Colloquium abstracts

N. Onoda
Fukui University, Japan
March 24, 2011

Some results on finite generation of algebras of certain type:  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 the fibre $k(P)\otimes_RA$ 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 focused on finite generation of $A$ over $R$.