曲面及算术曲面上的赋值
摘要
赋值理论具有很长的历史，他在很多领域都有重要的应用。赋值是许多代数对象的研究工具。令K为k的有限生成扩域，tr．deg(K／k)=n<+∞，v是K的一个k-赋值。如果n=1，则K是k上曲线C的函数域，v可以由C的一个素除子来定义，我们称它为除子类型。每个除子类型的赋值都是离散的。我们把它推广到高维(即n≥2)情形，它对K-群的计算以及刚性解析空间的研究都有帮助。
本论文主要研究n=2的情形，它比n=1的情形要复杂得多。我们定义了赋值的高度以及单项式ax~sy~n，s∈Q，n∈N，a∈k的(?)-次数，其中(?)(x)=sum from i=1 to +∞a_ix~(r_i),a_i∈k，r_i∈Q，0 一个平行的问题是算术曲面上赋值的分类，也就是对Q的超越度为1的有限生成扩域上的赋值的分类。在本论文中，我们给出了赋值的高度的定义以及大域C_(p,G)的定义，其中p为素数，G(?)R为包含1的加法子群。我们得出C_(p,G)是一个域并且C_(p,Q)是代数闭的。从而得到算术曲面上赋值的完整分类。进一步地，对任意m≤n∈Z，令V_(m,n)为n-m+1维R-向量空间，坐标指数从m到n。我们推广C_(p,G)的定义，使得其中p为素数，G(?)V_(m,n)为包含1的加法子群。我们得出如果m≤0≤n，则C_(p,G)是一个域。
The valuation theory has a long history. It has important applications in various areas. Valuation is a researching tool for many algebraic objects. Let K be a finitely generated field extension of k with tr.deg(K/k) = n <+∞, v be a k-valuation ofK. When n = 1, it is well known that K is the function field of a curve C over k,then v can be defined by a prime divisor of C. We call it the divisor type. Each valuation of divisor type is discrete. There have been attempts to generalize this method to higher dimensional (i.e. n≥2) cases. It is helpful to the calculations of K-groups and the study of rigid analytic spaces.
In this paper we study the case when n = 2, it is much more difficult than the casewhen n = 1. We define the height of a valuation and the (?) -degree of a monomialax~sy~n, s∈Q, n∈N, a∈k , for any given formal series(?)(x) = sum from i=1 to +∞a_ix~(r_i),a_i∈k,r_i∈Q, 0 < r_1 < r_2 <…, (?)r_i= r < +∞. Based on this the author obtains thecomplete classification of k -valuations on surfaces. In addition, we get the relationship between the valuation and transcendental series. Furthermore, we show that all the nontrivial k -valuations can be given by the infinite sequences of blowing-ups and give the process of blowing-ups.
A parallel problem is the classification of valuations on arithmetic surfaces, i.e. theclassification of all the valuations of a finitely generated field extension of Q withtranscendental degree 1.
In this paper, we give the definition of the height of a valuation and the definition ofthe big field C_(p,G), where p is a prime and G(?)R is an additive subgroupcontaining 1. We conclude that C_(p,G) is a field and C_(p,G) is algebraically closed. Based on this the author obtains the complete classification of valuations on arithmetic surfaces. Furthermore, for any m≤n∈Z, let V_(m,n) be an R -vector spaceof dimension n-m + 1 , whose coordinates are indexed from m to n . We generalize the definition of C_(p,G), where p is a prime and G(?)V_(m,n) is an additivesubgroup containing 1. We also conclude that C_(p,G) is a field if m≤0≤n.
本论文主要研究n=2的情形，它比n=1的情形要复杂得多。我们定义了赋值的高度以及单项式ax~sy~n，s∈Q，n∈N，a∈k的(?)-次数，其中(?)(x)=sum from i=1 to +∞a_ix~(r_i),a_i∈k，r_i∈Q，0
The valuation theory has a long history. It has important applications in various areas. Valuation is a researching tool for many algebraic objects. Let K be a finitely generated field extension of k with tr.deg(K/k) = n <+∞, v be a k-valuation ofK. When n = 1, it is well known that K is the function field of a curve C over k,then v can be defined by a prime divisor of C. We call it the divisor type. Each valuation of divisor type is discrete. There have been attempts to generalize this method to higher dimensional (i.e. n≥2) cases. It is helpful to the calculations of K-groups and the study of rigid analytic spaces.
