Henselian Local Rings: Around a Work in Progress
نویسندگان
چکیده
An introduction to the theme of local rings and Henselian local rings is given through numerous examples. We outline an elementary and effective construction of the Henselization of a local ring and an effective proof of about a classical result in Henselian local rings. This paper anounces a joint work of the three authors.
منابع مشابه
Elementary constructive theory of Henselian local rings
We give an elementary theory of Henselian local rings and construct the Henselization of a local ring. All our theorems have an algorithmic content.
متن کاملSums of squares in excellent henselian local rings
This is a survey about the representation of positive semidefinite elements as sums of squares from a local viewpoint, in the sense of analytic geometry. Here, we present the most recent results in the more general frame of excellent henselian local rings with real closed residue field. 1
متن کاملStrong cleanness of matrix rings over commutative rings
Let R be a commutative local ring. It is proved that R is Henselian if and only if each R-algebra which is a direct limit of module finite R-algebras is strongly clean. So, the matrix ring Mn(R) is strongly clean for each integer n > 0 if R is Henselian and we show that the converse holds if either the residue class field of R is algebraically closed or R is an integrally closed domain or R is ...
متن کاملDefinable Henselian Valuation Rings
We give model theoretic criteria for the existence of∃∀ and∀∃formulas in the ring language to define uniformly the valuation rings O of models (K,O) of an elementary theory Σ of henselian valued fields. As one of the applications we obtain the existence of an ∃∀-formula defining uniformly the valuation rings O of valued henselian fields (K,O) whose residue class field k is finite, pseudofinite,...
متن کاملQuadratic Forms Over Fraction Fields of Two-dimensional Henselian Rings and Brauer Groups of Related Schemes
Let A be an excellent henselian two-dimensional local domain (for the definition of excellent rings, see [EGA IV2, 7.8.2]). Let K be its field of fractions and k its residue field. Assume that k is separably closed (of arbitrary characteristic). We show that the unramified Brauer group of K (with respect to all rank 1 discrete valuations of K) is trivial. Under some more restrictive conditions ...
متن کامل