Construction B, I, D et anneaux localement ou residuellement de Jaffard. (B, I, D construction and locally or residually Jaffard rings)
Let R be a commutative ring with unit, of finite Krull dimension. Then one has inequalities \(n+\dim (R)\leq \dim (R[\underline X])\leq n\cdot \dim (R)+n+1\) \((\dim=\) Krull dimension, \(\underline X=(X_ 1,...,X_ n)\) a set of n indeterminates). The ring R is called a Jaffard ring if \(\dim(R[\underline b])=n+\dim(R)\) for all \(n\geq 1\). The ring R is locally Jaffard if\(R_ P\) is Jaffard for all prime ideals P of R, R is residually Jaffard if \(R/P\) is Jaffard for all primes P of R, and totally Jaffard if every localization of R is residually Jaffard. One has implications: Totally Jaffard \(\Rightarrow\) Locally Jaffard \(\Rightarrow\) Jaffard and: Totally Jaffard \(\Rightarrow\) Residually Jaffard \(\Rightarrow\) Jaffard. There are also implications relating the various versions of Jaffard to the concept ``S-strong'' (``S-fort'' in French). The author proves that these properties are independent, except for the implications listed above, by producing examples satisfying all permitted combinations of these properties. The method is as follows: Let K be a field. Then one can produce a semilocal K-algebra B with maximal ideals \(Q_ i\) of arbitrarily prescribed heights by localizing a suitable K-algebra of finite type. Let I be the intersection of the maximal ideals of B. Then \(B_ I\) is the product of fields \(B/Q_ i\). By suitably choosing K (\(=D)\) one can produce inclusions \(D\to B/Q_ i\) of arbitrarily prescribed transcendence degree. These induce an inclusion \(D\to B/I\). One defines A to be the Cartesian product \[ \begin{matrix} A & \rightarrow & B \\ \downarrow & & \downarrow \\ D & \rightarrow & B/I\end{matrix} \] The various Jaffard and strong properties of A were characterized earlier in the paper in terms of the heights of the \(Q_ i\) and the transcendence degrees \([B/Q_ i:D]\), yielding the desired examples.
scientific article; zbMATH DE number 4160899
scientific article