Local Log-Regular Rings

• Local log-regular rings are Noetherian local rings equipped with a monoid and logarithmic chart ensuring that the quotient by the non-unit ideal is regular and the dimension splits additively.
• They admit toric-type completions or hypersurface quotient representations, are normal Cohen–Macaulay, and support explicit formulas for canonical modules and divisor class groups.
• Applications include precise two-dimensional desingularization criteria, computation of divisor class groups, and mixed-characteristic arithmetic via perfectoid towers.

Local log-regular rings are Noetherian local rings equipped with logarithmic chart data that make them the local algebraic models of Kato’s log-regular schemes. In ring-theoretic form, a local log ring is a triple $(R,Q,\alpha)$ consisting of a commutative ring $R$, a commutative monoid $Q$, and a monoid homomorphism $\alpha:Q\to R$, and it is local when $R$ is local and $\alpha^{-1}(R^\times)=Q^\times$. The associated ideal

$$I_\alpha=\langle \alpha(Q^+)\rangle,\qquad Q^+=Q\setminus Q^\times,$$

records the non-unit part of the logarithmic structure. Local log-regularity requires that $R/I_\alpha$ be regular and that

$\dim(R)=\dim(R/I_\alpha)+\dim(Q),$

with $\overline Q:=Q/Q^\times$ finitely generated, cancellative, reduced, and root closed. These rings are normal Cohen–Macaulay, admit toric-style formal charts after completion, and support explicit descriptions of canonical modules and divisor class groups in terms of monoid data; later work has supplied geometric criteria in dimension two, a positive-characteristic criterion via logarithmic Frobenius–Witt differentials, constructions from monoid algebras, and mixed-characteristic applications using perfectoid towers (Ishiro, 2022, Nagamachi, 2022, Takeuchi, 19 Apr 2026, Ishiro et al., 2022, Ishiro, 1 Sep 2025).

1. Definition and local formulations

The basic local object is the triple $R$0, where $R$1 is a commutative local ring and $R$2 is a commutative monoid. In the ring-theoretic formulation used for local log-regular rings, one assumes $R$3 is Noetherian and that $R$4 is finitely generated, cancellative, reduced, and saturated. By Lemma 6.2.10 of Gabber–Ramero, such a monoid decomposes as $R$5, so one may work with reduced $R$6. The defining conditions are that $R$7 is a regular local ring and that

$R$8

Assuming $R$9 is regular, this dimension condition is equivalent to several structural properties, notably the existence of very solid charts, $Q$0-flatness, and exactness of the induced monoid map $Q$1; in particular, $Q$2 is injective and $Q$3 (Ishiro, 2022).

The pointwise scheme-theoretic form is parallel. For an fs log scheme $Q$4 and a geometric point $Q$5, the log-ideal

$Q$6

is generated by the image of $Q$7, and log regularity at $Q$8 means that $Q$9 is regular and

$\alpha:Q\to R$0

A log regular log scheme is normal, its nontrivial locus is a pure codimension-$\alpha:Q\to R$1 subset, and its localizations are toric precisely when the associated local pairs are toric. This pointwise formulation is the geometric source of the local ring definition and is the form used in the two-dimensional desingularization criterion (Nagamachi, 2022).

A common misconception is to identify local log-regular rings with ordinary toric rings. The monoidal structure is indeed toric in flavor, but the class includes mixed-characteristic and non-complete local rings, and the logarithmic framework requires passage from $\alpha:Q\to R$2 to $\alpha:Q\to R$3, so units in the monoid are part of the intrinsic data rather than an inessential embellishment (Ishiro, 2022).

2. Formal structure and comparison with toric rings

The central structural result is a Cohen-type theorem. If $\alpha:Q\to R$4 is local log-regular and $\alpha:Q\to R$5 lift a regular system of parameters of $\alpha:Q\to R$6, then in equal characteristic there is an isomorphism of complete local rings

$\alpha:Q\to R$7

compatible with the natural injection of $\alpha:Q\to R$8, where $\alpha:Q\to R$9 is the residue field of $R$0. In mixed characteristic, with $R$1 a Cohen ring of the residue field, there is a surjection

$R$2

whose kernel is principal, generated by an element $R$3 with constant term $R$4. Thus the completion is either a complete monoid algebra or a hypersurface quotient of one (Ishiro, 2022).

