∈ P gp | np ∈ P for some n ∈ N + }. R[P ] the monoid algebra of P over a ring R. The elements of R[P ] are written as "polynomials". That is, they are finite sums r p t p with coefficients in R and exponents in P . (K)the ideal β(K)A, where β : P → A is a monoid homomorphism with respect to multiplication on A and K is an ideal of P . We say such an ideal is a log ideal of A. R [[P ]] the (P + )-adic completion of R[P ]. P p the localization of the monoid P at the prime ideal p ⊆ P . dim P the (Krull) dimension of the monoid P .

Logarithmic geometry, minimal free resolutions and toric algebraic stacks

“…By an elementary observation, we see that C(P − p) is a face of C(P ) and it gives rise to a bijective correspondence between the set of prime ideals of P and the set of faces of C(P ) (cf. [25,Proposition 1.10]).…”