In this paper we study the case when n = 2, it is much more difficult than the casewhen n = 1. We define the height of a valuation and the (?) -degree of a monomialax~sy~n, s∈Q, n∈N, a∈k , for any given formal series(?)(x) = sum from i=1 to +∞a_ix~(r_i),a_i∈k,r_i∈Q, 0 < r_1 < r_2 <…, (?)r_i= r < +∞. Based on this the author obtains thecomplete classification of k -valuations on surfaces. In addition, we get the relationship between the valuation and transcendental series. Furthermore, we show that all the nontrivial k -valuations can be given by the infinite sequences of blowing-ups and give the process of blowing-ups.
A parallel problem is the classification of valuations on arithmetic surfaces, i.e. theclassification of all the valuations of a finitely generated field extension of Q withtranscendental degree 1.
In this paper, we give the definition of the height of a valuation and the definition ofthe big field C_(p,G), where p is a prime and G(?)R is an additive subgroupcontaining 1. We conclude that C_(p,G) is a field and C_(p,G) is algebraically closed. Based on this the author obtains the complete classification of valuations on arithmetic surfaces. Furthermore, for any m≤n∈Z, let V_(m,n) be an R -vector spaceof dimension n-m + 1 , whose coordinates are indexed from m to n . We generalize the definition of C_(p,G), where p is a prime and G(?)V_(m,n) is an additivesubgroup containing 1. We also conclude that C_(p,G) is a field if m≤0≤n.
引文
[1] F.J.Herrera, M.A.Olalla, J.L.Vicente. Valuations in fields of power series. Rev. Mat. Iberoamericana 19, 2003: 467-482. projecteuclid.org
[2] F.J.Herrera Govantes, M.A.Olalla Acosta, J.L.Vicente Cordoba. Rank one discrete valuations of k((X_1,…X_n)). Comm. Algebra, 2007(35) arXiv:math/0605225v3
[3] M.A.Olalla-Acosta. On the dimension of discrtete valuations of k((X_1,…X_n)).Comptes Rendus de l'Academie des Sciences Series I Mathematics, Volume 333, Number 1, 1 July 2001(6): 27-32
[4]李克正．代数几何初步．第一版．北京：科学出版社，2004
[5]李克正．交换代数和同调代数．第一版．北京：科学出版社，1998
[6]莫宗坚，蓝以中，赵春来等．代数学下册．北京：北京大学出版社，1986，92-107
[7] O.Endler. Valuation theory. Berlin, New York: Springer-Verlag, 1972, 81-82
[8] P.Roquette. History of valuation theory Part I. Fields Institute communicationseries, 2002-rzuser.uni-heidelberg.de
[9] C.Favre, M.Jonsson. The valuative tree (LNM1853). Berlin Heidelberg,Springer-Verlag, 2004: 9-24
[10]李克正．Lectures on moduli theory. Morning Centre Lecture Series, 2001
[11] B.Green. Automorphisms of formal power series rings over a valuation ring. Fields Institute Communications, International Conference and Workshop on Valuation Theory, July 26 - August 11, 1999
[12]李克正．Geometric proof of Modell's conjecture for function fields.arxiv.org,math.AG/0701407.http://arxiv.org/abs/math.AG/0701407
[13]李克正．Automorphism group schemes of finite field extensions.Max-Planck-Institut fur Mathematik Preprint Series, 2000(28)
[14] R.Hartshorne. Algerbraic geometry. New York: Springer-Verlag, 1977, 42-46
[15] Franz-Viktor Kuhlmann. Additive polynomials and their role in the model theoryof valued fields. preprint
[16] Steven Dale Cutkosky, Olivier Piltant. Ramifications of valuations. Advances in Mathematics. Volume 183, Issue 1, 20 March 2004: 1-79
[17] Masayoshi Nagata. A theorem on valuation rings and its appilications. Nagoya Math. J. 1967 (29): 85-91
[18] Steven Dale Cutkosky, Laura Ghezzi. Completions of valuation rings. Commutative Algebra, arXiv:math/0310192v1
