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Register a new userThe Chow rings of the algebraic groups E_6, E_7, and E_8
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We determine the Chow rings of the complex algebraic groups of the exceptional type E_6, E_7, and E_8, giving the explicit generators represented by the pull-back images of Schubert varieties of the corresponding flag varieties. This is a continuation of the work of R. Marlin on the computation of the Chow rings of SO_n, Spin_n, G_2, and F_4. Our method is based on Schubert calculus of the corresponding flag varieties, which has its own interest.
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In these lectures, we provide a toolkit to work with Chow-Witt groups, and more generally with the homology and cohomology of the Rost-Schmid complex associated to Milnor-Witt $K$-theory.
Let $X$ be a smooth projective quadric defined over a field of characteristic 2. We prove that in the Chow group of codimension 2 or 3 of $X$ the torsion subgroup has at most two elements. In codimension 2, we determine precisely when this torsion subgroup is nontrivial. In codimension 3, we show that there is no torsion if {$dim Xge 11$.} This extends the analogous results in characteristic different from 2, obtained by Karpenko in the nineteen-nineties.
For each prime $p$, we define a $t$-structure on the category $widehat{S^{0,0}}/tautext{-}mathbf{Mod}_{harm}^b$ of harmonic $mathbb{C}$-motivic left module spectra over $widehat{S^{0,0}}/tau$, whose MGL-homology has bounded Chow-Novikov degree, such that its heart is equivalent to the abelian category of $p$-completed $BP_*BP$-comodules that are concentrated in even degrees. We prove that $widehat{S^{0,0}}/tautext{-}mathbf{Mod}_{harm}^b$ is equivalent to $mathcal{D}^b({{BP}_*{BP}text{-}mathbf{Comod}}^{{ev}})$ as stable $infty$-categories equipped with $t$-structures. As an application, for each prime $p$, we prove that the motivic Adams spectral sequence for $widehat{S^{0,0}}/tau$, which converges to the motivic homotopy groups of $widehat{S^{0,0}}/tau$, is isomorphic to the algebraic Novikov spectral sequence, which converges to the classical Adams-Novikov $E_2$-page for the sphere spectrum $widehat{S^0}$. This isomorphism of spectral sequences allows Isaksen and the second and third authors to compute the stable homotopy groups of spheres at least to the 90-stem, with ongoing computations into even higher dimensions.
We consider generalized $Lambda$-structures on algebras and schemes over the ring of integers $mathit{O}_K$ of a number field $K$. When $K=mathbb{Q}$, these agree with the $lambda$-ring structures of algebraic K-theory. We then study reduced finite flat $Lambda$-rings over $mathit{O}_K$ and show that the maximal ones are classified in a Galois theoretic manner by the ray class monoid of Deligne and Ribet. Second, we show that the periodic loci on any $Lambda$-scheme of finite type over $mathit{O}_K$ generate a canonical family of abelian extensions of $K$. This raises the possibility that $Lambda$-schemes could provide a framework for explicit class field theory, and we show that the classical explicit class field theories for the rational numbers and imaginary quadratic fields can be set naturally in this framework. This approach has the further merit of allowing for some precise questions in the spirit of Hilberts 12th Problem. In an interlude which might be of independent interest, we define rings of periodic big Witt vectors and relate them to the global class field theoretical mathematics of the rest of the paper.
The central aim of this monograph is to provide decomposition results for quasi-coherent sheaves on the moduli stack of one-dimensional formal groups. These results will be based on the geometry of the stack itself, particularly the height filtration and an analysis of the formal neighborhoods of the geometric points. The main theorems are algebraic chromatic convergence results and fracture square decompositions. There is a major technical hurdle in this story, as the moduli stack of formal groups does not have the finitness properties required of an algebraic stack as usually defined. This is not a conceptual problem, but in order to be clear on this point and to write down a self-contained narrative, I have included a great deal of discussion of the geometry of the stack itself, giving various equivalent descriptions.
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