Assume $φ : R\to S$ is a ring homomorphism between commutative rings sending unity to unity. Taking any $S$-module $M$ as an $R$-module, is it true that always $$\operatorname{pd}_R (M)\le \operatorname{pd}_R (S)+\operatorname{pd}_S (M)?$$ (Here by $\operatorname{pd}$ I mean projective dimension.)
2026-04-02 14:33:40.1775140420
Inequality amongst projective dimensions!?
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See Weibel, "An Introduction to Homological Algebra", Theorem 4.3.1.