The aim of this paper is to generalize the‎‎notion of pseudo-almost valuation domains to arbitrary‎ ‎commutative rings‎. ‎It is shown that the classes of chained rings‎ ‎and pseudo-valuation rings are properly contained in the class of‎ ‎pseudo-almost valuation rings; also the class of pseudo-almost‎ ‎valuation rings is properly contained in the class of quasi-local‎ ‎rings with linearly ordere...

We prove an analogue of Lowrey--Sch\"urg's algebraic Spivak's theorem when working over a base ring $A$ that is either field or nice enough discrete valuation ring, and after inverting the residual characteristic exponent $e$ in coefficients. By this result bordism groups quasi-projective derived $A$-schemes can be generated by classical cycles, leading to vanishing results for low degree $e$-i...

We provide an irreducibility test in the ring $$\mathbb{K}[[x]][y]$$ whose complexity is quasi-linear with respect to discriminant valuation, assuming input polynomial $$F$$ square-free and $$\mathbb{K}$$ a perfect field of characteristic not dividing $$\deg(F)$$ . The algorithm uses theory approximate roots may be seen as generalisation Abhyankar's criterion case non-algebraically closed resid...

The idea is simple: we want to develop a theory of analytic manifolds and spaces over fields equipped with an arbitrary complete valuation. Of course, it is a standard fact that such a field must be either R, C, or a field with a nonarchimedean valuation, so what we really mean is that we want to develop a theory of nonarchimedean analytic spaces. Doing this näıvely (i.e., defining manifolds in...

We consider the Whitehead problem for principal ideal domains of large size. It is proved, in ZFC, that some p.i.d.’s of size ≥ א2 have nonfree Whitehead modules even though they are not complete discrete valuation rings. A module M is a Whitehead module if ExtR(M,R) = 0. The second author proved that the problem of whether every Whitehead Z-module is free is independent of ZFC + GCH (cf. [5], ...

It is proved that if R is a valuation domain with maximal ideal P and if RL is countably generated for each prime ideal L, then R R is separable if and only RJ is maximal, where J = ∩n∈NP . When R is a valuation domain satisfying one of the following two conditions: (1) R is almost maximal and its quotient field Q is countably generated (2) R is archimedean Franzen proved in [2] that R is separ...

Let R be a real closed field and A = R[x1, . . . , xn]. Let Sper A denote the real spectrum of A. There are two kinds of points in Sper A: finite points (those for which all of |x1|,. . . ,|xn| are bounded above by some constant in R) and points at infinity. In this paper we study the structure of the set of points at infinity of Sper A and their associated valuations. Let T be a subset of {1, ...