Abstract
Some multiplication is constructed for algebraic K-functors of the crossed products of a commutative algebra and a Hopf cocommutative algebra; the question on these functors to be Frobenius functors with respect to the constructed multiplication is studied.
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References
A. Bartels and W. Luck, “Induction theorems and isomorphism conjectures for K- and L-theory,” Forum Math., 19, 1–28 (2007).
S. M. Gersten, “On the functor K2,” J. Algebra, 17, 212–237 (1971).
R. G. Heyneman and D. E. Radford, “Reflexivity and coalgebras of finite type,” J. Algebra, 28, 215–246 (1974).
K. Kawakubo, “Induction theorems for equivariant K-theory and J-theory,” J. Math. Soc. Jpn., 38, 173–198 (1986).
T. Y. Lam, “Induction theorems for Grothendieck groups and Whitehead groups of finite groups,” Ann. Sci. Ec. Norm. Super., 1, 91–148 (1968).
J. G. McConnel and M. E. Sweedler, “Simplicity of smash products,” Proc. London Math. Soc., 23, 251–266 (1971).
A. I. Nemytov, “Kn(Rπ) functors as Frobenius modules on the functor GR0(π),” Usp. Mat. Nauk, 28, 187–188 (1973).
B. Pachuashvili, “Cohomologies in monoidal categories,” Bull. Georgian Acad. Sci., 106, 485—488 (1982).
D. Quillen, “Higher algebraic K-Theory, I,” Lect. Notes Math., 341, 85–147 (1972).
D. E. Radford, “Pointed Hopf algebras are free over Hopf subalgebras,” J. Algebra, 45, 266–273 (1977).
G. Rakviashvili, “Generalization of the Artin theorem for semisimple algebras and inductive theorems for orders and crossed group rings,” Bull. Georgian Acad. Sci., 96, 25—28 (1979).
G. Rakviashvili, “Inductive theorems and projective modules over crossed group rings,” Proc. A. Razmadze Math. Inst., 3, 92–107 (1982).
G. Rakviashvili, “On the crossed enveloping algebra of Lie p-algebra,” Bull. Georgian Acad. Sci., 96, No. 1, 265–268 (1979).
G. Rakviashvili, “On the K-theory of the crossed product of a commutative algebra and a Hopf algebra,” Proc. A. Razmadze Math. Inst., 5, 79–95 (1986).
G. Rakviashvili, “On algebraic K-functors of crossed group rings and its applications,” Tbilisi Math. J., 11, 1–15 (2018).
R. G. Swan, “Induced representations and projective modules,” Ann. Math., 71, 552–578 (1960).
R. G. Swan, “Nonabelian homological algebra and K-theory,” Proc. Symp. Pure Math., 17, 88–123 (1970).
M. Sweedler, “Cohomology of algebras over Hopf algebras,” Trans. Am. Math. Soc., 133, 205–239 (1968).
S. M. J. Wilson, “K-Theory for twisted group rings,” Proc. London Math. Soc., 29, 257–271 (1974).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 177, Algebra, 2020.
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Rakviashvili, G. On Products in Algebraic K-Theory of Crossed Hopf Algebras. J Math Sci 275, 758–766 (2023). https://doi.org/10.1007/s10958-023-06718-1
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DOI: https://doi.org/10.1007/s10958-023-06718-1