[19] Paul-Jean Cahen, Evan G.Houston, Thomas G.Lucas. Discrete valuation overrings of noetherian domains. Proceedings of the American Mathematical Society, Vol. 124, No. 6, Jun., 1996: 1719-1721
[20] Reinhold Hubl, Irena Swanson. Discrete valuations centerd on local domains. Journal of pure and applied algebra, 2001 (161): 145-166
[21] Jo-Ann D.Cohen. Extensions of valuation and absolute valued topologies. Pacific J. Math. Volume 125, Number 1, 1986: 39-44
[22] Thomas S.Shores. On generalized valuation rings. Michigan Math. J. Volume 21,Issue 4, 1975: 405-409
[23] Hagen Knaf. On the integral closure of polynomial rings over a valuation domain. Journal of Algebra, Volume 209, Issue 1, 1 November 1998: 108-128
[24] Murray Marshall, Yufei Zhang. Orders, real places, and valuations on noncommutative integral domains. Journal of Algebra, Volume 212, Issue 1, 1 February 1999: 190-207
[25] F.J.Herrera Govantes, M.A.Olalla Acosta, M. Spivakovsky. Valuations in algebraic field extensions. Commutative Algebra, arXiv:math/0605193v1
[26] Igor Zhukov. Explicit abelian extensions of complete discrete valuation fields. Geometry & Topology Monographs, 2000 (3)-Invitaion to higher local fields, Part Ⅰ, section 14: 117-122
[27] M.Kurihara. Abelian extensions of absolutely unramified complete discrete valuation fields. Geometry & Topology Monographs, 2000 (3) - Invitation to higher local fields, Part Ⅰ, section 13: 113-116
[28] J.Herrera, M.A.Olalla, M.Spivakovsky, B.Teissier. Extending a valuation centered in a local domain to the formal completion. inpreparation
[29] Reinhold Hubl. Completions of local morphisms and valuations. Mathematische Zeitschrift, Volume 236, Number 1, 201-214
[30] William Heinzer, Christel Rotthaus, Sylvia Wiegand. Approximating Discrete Valuation Rings by Regular Local Rings. Proceedings of the American Mathematical Society, Vol. 129, No. 1, Jan., 2001: 37-43
[31] Mark Spivakovsky. Valuations in Function Fields of Surfaces. American Journal of Mathematics, Vol. 112, No. 1,Feb., 1990: 107-156
[32] Saunders MacLane. A construction for absolute values in polynomial rings. Transactions of the American Mathematical Society, Vol. 40, No. 3, Dec, 1936: 363-395
[33] Saunders MacLane. A construction for prime ideals as absolute values of an algebraic field. Duke Math. J. Volume 2, Number 3, 1936: 492-510
[34] O.Zariski, P.Samuel. Commutative algebra. Volume II, Springer-Verlag, 1960
[35] Shreeram Abhyankar. On the valuations centered in a domain. American Journal of Mathematics, Vol. 78, No. 2, Apr., 1956: 321-348
[36] O.Zariski. Local uniformization of algebraic varieties. Ann.Math.41, 1940
[37] R.Brown. Real places and ordered fields. Rockey Mountain J.Math 1971 (1): 633-636
[38] T.Craven. Witt rings and orderings on skew fields. J.Algebra. 1982 (77): 74-96
[39] D.Dubois. Infinite primes and ordered fields. Dissertationes Math. 1970 (69): 1-43
[40] D.K.Harrison, Hoyt D.Warner. Infinite primes of fields and completions. Pacific J. Math. Volume 45, Number 1, 1973: 201-216
[41] T.Y.Lam. A first course in noncommutative rings. Graduate Texts in Mathematics, Vol. 131, Springer-Verlag, NewYork/Berlin, 1991
[42] M.Marshall. Spaces of orderings and abstract real spectra. Lecture Notes in Math., Vol. 1636, Springer-Verlag, NewYork/Berlin, 1996
[43] B.Green. On curves over valuation rings and morphisms to (?)~1. J. Number Theory. 1996 (59): 262-290
[44] B.W.Green, M.Matignon, F.Pop. On valued function fields I. Manuscripta Math. 1989 (65): 257-276
