Email this article Generalization of the Artin-Hasse logarithm for the Milnor $K$-groups of $\delta$-rings
- Authors: Tyurin D.N.1
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Affiliations:
- Steklov Mathematical Institute of Russian Academy of Sciences
- Issue: Vol 212, No 12 (2021)
- Pages: 95-114
- Section: Articles
- URL: https://journal-vniispk.ru/0368-8666/article/view/133416
- DOI: https://doi.org/10.4213/sm9520
- ID: 133416
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Abstract
Let $R$ be a $p$-adically complete ring equipped with a $\delta$-structure. We construct a functorial group homomorphism from the Milnor $K$-group $K^{M}_{n}(R)$ to the quotient of the $p$-adic completion of the module of differential forms $\widehat{\Omega}^{n-1}_{R}/d\widehat{\Omega}^{n-2}_{R}$. This homomorphism is a $p$-adic analogue of the Bloch map defined for the relative Milnor $K$-groups of nilpotent extensions of rings of nilpotency degree $N$ for which the number $N!$ is invertible. Bibliography: 12 titles.
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About the authors
Dimitrii Nikolaevich Tyurin
Steklov Mathematical Institute of Russian Academy of Scienceswithout scientific degree
References
- B. Bhatt, P. Scholze, Prisms and prismatic cohomology
- S. Bloch, “$K_2$ of Artinian $Q$-algebras, with application to algebraic cycles”, Comm. Algebra, 3:5 (1975), 405–428
- A. Buium, “Arithmetic analogues of derivations”, J. Algebra, 198:1 (1997), 290–299
- T. tom Dieck, “The Artin–Hasse logarithm for $lambda$-rings”, Algebraic topology (Arcata, CA, 1986), Lecture Notes in Math., 1370, Springer, Berlin, 1989, 409–415
- B. F. Dribus, A Goodwillie-type theorem for Milnor $K$-Theory
- С. О. Горчинский, Д. Н. Тюрин, “Относительные $K$-группы Милнора и дифференциальные формы расщепимых нильпотентных расширений”, Изв. РАН. Сер. матем., 82:5 (2018), 23–60
- A. Joyal, “$delta$-anneaux et vecteurs de Witt”, C. R. Math. Rep. Acad. Sci. Canada, 7:3 (1985), 177–182
- W. van der Kallen, “The $K_2$ of rings with many units”, Ann. Sci. Ecole Norm. Sup. (4), 10:4 (1977), 473–515
- K. Kato, “On $p$-adic vanishing cycles (application of ideas of Fontaine–Messing”, Algebraic geometry (Sendai, 1985), Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987, 207–251
- K. Kato, “The explicit reciprocity law and the cohomology of Fontaine–Messing”, Bull. Soc. Math. France, 119:4 (1991), 397–441
- H. Maazen, J. Stienstra, “A presentation for $K_2$ of split radical pairs”, J. Pure Appl. Algebra, 10:3 (1977/1978), 271–294
- С. В. Востоков, “Явная форма закона взаимности”, Изв. АН СССР. Сер. матем., 42:6 (1978), 1288–1321
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