This formal description makes the analogy with affine toric rings precise. For a finitely generated, cancellative, saturated affine semigroup $R$5, the toric ring $R$6 is normal Cohen–Macaulay and has monomially controlled invariants. Local log-regular rings share these properties, and after completion they become monoid algebras in equal characteristic, or hypersurface quotients thereof in mixed characteristic. Conversely, every toric ring $R$7 becomes local log-regular by taking $R$8 and using the natural monoid algebra embedding; every local log-regular ring has a completion of toric type on $R$9 (Ishiro, 2022).

The class is stable under quotient by regular sequences arising from the regular quotient $\alpha^{-1}(R^\times)=Q^\times$0. If $\alpha^{-1}(R^\times)=Q^\times$1 map to a regular system of parameters of $\alpha^{-1}(R^\times)=Q^\times$2, then they form a regular sequence on $\alpha^{-1}(R^\times)=Q^\times$3, and each quotient

$\alpha^{-1}(R^\times)=Q^\times$4

with the induced map $\alpha^{-1}(R^\times)=Q^\times$5 is again local log-regular. This stability is important both for inductive arguments and for the reduction of local questions to the purely monoidal part of the singularity (Ishiro, 2022).

These results place local log-regular rings at the intersection of commutative algebra and logarithmic geometry: the quotient $\alpha^{-1}(R^\times)=Q^\times$6 behaves like a regular base, while $\alpha^{-1}(R^\times)=Q^\times$7 records the logarithmic boundary. The local singularity is therefore controlled by a regular piece and a combinatorial monoid piece.

3. Canonical modules, Gorensteinness, and divisor class groups

One of the main ring-theoretic advances is the explicit description of the canonical module. If $\alpha^{-1}(R^\times)=Q^\times$8 is local log-regular and $\alpha^{-1}(R^\times)=Q^\times$9 lift a regular system of parameters of $$I_\alpha=\langle \alpha(Q^+)\rangle,\qquad Q^+=Q\setminus Q^\times,$$0, then $$I_\alpha=\langle \alpha(Q^+)\rangle,\qquad Q^+=Q\setminus Q^\times,$$1 has a canonical module

$$I_\alpha=\langle \alpha(Q^+)\rangle,\qquad Q^+=Q\setminus Q^\times,$$2

where $$I_\alpha=\langle \alpha(Q^+)\rangle,\qquad Q^+=Q\setminus Q^\times,$$3 denotes the relative interior of $$I_\alpha=\langle \alpha(Q^+)\rangle,\qquad Q^+=Q\setminus Q^\times,$$4 inside $$I_\alpha=\langle \alpha(Q^+)\rangle,\qquad Q^+=Q\setminus Q^\times,$$5. Equivalently, multiplication by $$I_\alpha=\langle \alpha(Q^+)\rangle,\qquad Q^+=Q\setminus Q^\times,$$6 identifies the “pure monoid interior” ideal

$$I_\alpha=\langle \alpha(Q^+)\rangle,\qquad Q^+=Q\setminus Q^\times,$$7

with $$I_\alpha=\langle \alpha(Q^+)\rangle,\qquad Q^+=Q\setminus Q^\times,$$8. In equal characteristic this is deduced from the completion

$$I_\alpha=\langle \alpha(Q^+)\rangle,\qquad Q^+=Q\setminus Q^\times,$$9

and the toric formulas for canonical modules; in mixed characteristic the same ideal remains a canonical module after quotient by the element $R/I_\alpha$0 with constant term $R/I_\alpha$1 (Ishiro, 2022).