[45] B.W.Green, M.Matignon, F.Pop. On valued function fields II, Regular functions and elements with the uniqueness property. J.Reine Angew. Math. 1990 (412): 128-149
[46] B.W.Green, M.Matignon, F.Pop. On valued function fields III, Reductions of algebraic curces. J.Reine Angew. Math. 1992 (432): 117-133
[47] Hagen Knaf. Divisors on varieties over valuation domains. Israel Journal of Mathematics, Volume 119, Number 1, 2000: 349-377
[48] M.Nagata. Finitely generated rings over a valuation ring. J. Math. Kyoto Univ. 1965(5): 163-169
[48] A.H.Clifford. Naturally totally ordered commutative semigroups. American Journal of Mathematics, Vol. 76, No. 3, Jul., 1954: 631-646
[49] Irving Kaplansky. Elementary divisors and modules. Transactions of the American Mathematical Society, Vol. 66, No. 2, Jul., 1949: 464-491
[50] E.J.Tully, Jr. A Class of Naturally Partly Ordered Commutative Archimedean Semi-Groups with Maximal Condition. Proceedings of the American Mathematical Society, Vol. 17, No. 5, Oct., 1966: 1133-1139
[51] T.Rigo, S.Warner. Topologies extending valuations. Canad. J. Math., 1978 (30): 164-169
[52] A.Prestel, M.Ziegler. Model theoretic methods in the theory of topological fields. J. Riene Angew. Math., 1978 (299/300): 318-341
[53] Leopoldo Nachbin. On strictly minimal topological division rings. Bull. Amer. Math. Soc. Volume 55, Number 12 , 1949: 1128-1136
[54] S.V.Vostokov. The pairing on K-groups in fields of valuation of rank n . Trudy S.-Peterb. Mat. Obsch, 1995; English translation in Amer. Math. Soc. Transl. (Sec. 2) 1995(165): 111-148
[55] I.B.Zhukov. Milnor and topological K -groups of multidimensional complete fields. Algebra I Analiz, 1997; English translation in St. Petersburg Math. J. 1998 (9): 69-105
[56] Shreeram Abhyankar. Local uniformization on algebraic surfaces over ground fields of characteristic p ≠ 0. The Annals of Mathematics, 2nd Ser., Vol. 63, No. 3, May, 1956: 491-526
[57] Shreeram Abhyankar. Ramification theoretic methods in algebraic geometry. Annals of Math. Studies 43, Princeton University Press, 1959
[58] B.G.Kang, M.H.Park. A localization of a power series ring over a valuation domain. Journal of Pure and Applied Algebra, Volume 140, Number 2, 28 July 1999: 107-124(18)
[59] Semyon Alesker. Algebraic sturctures on valuations, their properties and applications. Proceedings of the ICM, Beijing 2002(2): 757—764. arXiv:math/0304338vl
[60] Jack Maney. Boundary valuation domains. Journal of Algebra, Volume 273, Issue 1,1 March 2004: 373-383
[61] B.Goldsmith, P.Zanardo. Minimal modules over valuation domains. Journal of Pure and Applied Algebra, Volume 199, Issues 1-3, 1 July 2005: 95-109
[62] L.Fuchs, L.Salce. Modules over valuation domains. Marcel Dekker, NewYork and Basel, 1985
[63] L.Salce, P.Zanardo. On two-generated modules over valuation domains. Archiv der Mathematik, Volume 46, Number 5, 1986: 408-418
[64] L.Salce, P.Zanardo. Rank-two torsion-free modules over valuation domains. Rend. Sem. Mat. Univ. Padova, 1988 (80): 175-201
[65] D.M.Arnold. A duality for torsion-free modules of finite rank over a discrete valuation ring. Proc. London Math. Soc, (3), 1972 (24): 204-216
[66] A.Facchini, P.Zanardo. Discrete valuation domains and ranks of their maximal extensions. Rend. Sem. Mat. Univ. Padova, 1986 (75): 143-156
[67] L.Fuchs. Infinite abelian groups. vol. II, Academic Press, London and New York, 1973
