On the n-perfectness under split nilpotent extensions

Abstract

Let $\pi:S\twoheadrightarrow R$ be a split ring epimorphism with nilpotent kernel $J$, and let $R\hookrightarrow S$ be a fixed ring-theoretic section. We prove a change-of-rings formula which is considerably stronger than an equality of global invariants: if $X$ is a left $S$-module satisfying $\Tor_i^S(R,X)=0$ for every $i>0$, then \[ \pd_S X=\pd_R(R\otimes_S X) \] in $\mathbb N\cup\{0,\infty\}$. In particular, for every flat left $S$-module $F$, \[ \pd_SF=\pd_R(F/JF), \] and $F/JF$ is flat over $R$. Reduction and induction of scalars therefore show that the same numerical set of projective dimensions is attained by flat modules over $S$ and over $R$. Consequently, left $n$-perfectness is invariant under every split-by-nilpotent extension. The right-handed analogue holds as well. For a commutative ring $R$ and an arbitrary $R$-module $M$, this applies to the idealization $S=R\ltimes M$ without any projectivity, flatness, finite generation, or finite flat-dimension assumption on $M$. We obtain a list of equivalent characterizations of the condition that $R\ltimes M$ is $n$-perfect, together with \[ \cotD(R\ltimes M)=\cotD(R). \] We further derive exact preservation of the flat-projective-dimension spectrum, strict $n$-perfectness, pointwise localization, arbitrary iterated idealizations, $n$-trivial extensions, truncated polynomial rings, and upper triangular matrix rings.