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Cancellation theorem for framed motives of algebraic varieties

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Preprint
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arXiv ID: 1601.06642
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arXiv
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Abstract

The machinery of framed (pre)sheaves was developed by Voevodsky. Based on the theory, framed motives of algebraic varieties are introduced and studied in [GP1]. An analog of Voevodsky's Cancellation Theorem is proved in this paper for framed motives stating that a natural map of framed $S^1$-spectra $M_{fr}(X)(n)\to\underline{Hom}(\mathbb{G},M_{fr}(X)(n+1))$, $n\geq 0,$ is a Nisnevich local stable equivalence, where $M_{fr}(X)(n)$ is the $n$th twisted framed motive of $X$. This result is reduced to the Cancellation Theorem for linear framed motives stating that the natural map of complexes of abelian groups $\mathbb{Z}F(\Delta^\bullet \times X,Y)\to\mathbb{Z}F((\Delta^\bullet \times X)\wedge (\mathbb{G}_m,1),Y\wedge(\mathbb{G}_m,1))$, $X,Y\in Sm/k,$ is a quasi-isomorphism, where $\mathbb{Z}F(X,Y)$ is the group of stable linear framed correspondences in the sense of [GP1,GP3].

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