[68] L.Fuchs, L.Salce. Prebasic submodules over valuation rings. Annali Mat. Pura e Appl., 1982 (32): 257-274
[69] L.Fuchs, L.Salce. Modules over valuation domains. Lecture Notes Pure Appl. Math., no. 97, Marcel Dekker, New York, Basel, 1985
[70] L.Fuchs, G.Viljoen. On finite rank torsion-free modules over almost maximal valuation domains. Comm. in Algebra, 1984, 12(3): 245-258
[71] I.Kaplansky. Modules over Dedekind and valuation rings. Trans. Amer. Math. Soc, 1952 (72): 327-340
[72] I.Kaplansky. Infinite abelian groups. Univ. of Michigan Press, Ann Arbor, Michigan, 1969
[73] L.Salce, P.Zanardo. Finitely generated modules over valuation rings. Comm. in Algebra, 1984, 12(15): 1795-1818
[74] L.Salce, P.Zanardo. Some cardinal invariants for valuation domains. Rend. Sem.Mat. Univ. Padova, 1985 (74): 205-217
[75] L.Salce, P.Zanardo. A duality for finitely generated modules over valuation domains. Proceedings Oberwolfach 1985, Conference on Abelian Groups [76] G.Viljoen. On torsion-free modules of rank two over almost maximal valuation domains. Lecture Notes no. 1006, Springer-Verlag, 1983. 607-616
[77] P.Zanardo. On the classification of indecomposable finitely generated modules overvaluation domains. Comm. in Algebra, 1985, 43 (11) : 2473-2491
[78] P.Zanardo. Indecomposable finitely generated modules over valuation domains. Annali Univ. Ferrara - Sc. Mat, 1985 (31): 71-89
[2] F.J.Herrera Govantes, M.A.Olalla Acosta, J.L.Vicente Cordoba. Rank one discrete valuations of k((X_1,…X_n)). Comm. Algebra, 2007(35) arXiv:math/0605225v3
[3] M.A.Olalla-Acosta. On the dimension of discrtete valuations of k((X_1,…X_n)).Comptes Rendus de l'Academie des Sciences Series I Mathematics, Volume 333, Number 1, 1 July 2001(6): 27-32
[4]李克正．代数几何初步．第一版．北京：科学出版社，2004
[5]李克正．交换代数和同调代数．第一版．北京：科学出版社，1998
[6]莫宗坚，蓝以中，赵春来等．代数学下册．北京：北京大学出版社，1986，92-107
[7] O.Endler. Valuation theory. Berlin, New York: Springer-Verlag, 1972, 81-82
[8] P.Roquette. History of valuation theory Part I. Fields Institute communicationseries, 2002-rzuser.uni-heidelberg.de
[9] C.Favre, M.Jonsson. The valuative tree (LNM1853). Berlin Heidelberg,Springer-Verlag, 2004: 9-24
[10]李克正．Lectures on moduli theory. Morning Centre Lecture Series, 2001
[11] B.Green. Automorphisms of formal power series rings over a valuation ring. Fields Institute Communications, International Conference and Workshop on Valuation Theory, July 26 - August 11, 1999
[12]李克正．Geometric proof of Modell's conjecture for function fields.arxiv.org,math.AG/0701407.http://arxiv.org/abs/math.AG/0701407
[13]李克正．Automorphism group schemes of finite field extensions.Max-Planck-Institut fur Mathematik Preprint Series, 2000(28)
[14] R.Hartshorne. Algerbraic geometry. New York: Springer-Verlag, 1977, 42-46
[15] Franz-Viktor Kuhlmann. Additive polynomials and their role in the model theoryof valued fields. preprint
[16] Steven Dale Cutkosky, Olivier Piltant. Ramifications of valuations. Advances in Mathematics. Volume 183, Issue 1, 20 March 2004: 1-79
[17] Masayoshi Nagata. A theorem on valuation rings and its appilications. Nagoya Math. J. 1967 (29): 85-91
[18] Steven Dale Cutkosky, Laura Ghezzi. Completions of valuation rings. Commutative Algebra, arXiv:math/0310192v1
[19] Paul-Jean Cahen, Evan G.Houston, Thomas G.Lucas. Discrete valuation overrings of noetherian domains. Proceedings of the American Mathematical Society, Vol. 124, No. 6, Jun., 1996: 1719-1721
[20] Reinhold Hubl, Irena Swanson. Discrete valuations centerd on local domains. Journal of pure and applied algebra, 2001 (161): 145-166
