Adams operations and power structures

The Riemann-Roch theorem for the Adams operations

We prove the classical Riemann-Roch theorems for the Adams operations $\,ψ^j\,$ on $K$-theory: a statement with coefficients on $\mathbb{Z}[j^{-1}]$, that holds for arbitrary projective morphisms, as well as another one with integral coefficients, that is valid for closed immersions. In presence of rational coefficients, we also analyze the relation between the corresponding Riemann-Roch formula for one Adams operation and the analogous formula for the Chern character. To do so, we complete the elementary exposition of the work of Panin-Smirnov that was initiated by the first author in a previ…

连通K-理论与Adams运算

Connective K-theory and Adams operations

该研究探讨代数簇的凝聚模Grothendieck群与Chow群的挠现象关联，借助Adams运算在相关谱序列及连通K₀-理论上的作用分析，以初等论证完成两类代数对象的连通K₀-理论计算。

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。

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…

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.

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…

Operations in connective K-theory

In this article we classify additive operations in connective K-theory with various torsion-free coefficients. We discover that the answer for the integral case requires understanding of the $\hat{\mathbb{Z}}$ one. Moreover, although integral additive operations are topologically generated by Adams operations, these are not reduced to infinite linear combinations of the latter ones. We describe a topological basis for stable operations and relate it to a basis of stable operations in graded K-theory. We classify multiplicative operations in both theories and show that homogeneous additive stab…

几何dg范畴的Grothendieck环上的幂结构

A power structure over the Grothendieck ring of geometric dg categories

该研究证明几何dg范畴的Grothendieck环上存在幂结构，将Galkin-Shinder猜想重新表述为动机与范畴幂结构的相容性，证明范畴zeta函数可表为以范畴自身为指数的幂，并给出相关应用。

Normed motivic spectra and power operations

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.