2018
DOI: 10.1016/j.aim.2018.01.001
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Riemann–Roch for homotopy invariant K-theory and Gysin morphisms

Abstract: We prove the Riemann-Roch theorem for homotopy invariant K-theory and projective local complete intersection morphisms between finite dimensional noetherian schemes, without smoothness assumptions. We also prove a new Riemann-Roch theorem for the relative cohomology of a morphism.In order to do so, we construct and characterize Gysin morphisms for regular immersions between cohomologies represented by spectra (examples include homotopy invariant K-theory, motivic cohomology, their arithmetic counterparts, real… Show more

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Cited by 10 publications
(9 citation statements)
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“…et (S Y , µ n (j)) → H i+2 et (S X , µ n (j + 1)) for i ≥ 0. Furthermore, it follows from the Cartesian square (2.9) and [50,Corollary 2.12] (see also [19,Proposition 1.1.3]) that the pull-back via the closed immersions ι ± : X ֒→ S X induces a commutative diagram (5.5) H Proof. The first isomorphism is a well known consequence of the weak Lefschetz theorem forétale cohomology.…”
mentioning
confidence: 99%
“…et (S Y , µ n (j)) → H i+2 et (S X , µ n (j + 1)) for i ≥ 0. Furthermore, it follows from the Cartesian square (2.9) and [50,Corollary 2.12] (see also [19,Proposition 1.1.3]) that the pull-back via the closed immersions ι ± : X ֒→ S X induces a commutative diagram (5.5) H Proof. The first isomorphism is a well known consequence of the weak Lefschetz theorem forétale cohomology.…”
mentioning
confidence: 99%
“…These two morphisms agree due to the same argument of [Dég14, 1.2.10.E7] replacing inverse images by exceptional images and using the analogous compatibility. A direct image for the cohomology defined by absolute modules over absolute oriented ring spectra was defined in [Nav16b,§2]. For the sake of completeness, let us prove here a uniqueness result.…”
Section: Direct Image For Proper Morphismsmentioning
confidence: 99%
“…All schemes we will consider throughout this paper are smooth over a finite dimensional noetherian base S. We use the same notations of [Nav16b] and recall the indispensable.…”
Section: Preliminariesmentioning
confidence: 99%
“…They are also an essential feature in the algebraic cobordism theory of Levine and Morel [LM07]. Such transfers were further constructed by Navarro [Nav16] on E-cohomology groups for any MGL-module E, and Déglise, Jin and Khan [DJK20] showed that these transfers exist at the level of spaces. Our main result implies that E-cohomology spaces admit coherent finite syntomic transfers, and that this structure even characterises MGL-modules.…”
Section: Related Workmentioning
confidence: 99%