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Let R be a commutative Noetherian domain and A an integral domain containing R. For a prime ideal P in R, we denote by $k(P)$ the residue field $R_p/PR_p$ of the local ring $R_p$. Then the ring $k(P) \otimes_R A$ is called the fibre of A over R at P. Let $\Delta$ be a subset of ${\rm Spec }(R)$, and suppose that the structure of fibre $k(P)\otimes_R A$ is given for every P in
$\Delta$. Then what can we say about the structure of A as an R-algebra? This is an interesting and important problem of commutative algebra. In this talk I shall present some results regarding this problem focussed on finite generation of A over R.
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