This note investigates two long-standing conjectures on the Krull dimension of integer-valued polynomial rings and of polynomial rings, respectively, in the context of (locally) essential domains.

— We prove a theorem of diophantine approximation between the field of formal power series in several variables and its completion for the Krull topology.

For any field F , we give an example of a prime finitely generated F -algebra of GK dimension 2 for which GK dimension is not finitely partitive. For each m ∈ {0, 1, 2, 3, . . .}∪{∞}, we give an example of a prime finitely generated F -algebra having GK dimension 2 which has classical Krull dimension m; the algebra of GK dimension 2 which has infinite classical Krull dimension has the additiona...

Let Γ denote a group of finite virtual cohomological dimension and p a prime. If the cohomology ring H∗(Γ;Fp) has Krull dimension one, the p-period of Γ is defined; it measures the periodicity of H∗(Γ;Fp) in degrees above the virtual cohomological dimension of Γ. The Yagita invariant p(Γ) of Γ is a natural generalization of the p-period to groups with H∗(Γ;Fp) of Krull dimension larger than one...

We investigate the class groups and distribution of prime divisors in affine monoid algebras over fields thereby extend result Kainrath that every finitely generated integral algebra Krull dimension at least 2 an infinite field has infinitely many all classes.

This note investigates two long-standing conjectures on the Krull dimension of integer-valued polynomial rings and of polynomial rings, respectively , in the context of (locally) essential domains.