This formula yields a sharp Gorenstein criterion. The following are equivalent: $R/I_\alpha$2 is Gorenstein; for some field $R/I_\alpha$3, the semigroup ring $R/I_\alpha$4 is Gorenstein; and there exists $R/I_\alpha$5 such that

$R/I_\alpha$6

In dimension two, if $R/I_\alpha$7 is two-dimensional, then $R/I_\alpha$8 is Gorenstein if and only if

$R/I_\alpha$9

for some $\dim(R)=\dim(R/I_\alpha)+\dim(Q),$0. The corresponding completions are

$\dim(R)=\dim(R/I_\alpha)+\dim(Q),$1

in equal characteristic, and

$\dim(R)=\dim(R/I_\alpha)+\dim(Q),$2

in mixed characteristic, again with $\dim(R)=\dim(R/I_\alpha)+\dim(Q),$3 of constant term $\dim(R)=\dim(R/I_\alpha)+\dim(Q),$4 (Ishiro, 2022).

The divisor class group is governed just as explicitly by the monoid. If $\dim(R)=\dim(R/I_\alpha)+\dim(Q),$5 is finitely generated, cancellative, and root closed, then it is a Krull monoid, its divisorial ideals form a free abelian group on the finite set of height-one prime ideals of $\dim(R)=\dim(R/I_\alpha)+\dim(Q),$6, and the chart induces an isomorphism

$\dim(R)=\dim(R/I_\alpha)+\dim(Q),$7

Hence $\dim(R)=\dim(R/I_\alpha)+\dim(Q),$8 is finitely generated. Combined with Chouinard’s theorem, this gives

$\dim(R)=\dim(R/I_\alpha)+\dim(Q),$9

which is the local log-regular analogue of the toric description of class groups by monomial prime divisors and principal relations (Ishiro, 2022).

Concrete examples underscore the scope of the theory. Jungian domains furnish two-dimensional normal local domains carrying local log-regular structures from explicit monoids. There are also mixed-characteristic non-Gorenstein examples such as

$\overline Q:=Q/Q^\times$0

with canonical module generated by $\overline Q:=Q/Q^\times$1, and examples with

$\overline Q:=Q/Q^\times$2

for which $\overline Q:=Q/Q^\times$3 (Ishiro, 2022).

4. Two-dimensional local theory and minimal desingularization

In dimension two, log regularity admits a geometric criterion in terms of the minimal desingularization. Let $\overline Q:=Q/Q^\times$4 be a two-dimensional Noetherian normal local domain, $\overline Q:=Q/Q^\times$5, and $\overline Q:=Q/Q^\times$6 a reduced closed subscheme of dimension $\overline Q:=Q/Q^\times$7. If $\overline Q:=Q/Q^\times$8 is a desingularization with exceptional divisor $\overline Q:=Q/Q^\times$9 and $R$00 the strict transform of $R$01, then the conditions (Exc) and (Str) characterize when the pair $R$02 is toric, hence when the associated log scheme is log regular. Condition (Exc) requires that $R$03 be a connected normal crossing divisor with a chain decomposition

$R$04

each $R$05, self-intersections $R$06, consecutive components meeting transversely in a single $R$07-rational point, and nonconsecutive components disjoint. Condition (Str) requires that $R$08 have normal crossings and that $R$09 meet only the end components $R$10 and $R$11 (Nagamachi, 2022).

The main local theorem states that for such a two-dimensional local domain the following are equivalent: $R$12 is a toric pair; either $R$13 is regular and $R$14 is a normal crossing divisor, or $R$15 is singular and some desingularization satisfies (Exc) and (Str). In the Zariski log regular case, the sharper forms (Excz) and (Str2) appear, and the criterion can be stated as the existence of a chain of rational curves with the boundary meeting the two ends (Nagamachi, 2022).

These conditions yield a detailed cohomological and intersection-theoretic package. Under (Exc) or its variants, the exceptional divisor is the fundamental cycle, the singularity is rational, and

$R$16

The monoid of functions invertible off $R$17 is fs and sharp of rank $R$18, and it can be described by valuation vectors $R$19 subject to the linear relations

$R$20

This ties the local logarithmic monoid directly to the intersection matrix of the exceptional chain (Nagamachi, 2022).

