2015/12/11 by William Heinzer, Heinzer, William, Mee-Kyoung Kim +3
Mathematics · #13A18 #13C05 (Primary) 13E05 #13H05 #13H15 (Secondary) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #msc:13A18 #msc:13C05 #msc:13E05 #msc:13H05 #msc:13H15
paper · pdf · doi:10.48550/arxiv.1512.03848
35 pages, comments welcome
arxiv created 2016/01/02 · arxiv updated 2016/01/05
We consider the directed union S of an infinite sequence (Rn, mn) of successive local quadratic transforms of a regular local ring (R, m). If dim R = 2, Abhyankar proves that S is a valuation ring. If dim R > 2, Shannon gives necessary and sufficient conditions for S to be a rank 1 valuation domain and Granja gives necessary and sufficient conditions that S be a rank 2 rational rank 2 valuation domain. Granja observes that these are the only cases where S is a valuation domain. If the sequence is along a rank 1 valuation ring V with valuation v, Granja, Martinez, and Rodriguez show that if the infinite sum of the values v(mn) diverges, then S = V. We prove that this infinite sum is finite if V has rational rank at least 2. We present an example of a sequence whose union S is a rank 2 valuation domain, but whose value group is not Z2. We also consider sequences of monomial local quadratic transforms and give necessary and sufficient conditions that the union be a rank 1 valuation domain. If it is, it has rational rank d. We string together finite sequences of monomial local quadratic transforms to construct examples where S is a rank 1 valuation domain with rational rank < d.