Finitistic dimensions of formal power series rings

Abstract

We consider the big finitistic projective and flat dimensions of a commutative ring and of its formal power series ring. When the ring is Noetherian, both invariants increase by one. Coherence of the power series ring does not force either equality to hold. For every \(n\ge0\) and every \(d\) with \(n+1\le d\le\infty\), the flat pair \((n,d)\) can be realized. The finite projective pairs are realized when \(n\ge1\), and also for \((0,1)\). For every \(n\ge0\), one family realizes \((n,\infty)\) for both invariants at the same time. The seed is a zero-dimensional perfect local ring. Its power series ring has countably presented modules whose flat dimension is \(r\) and whose projective dimension lies between \(r\) and \(r+1\), for every \(r\ge1\). The remaining finite projective case is controlled by a dichotomy for perfect coefficient rings. If the Jacobson radical is nilpotent, the power series value is one; if it is not nilpotent, there is a module of projective dimension exactly three. In particular, the pair \((0,2)\) is impossible. For one fixed countable perfect ring, the power series ring has finitistic flat dimension one. Its finitistic projective dimension is \(m+2\) when \(2^{\aleph_0}=\aleph_m\) with \(1\le m<\infty\), and is infinite when \(2^{\aleph_0}>\aleph_ω\).