A precise formula is derived for
the (noncommutative) Krull dimension of a skew-Laurent extension R[𝜃1±1,…,𝜃u±1],
where R is a commutative noetherian ring of finite Krull dimension, equipped
with u commuting automorphisms σ1,…,σu. The formula is given in terms
of heights and automorphian dimensions of prime ideals of R, where the
automorphian dimension of a prime ideal P is a positive integer that measures
the invariance of P relative to products of powers of the σi. As part of the
development of this formula, the Krull dimension of a skew-Laurent extension
R[𝜃1±1] over a right noetherian ring R of finite right Krull dimension is
determined. Also, some partial results are obtained for an iterated skew-Laurent
extension R[𝜃1±1,…,𝜃u±1] over a right noetherian ring R of finite right Krull
dimension. In particular, a criterion is derived that indicates when such an
iterated skew-Laurent extension can achieve the maximum possible Krull
dimension.
