arXiv subjects
Publications and source records attributed to Jeremiah Heller.
Over the complex numbers, we compute the $C_2$-equivariant Bredon motivic cohomology ring with $\mathbb{Z}/2$ coefficients. By rigidity, this extends Suslin's calculation of the motivic cohomology ring of algebraically closed fields of characteristic zero to the $C_2$-equivariant motivic setting.
We prove that the universal normed motivic spectrum of characteristic 2 over a scheme on which 2 is a unit, splits into a sum of motivic Eilenberg--MacLane spectra.
We use motivic colimits to construct power operations on the homotopy groups of normed motivic spectra admitting a (normed) map from HF_2. We establish enough of their standard properties to prove that the motivic dual Steenrod algebra is generated by one element under ring and power operations, establishing a motivic analog of Steinberger's theorem.
We define a notion of colimit for diagrams in a motivic category indexed by a presheaf of spaces (e.g. an étale classifying space), and we study basic properties of this construction. As a case study, we construct the motivic analogs of the classical extended and generalized powers, which refine the categorical versions of these constructions as special cases. We also offer more computationally tractable models of these constructions using equivariant motivic homotopy theory. This is the first in a series of papers on power operations in motivic stable homotopy theory.
We reformulate the problem of bounding the total rank of the homology of perfect chain complexes over the group ring $\mathbb{F}_p[G]$ of an elementary abelian $p$-group $G$ in terms of commutative algebra. This extends results of Carlsson for $p=2$ to all primes. As an intermediate step, we construct an embedding of the derived category of perfect chain complexes over $\mathbb{F}_p[G]$ into the derived category of $p$-DG modules over a polynomial ring.
We establish, in the setting of equivariant motivic homotopy theory for a finite group, a version of tom Dieck's splitting theorem for the fixed points of a suspension spectrum. Along the way we establish structural results and constructions for equivariant motivic homotopy theory of independent interest. This includes geometric fixed point functors and the motivic Adams isomorphism.
Schlichting conjectured that the negative K-groups of small abelian categories vanish and proved this for noetherian abelian categories and for all abelian categories in degree $-1$. The main results of this paper are that $K_{-1}(E)$ vanishes when $E$ is a small stable $\infty$-category with a bounded t-structure and that $K_{-n}(E)$ vanishes for all $n\geq 1$ when additionally the heart of $E$ is noetherian. It follows that Barwick's theorem of the heart holds for nonconnective K-theory spectra when the heart is noetherian. We give several applications, to non-existence results for bounded t-structures and stability conditions, to possible K-theoretic obstructions to the existence of the motivic t-structure, and to vanishing results for the negative K-groups of a large class of dg algebras and ring spectra.
We prove versions of the Suslin and Gabber rigidity theorems in the setting of equivariant pseudo pretheories of smooth schemes over a field with an action of a finite group. Examples include equivariant algebraic $K$-theory, presheaves with equivariant transfers, equivariant Suslin homology, and Bredon motivic cohomology.
We prove equivariant versions of the Beilinson-Lichtenbaum conjecture for Bredon motivic cohomology of smooth complex and real varieties with an action of the group of order two. This identifies equivariant motivic and topological invariants in a large range of degrees.
Using techniques from motivic homotopy theory, we prove a conjecture of Anthony Blanc about semi-topological K-theory of dg categories with finite coefficients. Along the way, we show that the connective semi-topological K-theories defined by Friedlander-Walker and by Blanc agree for quasi-projective complex varieties and we study étale descent of topological K-theory of dg categories.
Let F be a field of characteristic different than 2. We establish surjectivity of Balmer's comparison map rho^* from the tensor triangular spectrum of the homotopy category of compact motivic spectra to the homogeneous Zariski spectrum of Milnor-Witt K-theory. We also comment on the tensor triangular geometry of compact cellular motivic spectra, producing in particular novel field spectra in this category. We conclude with a list of questions about the structure of the tensor triangular spectrum of the stable motivic homotopy category.
We define an unstable equivariant motivic homotopy category for an algebraic group over a Noetherian base scheme. We show that equivariant algebraic $K$-theory is representable in the resulting homotopy category. Additionally, we establish homotopical purity and blow-up theorems for finite abelian groups.
We introduce a Bredon motivic cohomology theory for smooth schemes defined over a field and equipped with an action by a finite group. These cohomology groups are defined for finite dimensional representations as the hypercohomology of complexes of equivariant correspondences in the equivariant Nisnevich topology. We generalize the theory of presheaves with transfers to the equivariant setting and prove a Cancellation Theorem.
We generalize several comparison results between algebraic, semi-topological and topological K-theories to the equivariant case with respect to a finite group.
We show that Shipley's "detection functor" for symmetric spectra generalizes to motivic symmetric spectra. As an application, we construct motivic strict ring spectra representing morphic cohomology, semi-topological $K$-theory, and semi-topological cobordism for complex varieties. As a further application to semi-topological cobordism, we show that it is related to semi-topological $K$-theory via a Conner-Floyd type isomorphism and that after inverting a lift of the Friedlander-Mazur $s$-element in morphic cohomology, semi-topological cobordism becomes isomorphic to periodic complex cobordism.
We construct an equivariant algebraic cobordism theory for schemes with an action by a linear algebraic group over a field of characteristic zero.
We demonstrate that a conjecture of Teh which relates the niveau filtration on Borel-Moore homology of real varieties and the images of generalized cycle maps from reduced Lawson homology is false. We show that the niveau filtration on reduced Lawson homology is trivial and construct an explicit class of examples for which Teh's conjecture fails by generalizing a result of Schulting. We compare various cycle maps and in particular we show that the Borel-Haeflinger cycle map naturally factors through the reduced Lawson homology cycle map.
We establish an Atiyah-Hirzebruch type spectral sequence relating real morphic cohomology and real semi-topological K-theory and prove it to be compatible with the Atiyah-Hirzebruch spectral sequence relating Bredon cohomology and Atiyah's KR-theory constructed by Dugger. An equivariant and a real version of Suslin's conjecture on morphic cohomology are formulated, proved to come from the complex version of Suslin conjecture and verified for certain real varieties. In conjunction with the spectral sequences constructed here this allows the computation of the real semi-topological K-theory of some real varieties. As another application of this spectral sequence we give an alternate proof of the Lichtenbaum-Quillen conjecture over $\R$, extending an earlier proof of Karoubi and Weibel.