[21] Jo-Ann D.Cohen. Extensions of valuation and absolute valued topologies. Pacific J. Math. Volume 125, Number 1, 1986: 39-44
[22] Thomas S.Shores. On generalized valuation rings. Michigan Math. J. Volume 21,Issue 4, 1975: 405-409
[23] Hagen Knaf. On the integral closure of polynomial rings over a valuation domain. Journal of Algebra, Volume 209, Issue 1, 1 November 1998: 108-128
[24] Murray Marshall, Yufei Zhang. Orders, real places, and valuations on noncommutative integral domains. Journal of Algebra, Volume 212, Issue 1, 1 February 1999: 190-207
[25] F.J.Herrera Govantes, M.A.Olalla Acosta, M. Spivakovsky. Valuations in algebraic field extensions. Commutative Algebra, arXiv:math/0605193v1
[26] Igor Zhukov. Explicit abelian extensions of complete discrete valuation fields. Geometry & Topology Monographs, 2000 (3)-Invitaion to higher local fields, Part Ⅰ, section 14: 117-122
[27] M.Kurihara. Abelian extensions of absolutely unramified complete discrete valuation fields. Geometry & Topology Monographs, 2000 (3) - Invitation to higher local fields, Part Ⅰ, section 13: 113-116
[28] J.Herrera, M.A.Olalla, M.Spivakovsky, B.Teissier. Extending a valuation centered in a local domain to the formal completion. inpreparation
[29] Reinhold Hubl. Completions of local morphisms and valuations. Mathematische Zeitschrift, Volume 236, Number 1, 201-214
[30] William Heinzer, Christel Rotthaus, Sylvia Wiegand. Approximating Discrete Valuation Rings by Regular Local Rings. Proceedings of the American Mathematical Society, Vol. 129, No. 1, Jan., 2001: 37-43
[31] Mark Spivakovsky. Valuations in Function Fields of Surfaces. American Journal of Mathematics, Vol. 112, No. 1,Feb., 1990: 107-156
[32] Saunders MacLane. A construction for absolute values in polynomial rings. Transactions of the American Mathematical Society, Vol. 40, No. 3, Dec, 1936: 363-395
[33] Saunders MacLane. A construction for prime ideals as absolute values of an algebraic field. Duke Math. J. Volume 2, Number 3, 1936: 492-510
[34] O.Zariski, P.Samuel. Commutative algebra. Volume II, Springer-Verlag, 1960
[35] Shreeram Abhyankar. On the valuations centered in a domain. American Journal of Mathematics, Vol. 78, No. 2, Apr., 1956: 321-348
[36] O.Zariski. Local uniformization of algebraic varieties. Ann.Math.41, 1940
[37] R.Brown. Real places and ordered fields. Rockey Mountain J.Math 1971 (1): 633-636
[38] T.Craven. Witt rings and orderings on skew fields. J.Algebra. 1982 (77): 74-96
[39] D.Dubois. Infinite primes and ordered fields. Dissertationes Math. 1970 (69): 1-43
[40] D.K.Harrison, Hoyt D.Warner. Infinite primes of fields and completions. Pacific J. Math. Volume 45, Number 1, 1973: 201-216
[41] T.Y.Lam. A first course in noncommutative rings. Graduate Texts in Mathematics, Vol. 131, Springer-Verlag, NewYork/Berlin, 1991
[42] M.Marshall. Spaces of orderings and abstract real spectra. Lecture Notes in Math., Vol. 1636, Springer-Verlag, NewYork/Berlin, 1996
[43] B.Green. On curves over valuation rings and morphisms to (?)~1. J. Number Theory. 1996 (59): 262-290
[44] B.W.Green, M.Matignon, F.Pop. On valued function fields I. Manuscripta Math. 1989 (65): 257-276
[45] B.W.Green, M.Matignon, F.Pop. On valued function fields II, Regular functions and elements with the uniqueness property. J.Reine Angew. Math. 1990 (412): 128-149
