Given that $R\to S$ is a flat local ring homomorphism of two Noetherian local rings. Then is $S$ always a finitely generated $R$-module?
This question stems from a small detail in a proof I am currently reading, which asserts that given the above hypothesis and $Y$ is a finitely generated $R$-module, then they conclude that $S\otimes_R Y$ is a finitely generated $R$-module. This is why I have the above question.
Can you explain this for me? Thank you
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