相对弗罗贝尼乌斯映射的Betti数与Bass数
Betti and Bass numbers of relative Frobenius maps
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中文总结 AI 辅助
该研究在局部环同态满足适当假设下，得到同调有限复形相对弗罗贝尼乌斯推前的Betti数与Bass数的增长界，将Betti数研究扩展到系数情形并通过Bass数提供对偶视角，助力环同态信息的探究。
中文摘要 AI 辅助
受涉及Betti数增长的Kunz定理改进结果的启发，我们研究与相对弗罗贝尼乌斯映射相关的Betti数和Bass数。在局部环同态φ:R→S满足适当假设的情况下，我们根据闭纤维S/mS的剩余域的对应不变量，得到同调有限复形的相对弗罗贝尼乌斯推前的Betti数与Bass数的增长界。这些结果将此前关于Betti数的研究扩展到了系数情形，并通过Bass数提供了对偶视角，为探究弗罗贝尼乌斯作用如何编码环同态信息的研究作出了贡献。
英文摘要
Motivated by refinements of Kunz's theorem involving the growth of Betti numbers, we study Betti and Bass numbers associated to the relative Frobenius. Under appropriate hypotheses on a local ring homomorphism $φ\colon R \to S$, we obtain bounds on the growth of the Betti and Bass numbers of relative Frobenius pushforwards of homologically finite complexes in terms of the corresponding invariants of the residue field of the closed fiber $S/\mathfrak m S$. These results extend previous work on Betti numbers to coefficients and provide a dual perspective via Bass numbers, contributing to the study of how Frobenius actions encode information about ring homomorphisms.
