Cohen structure theorem

Abstract: A relative form of Gersten's Conjecture is established for a ring of formal power series over a complete discrete valuation ring. The main corol- laries are that the absolute version of Gersten's Conjecture is valid for such a ring if it is valid for arbitrary discrete valuation rings, and, consequently, that the conjecture is true for such a ring if we use K-theory with finite coefficients of order prime to the characteristic of the residue field. In (GL) , Gillet and Levine established the relative version of Gersten's Con- jecture for a regular local ring essentially of finite type and smooth over a dis- crete valuation ring (DVR). The main consequence is that the absolute version of Gersten's Conjecture is valid for such a ring if it is true for DVRs. These results were a relative analog of Quillen's theorem on the absolute version of the Conjecture for regular local rings essentially of finite type and smooth over a field (from which Quillen then deduced the Conjecture for all regular local rings essentially of finite type over a field (Q, Theorem 7.5.1 1)). Quillen also proved the absolute version of the Conjecture for a ring of formal power series over a field (Q, Theorem 5.13) (in other words, by the Cohen Structure Theorem, the complete equicharacteristic case). In this note we show that the analogs of Gillet and Levine's results are valid for a ring of formal power series over a complete DVR. (By the Cohen Structure Theorem, in the unequal characteristic situation this includes the case of a complete unramified regular local ring.) The proof bears the same relation to Quillen's as Gillet and Levine's proof bears to the proof of Quillen's geometric result. We will adopt the following notation: R will denote a complete DVR, with maximal ideal generated by 7, and residue field k. We will put A =

Abstract: In this note we will make a few observations on the structure of fields and local rings. The main point is to show that a weaker version of Cohen structure theorem for complete local rings holds for any (not necessarily complete) local ring. The consideration of non-complete case makes the meaning of Cohen’s theorem itself clearer. Moreover, quasi-coefficient fields (or rings) are handy when we consider derivations of a local ring.