发表机构
School of Mathematics and Statistics, Tianshui Normal University（天水师范学院数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明分裂幂零扩张下投射维数换环公式，从而得出左 n-完美性在该扩张下不变，并应用于理想化环，给出等价刻画及平坦-投射维数谱等性质的精确保持。
AI 中文摘要
设 $\pi:S\twoheadrightarrow R$ 为具有幂零核 $J$ 的分裂环满态射，并设 $R\hookrightarrow S$ 为固定的环论截面。我们证明了一个比全局不变量等式强得多的换环公式：若 $X$ 是满足对每个 $i>0$ 有 $\Tor_i^S(R,X)=0$ 的左 $S$-模，则 $\pd_S X=\pd_R(R\otimes_S X)$ 在 $\mathbb N\cup\{0,\infty\}$ 中成立。特别地，对每个平坦左 $S$-模 $F$，有 $\pd_SF=\pd_R(F/JF)$，且 $F/JF$ 在 $R$ 上平坦。因此，标量的限制与诱导表明，$S$ 与 $R$ 上的平坦模所达到的投射维数的数值集合相同。由此，左 $n$-完美性在每次分裂幂零扩张下保持不变。右手的类似结论同样成立。对于交换环 $R$ 与任意 $R$-模 $M$，此结果适用于理想化 $S=R\ltimes M$，且对 $M$ 无需任何投射性、平坦性、有限生成或有限平坦维数假设。我们得到了 $R\ltimes M$ 为 $n$-完美的等价刻画列表，以及 $\cotD(R\ltimes M)=\cotD(R)$。我们进一步推导出平坦-投射维数谱的精确保持、严格 $n$-完美性、逐点局部化、任意迭代理想化、$n$-平凡扩张、截断多项式环以及上三角矩阵环的相应结果。
英文摘要
Let $π:S\twoheadrightarrow R$ 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.
