Frobenius line invariance of algebraic 𝐾-theory

Braunling, Oliver

doi:10.1090/tran/8231

Frobenius line invariance of algebraic $K$-theory
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Trans. Amer. Math. Soc. 373 (2020), 8197-8217 Request permission

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.

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