The two-dimensional local theory is also the local engine behind the global theory of minimal log regular models of curves over discrete valuation fields. Contractibility criteria for divisors in log regular models are expressed in terms of local versions of (Exc) and (Str), leading to smallest minimal log regular models and, in the hyperbolic case, to the equivalence between stable reduction of the curve and stable reduction of its Jacobian without assuming an algebraically closed residue field (Nagamachi, 2022).

5. Alternative criteria and canonical constructions

A distinct positive-characteristic characterization is given by logarithmic Frobenius–Witt differentials. For a prelog ring $R$21 and an $R$22-module $R$23, a log FW-derivation is a pair $R$24 where $R$25 is an FW-derivation, $R$26 is a monoid homomorphism, and

$R$27

The representing module is denoted $R$28, and its reduction modulo $R$29 is

$R$30

If $R$31 is a local prelog ring with residue field $R$32 of characteristic $R$33, $R$34 is $R$35-finite with $R$36, $R$37 is integral, and $R$38 is fine and saturated, then

$R$39

is equivalent to log regularity. If $R$40 is $R$41-finite, this is also equivalent to $R$42 being a free $R$43-module of rank $R$44. In the sharp case the key splitting is

$R$45

so the FW criterion packages the regular quotient and the monoidal rank in a single module (Takeuchi, 19 Apr 2026).

A second major source of examples comes from monoid algebras over regular rings. Let $R$46 be cancellative, finitely generated, root closed, and with torsion-free $R$47, let $R$48 be a regular ring, and put $R$49. For any prime ideal $R$50 with contraction $R$51, the localization

$R$52

is a local log-regular ring. At a point corresponding to $R$53, where $R$54 is the face complementary to the prime ideal in the monoid, the characteristic monoid is

$R$55

and

$R$56

The dimension formula becomes

$R$57

which is exactly Kato’s criterion. This provides a broad class of non-complete local log-regular rings and shows that localizations of monoid algebras over regular bases are automatically within the theory (Ishiro, 1 Sep 2025).

Together, these developments show that log regularity can be detected either by toroidal completion data, by differential modules in positive characteristic, or by explicit monoid algebra constructions over regular bases.

6. Mixed characteristic, perfectoid towers, and arithmetic consequences

Perfectoid methods provide a mixed-characteristic approach to invariants of local log-regular rings. For a complete local log-regular ring $R$58 with perfect residue field and $R$59 fs sharp saturated, one constructs a perfectoid tower by adjoining compatible $R$60-th roots of the regular parameters of $R$61 and of the monoid elements. The resulting rings $R$62 again carry local log-regular structures $R$63, and the inverse perfection of the tower yields a tilt preserving principal ideals, torsion behavior, Noetherianness, module-finiteness properties, and, under regularity hypotheses on the distinguished generator, dimension (Ishiro et al., 2022).

This tower formalism supports an étale cohomology comparison under tilting. Combined with the Kummer exact sequence

$R$64

it gives control over Picard groups of punctured spectra and hence over divisor class groups. The main arithmetic consequence is a finiteness theorem for prime-to-$R$65 torsion in divisor class groups: if $R$66 is a local log-regular ring of mixed characteristic with perfect residue field $R$67 of characteristic $R$68, and if the completion of the strict henselization $R$69 is locally factorial, then for every prime $R$70 the subgroup $R$71 is finite and vanishes for almost all such $R$72 (Ishiro et al., 2022).

The positive-characteristic side of this argument is governed by the tilt. The tilt of the perfectoid tower is $R$73-finite and log-regular in characteristic $R$74, so finiteness results for torsion in class groups on the tilted side can be transported back to mixed characteristic. In this way, local log-regular rings behave as a class for which toric-style divisor theory survives across the mixed-characteristic divide (Ishiro et al., 2022).

These arithmetic applications reinforce a broader perspective. Local log-regular rings are not merely formal analogues of affine toric singularities: they form a robust class of normal Cohen–Macaulay local domains, stable under logarithmic and homological operations, accessible through monoid charts, and rich enough to support canonical module formulas, class-group computations, two-dimensional desingularization criteria, and perfectoid comparison arguments.