连通K-理论与Adams运算

Adams operations and power structures

We construct a family of additive endomorphisms $Ψ_k, k=1, 2...$ of the Grothendieck ring of quasiprojective varieties and the Grothendieck ring of Chow motives similar to the Adams operations in the K-theory. The speciality of the $λ$-structure on the Grothendieck ring of motives (proved by F. Heinloth) gives a set of natural equations for these operations. We discuss this construction in a general setting and relate it to the concept of power structures introduced by S. Gusein-Zade, I. Luengo and A. Melle-Hernandez. Some interpretation of the E. Getzler's formula for the equivariant Hodge-De…

Adams operations and Galois structure

We present a new method for determining the Galois module structure of the cohomology of coherent sheaves on varieties over the integers with a tame action of a finite group. This uses a novel Adams-Riemann-Roch type theorem obtained by combining the Kunneth formula with localization in equivariant K-theory and classical results about cyclotomic fields. As an application, we show two conjectures of Chinburg-Pappas-Taylor, in the case of curves.

Adams operations on the virtual K-theory of P(1,n)

We analyze the structure of the virtual (orbifold) K-theory ring of the complex orbifold P(1,n) and its virtual Adams (or power) operations, by using the non-Abelian localization theorem of Edidin-Graham. In particular, we identify the group of virtual line elements and obtain a natural presentation for the virtual K-theory ring in terms of these virtual line elements. This yields a surjective homomorphism from the virtual K-theory ring of P(1,n) to the ordinary K-theory ring of a crepant resolution of the cotangent bundle of P(1,n) which respects the Adams operations. Furthermore, there is a …

Duality and the topological filtration

We investigate some relations between the duality and the topological filtration in algebraic K-theory. As a result, we obtain a construction of the first Steenrod square for Chow groups modulo two of varieties over a field of arbitrary characteristic. This improves previously obtained results, in the sense that it is not anymore needed to mod out the image modulo two of torsion integral cycles. Along the way we construct a lifting of the first Steenrod square to algebraic connective K-theory with integral coefficients, and homological Adams operations in this theory. Finally we provide some a…

准映射K-理论中的量子Adams运算

Quantum Adams operations in quasimap K-theory

本文在准映射量子K-理论中定义量子Adams运算，作为量子Steenrod运算的K-理论类比，验证其与q-差分结构相容，并通过局部化计算识别为p-曲率算子，同时提出并验证K-理论量子Hikita猜想。

On uniqueness of characteristic classes

We give an axiomatic characterization of maps from algebraic K-theory. The results apply to a class of maps from algebraic K-theory to any suitable cohomology theory or to algebraic K-theory, which includes all group morphisms. In particular, we obtain comparison theorems for the Chern character and Chern classes and for the Lambda and Adams operations on higher algebraic K-theory. We show that the Adams operations defined by Grayson agree with the ones defined by Gillet and Soule.

Moduli stack of oriented formal groups and periodic complex bordism

We introduce and study the non-connective spectral stack $\mathcal M_\mathrm{FG}^\mathrm{or}$, the moduli stack of oriented formal groups. We realize some results of chromatic homotopy theory in terms of the geometry of this stack. For instance, we show that its descent spectral sequence recovers the Adams-Novikov spectral sequence. For two $\mathbb E_\infty$-forms of periodic complex bordism $\mathrm{MP}$, the Thom spectrum and Snaith construction model, we describe the universal property of the cover $\mathrm{Spec}(\mathrm{MP})\to\mathcal M_\mathrm{FG}^\mathrm{or}$. We show that Quillen's ce…

Existence and uniqueness of E-infinity structures on motivic K-theory spectra

We show that algebraic K-theory KGL, the motivic Adams summand ML and their connective covers acquire unique E-infinity structures refining naive multiplicative structures in the motivic stable homotopy category. The proofs combine Gamma-homology computations and work due to Robinson giving rise to motivic obstruction theory. As an application we employ a motivic to simplicial delooping argument to show a uniqueness result for E-infinity structures on the K-theory Nisnevich presheaf of spectra.

Operations on locally free classgroups

Let G be a finite group and K a number field. We show that the "k-th Adams operation" defined by Cassou-Nogues and Taylor on the locally free class group Cl(Z_K G) is a symmetric power operation, if k is coprime to the order of G. Using the equivariant Adams-Riemann-Roch theorem, we furthermore give a geometric interpretation of a formula established by Burns and Chinburg for these operations.

Simplification of $λ$-ring expressions in the Grothendieck ring of Chow motives

The Grothendieck ring of Chow motives admits two natural opposite $λ$-ring structures, one of which is a special structure allowing the definition of Adams operations on the ring. In this work I present algorithms which allow an effective simplification of expressions that involve both $λ$-ring structures, as well as Adams operations. In particular, these algorithms allow the symbolic simplification of algebraic expressions in the sub-$λ$-ring of motives generated by a finite set of curves into polynomial expressions in a small set of motivic generators. As a consequence, the explicit computat…

Stable and Unstable operations in Algebraic Cobordism

We describe additive (unstable) operations from a theory A^* obtained from Algebraic Cobordism of M.Levine-F.Morel by change of coefficients to any oriented cohomology theory B^*. We prove that there is 1-to-1 correspondence between the set of operations, and the set of transformations: A^n((P^{\infty})^{\times r}) ---> B^m((P^{\infty})^{\times r}) satisfying certain simple properties. This provides an effective tool of constructing such operations. As an application, we prove that (unstable) additive operations in Algebraic Cobordism are in 1-to-1 correspondence with the L\otimes_Z Q-linear c…

等特征方案的motivic cohomology理论

Motivic cohomology of equicharacteristic schemes

本文构建了等特征方案的motivic cohomology理论，关联到非连通代数K-理论及etale上同调，并在光滑变元上恢复经典motivic cohomology。