[46] B.W.Green, M.Matignon, F.Pop. On valued function fields III, Reductions of algebraic curces. J.Reine Angew. Math. 1992 (432): 117-133
[47] Hagen Knaf. Divisors on varieties over valuation domains. Israel Journal of Mathematics, Volume 119, Number 1, 2000: 349-377
[48] M.Nagata. Finitely generated rings over a valuation ring. J. Math. Kyoto Univ. 1965(5): 163-169
[48] A.H.Clifford. Naturally totally ordered commutative semigroups. American Journal of Mathematics, Vol. 76, No. 3, Jul., 1954: 631-646
[49] Irving Kaplansky. Elementary divisors and modules. Transactions of the American Mathematical Society, Vol. 66, No. 2, Jul., 1949: 464-491
[50] E.J.Tully, Jr. A Class of Naturally Partly Ordered Commutative Archimedean Semi-Groups with Maximal Condition. Proceedings of the American Mathematical Society, Vol. 17, No. 5, Oct., 1966: 1133-1139
[51] T.Rigo, S.Warner. Topologies extending valuations. Canad. J. Math., 1978 (30): 164-169
[52] A.Prestel, M.Ziegler. Model theoretic methods in the theory of topological fields. J. Riene Angew. Math., 1978 (299/300): 318-341
[53] Leopoldo Nachbin. On strictly minimal topological division rings. Bull. Amer. Math. Soc. Volume 55, Number 12 , 1949: 1128-1136
[54] S.V.Vostokov. The pairing on K-groups in fields of valuation of rank n . Trudy S.-Peterb. Mat. Obsch, 1995; English translation in Amer. Math. Soc. Transl. (Sec. 2) 1995(165): 111-148
[55] I.B.Zhukov. Milnor and topological K -groups of multidimensional complete fields. Algebra I Analiz, 1997; English translation in St. Petersburg Math. J. 1998 (9): 69-105
[56] Shreeram Abhyankar. Local uniformization on algebraic surfaces over ground fields of characteristic p ≠ 0. The Annals of Mathematics, 2nd Ser., Vol. 63, No. 3, May, 1956: 491-526
[57] Shreeram Abhyankar. Ramification theoretic methods in algebraic geometry. Annals of Math. Studies 43, Princeton University Press, 1959
[58] B.G.Kang, M.H.Park. A localization of a power series ring over a valuation domain. Journal of Pure and Applied Algebra, Volume 140, Number 2, 28 July 1999: 107-124(18)
[59] Semyon Alesker. Algebraic sturctures on valuations, their properties and applications. Proceedings of the ICM, Beijing 2002(2): 757—764. arXiv:math/0304338vl
[60] Jack Maney. Boundary valuation domains. Journal of Algebra, Volume 273, Issue 1,1 March 2004: 373-383
[61] B.Goldsmith, P.Zanardo. Minimal modules over valuation domains. Journal of Pure and Applied Algebra, Volume 199, Issues 1-3, 1 July 2005: 95-109
[62] L.Fuchs, L.Salce. Modules over valuation domains. Marcel Dekker, NewYork and Basel, 1985
[63] L.Salce, P.Zanardo. On two-generated modules over valuation domains. Archiv der Mathematik, Volume 46, Number 5, 1986: 408-418
[64] L.Salce, P.Zanardo. Rank-two torsion-free modules over valuation domains. Rend. Sem. Mat. Univ. Padova, 1988 (80): 175-201
[65] D.M.Arnold. A duality for torsion-free modules of finite rank over a discrete valuation ring. Proc. London Math. Soc, (3), 1972 (24): 204-216
[66] A.Facchini, P.Zanardo. Discrete valuation domains and ranks of their maximal extensions. Rend. Sem. Mat. Univ. Padova, 1986 (75): 143-156
[67] L.Fuchs. Infinite abelian groups. vol. II, Academic Press, London and New York, 1973
[68] L.Fuchs, L.Salce. Prebasic submodules over valuation rings. Annali Mat. Pura e Appl., 1982 (32): 257-274
[69] L.Fuchs, L.Salce. Modules over valuation domains. Lecture Notes Pure Appl. Math., no. 97, Marcel Dekker, New York, Basel, 1985
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