On t - Spec( R [[ X ]])

Source abstract

Abstract Let D be an integral domain, and let X be an analytic indeterminate. As usual, if I is an ideal of D , set I t = ∪{ J V = ( J -1 ) -1 | J is a nonzero finitely generated subideal of I }; this defines the t -operation, a particularly useful star-operation on D . We discuss the t -operation on R [[ X ]], paying particular attention to the relation between t - dim( R ) and t - dim( R [[ X ]]). We show that if P is a t -prime of R , then P [[ X ]] contains a t -prime which contracts to P in R , and we note that this does not quite suffice to show that t - dim( R [[ X ]]) ≥ t - dim( R ) in general. If R is Noetherian, it is easy to see that t - dim( R [[ X ]]) ≥ t - dim( R ), and we show that we have equality in the case of t -dimension 1. We also observe that if V is a valuation domain, then t -dim( V [[ X ]]) ≥ t - dim( V ), and we give examples to show that the inequality can be strict. Finally, we prove that if V is a finite-dimensional valuation domain with maximal ideal M , then MV [[ X ]] is a maximal t -ideal of V [[ X ]].