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Milnor-Witt Motives
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abstract
We develop the theory of Milnor-Witt motives and motivic cohomology. Compared to Voevodsky's theory of motives and his motivic cohomology, the first difference appears in our definition of Milnor-Witt finite correspondences, where our cycles come equipped with quadratic forms. This yields a weaker notion of transfers and a derived category of motives that is closer to the stable homotopy theory of schemes. We prove a cancellation theorem when tensoring with the Tate object, we compare the diagonal part of our Milnor-Witt motivic cohomology to Minor-Witt K-theory and we provide spectra representing various versions of motivic cohomology in the $\mathbb{A}^1$-derived category or the stable homotopy category of schemes.
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Cellular $\mathbb{A}^1$-Homology of Smooth Toric Varieties
For smooth pure shellable toric varieties, the cellular A1-chain complex is computed explicitly in terms of homology of restriction subcomplexes Kω with Milnor-Witt K-theory coefficients, yielding MW-motivic decomposi...
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