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We extend some major theorems in commutative algebra to the class
of modules that are not necessarily finitely generated. The
novelty of our extension is that the hypothesis of finite
generation over $R$ is replaced by one over $S$, where $R$ and $S$
are commutative Noetherian local rings and there is a local
homomorphism $\varphi: R\rightarrow S$. Among the results that we
extend are: Grothendieck's Non-Vanishing Theorem, Intersection
Theorem and Intersection dimension formula.
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