## Frobenius line invariance of algebraic $K$-theory

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- by Oliver Braunling PDF
- Trans. Amer. Math. Soc.
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## Abstract:

The $K$-theory of smooth schemes is $\mathbf {A}^{1}$-invariant. We show that this remains true over finite fields if one replaces the affine line by the Frobenius line, i.e., the non-commutative algebra where multiplication with the variable behaves like the Frobenius. Emerton had shown that over regular rings the Frobenius line is left coherent. As a technical ingredient for our theorem, but also of independent interest, we extend this and show that merely assuming finite type (or just $F$-finite), the Frobenius line is right coherent.## References

- Manuel Blickle and Gebhard Böckle,
*Cartier modules: finiteness results*, J. Reine Angew. Math.**661**(2011), 85–123. MR**2863904**, DOI 10.1515/CRELLE.2011.087 - Kenneth S. Brown and Stephen M. Gersten,
*Algebraic $K$-theory as generalized sheaf cohomology*, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 266–292. MR**0347943** - P. M. Cohn,
*Algebra. Vol. 3*, 2nd ed., John Wiley & Sons, Ltd., Chichester, 1991. MR**1098018** - P. M. Cohn,
*Skew fields*, Encyclopedia of Mathematics and its Applications, vol. 57, Cambridge University Press, Cambridge, 1995. Theory of general division rings. MR**1349108**, DOI 10.1017/CBO9781139087193 - Rankeya Datta and Karen E. Smith,
*Excellence in prime characteristic*, Local and global methods in algebraic geometry, Contemp. Math., vol. 712, Amer. Math. Soc., [Providence], RI, [2018] ©2018, pp. 105–116. MR**3832401**, DOI 10.1090/conm/712/14344 - Matthew Emerton and Mark Kisin,
*The Riemann-Hilbert correspondence for unit $F$-crystals*, Astérisque**293**(2004), vi+257 (English, with English and French summaries). MR**2071510** - M. Emerton,
*On a class of coherent rings, with applications to the smooth representation theory of $GL_{2}$ in characteristic $p$*, unpublished (2008), available as: http://www.math.uchicago.edu/\symbol {126}emerton/pdffiles/frob.pdf. - Pierre Gabriel,
*Des catégories abéliennes*, Bull. Soc. Math. France**90**(1962), 323–448 (French). MR**232821**, DOI 10.24033/bsmf.1583 - S. M. Gersten,
*$K$-theory of free rings*, Comm. Algebra**1**(1974), 39–64. MR**396671**, DOI 10.1080/00927877408548608 - Daniel R. Grayson,
*$K$-theory and localization of noncommutative rings*, J. Pure Appl. Algebra**18**(1980), no. 2, 125–127. MR**585217**, DOI 10.1016/0022-4049(80)90123-1 - Daniel R. Grayson,
*The $K$-theory of semilinear endomorphisms*, J. Algebra**113**(1988), no. 2, 358–372. MR**929766**, DOI 10.1016/0021-8693(88)90165-2 - J. F. Jardine,
*Generalised sheaf cohomology theories*, Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131, Kluwer Acad. Publ., Dordrecht, 2004, pp. 29–68. MR**2061851**, DOI 10.1007/978-94-007-0948-5_{2} - Ernst Kunz,
*Characterizations of regular local rings of characteristic $p$*, Amer. J. Math.**91**(1969), 772–784. MR**252389**, DOI 10.2307/2373351 - T. Y. Lam,
*Lectures on modules and rings*, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999. MR**1653294**, DOI 10.1007/978-1-4612-0525-8 - Linquan Ma,
*The category of $F$-modules has finite global dimension*, J. Algebra**402**(2014), 1–20. MR**3160413**, DOI 10.1016/j.jalgebra.2013.12.008 - J. C. McConnell and J. C. Robson,
*Noncommutative Noetherian rings*, Revised edition, Graduate Studies in Mathematics, vol. 30, American Mathematical Society, Providence, RI, 2001. With the cooperation of L. W. Small. MR**1811901**, DOI 10.1090/gsm/030 - Daniel Quillen,
*Higher algebraic $K$-theory. I*, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR**0338129** - Jean-Pierre Soublin,
*Anneaux et modules cohérents*, J. Algebra**15**(1970), 455–472 (French). MR**260799**, DOI 10.1016/0021-8693(70)90050-5 - Christophe Soulé,
*Les variétés sur le corps à un élément*, Mosc. Math. J.**4**(2004), no. 1, 217–244, 312 (French, with English and Russian summaries). MR**2074990**, DOI 10.17323/1609-4514-2004-4-1-217-244 - Stacks,
*Stacks Project*, http://stacks.math.columbia.edu, 2016. - Richard G. Swan,
*K-theory of coherent rings*, J. Algebra Appl.**18**(2019), no. 9, 1950161, 16. MR**3981694**, DOI 10.1142/S0219498819501615 - Charles A. Weibel,
*The $K$-book*, Graduate Studies in Mathematics, vol. 145, American Mathematical Society, Providence, RI, 2013. An introduction to algebraic $K$-theory. MR**3076731**, DOI 10.1090/gsm/145 - Charles Weibel and Dongyuan Yao,
*Localization for the $K$-theory of noncommutative rings*, Algebraic $K$-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989) Contemp. Math., vol. 126, Amer. Math. Soc., Providence, RI, 1992, pp. 219–230. MR**1156514**, DOI 10.1090/conm/126/00513 - Yuji Yoshino,
*Skew-polynomial rings of Frobenius type and the theory of tight closure*, Comm. Algebra**22**(1994), no. 7, 2473–2502. MR**1271618**, DOI 10.1080/00927879408824972

## Additional Information

**Oliver Braunling**- Affiliation: Freiburg Institute for Advanced Studies (FRIAS), University of Freiburg, D-79104 Freiburg im Breisgau, Germany
- MR Author ID: 1036829
- ORCID: 0000-0003-4845-7934
- Received by editor(s): March 7, 2018
- Received by editor(s) in revised form: October 10, 2019, and April 10, 2020
- Published electronically: August 28, 2020
- Additional Notes: The author was supported by the GK1821 “Cohomological Methods in Geometry”and the Freiburg Institute for Advanced Studies (FRIAS)
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**373**(2020), 8197-8217 - MSC (2010): Primary 19F27, 14F30
- DOI: https://doi.org/10.1090/tran/8231
- MathSciNet review: 4169686