Hermitian K-理论上的稳定 Adams 运算

Adams operations on classical compact Lie groups

Let $G$ be $U(n)$, $SU(n)$, $Sp(n)$ or $Spin(n)$. In this short note we give explicit general formulas for Adams operations on $K^*(G)$, and eigenvectors of Adams operations on $K^*(U(n))$.

Cyclic Adams Operations

Let $Q$ be a commutative, Noetherian ring and $Z \subseteq \operatorname{Spec}(Q)$ a closed subset. Define $K_0^Z(Q)$ to be the Grothendieck group of those bounded complexes of finitely generated projective $Q$-modules that have homology supported on $Z$. We develop "cyclic" Adams operations on $K_0^Z(Q)$ and we prove these operations satisfy the four axioms used by Gillet and Soulé in their paper "Intersection Theory Using Adams Operations". From this we recover a shorter proof of Serre's Vanishing Conjecture. We also show our cyclic Adams operations agree with the Adams operations defined by…

紧 Lie 群扭曲 $K$ 理论上的 Adams 运算

Adams operations on twisted $K$-theory of compact Lie groups

本文扩展已有结果，在等变与非等变设定下，计算连通、单连通且单紧 Lie 群 $G$ 扭曲 $K$ 理论上的 Adams 运算。

From algebraic K-theory to hermitian K-theory

We introduce a new morphism between algebraic and hermitian K-theory. The topological analog is the Adams operation in real K-theory. From this morphism, we deduce a lower bound for the higher algebraic K-theory of a ring A in terms of the classical Witt group of the ring A tensored by its opposite ring.

A note on spaces of symmetric matrices

We calculate the maximal dimension of linear spaces of symmetric and hermitian matrices with given high rank, generalizing a well-known result of Adams {\em et al.}

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.

有限域上厄米特K-理论的合作环

Rings of cooperations for hermitian K-theory over finite fields

计算了所有特征不为2的有限域上非常有效厄米特K-理论的合作环，利用动机亚当斯谱序列，并证明了所有微分由有限域的积分动机上同调决定，作为应用计算了kq-分辨的E_1-页。

Algebras of operations in K-theory

We describe explicitly the algebras of degree zero operations in connective and periodic p-local complex K-theory. Operations are written uniquely in terms of certain infinite linear combinations of Adams operations, and we give formulas for the product and coproduct structure maps. It is shown that these rings of operations are not Noetherian. Versions of the results are provided for the Adams summand and for real K-theory.

关于实Hermite K-理论的共作环

On the ring of cooperations for real hermitian K-theory

本文研究实Hermite K-理论的共作环，通过分解motivic Adams谱序列的E₂页并证明其坍塌，利用Brown-Gitler余模描述该环的结构。

From Representation Theory to Homotopy Groups

Bousfield recently gave a formula for the odd-primary v1-periodic homotopy groups of a finite H-space in terms of its K-theory and Adams operations. We apply Bousfield's theorem to give explicit determinations of the v1-periodic homotopy groups of (E8,5) and (E8,3), thus completing the determination of all odd-primary v1-periodic homotopy groups of compact simple Lie groups, a project suggested by Mimura in 1989. The method involves no homotopy theoretic input and no spectral sequences. The input is the second eexterior power operation in the representation ring of E8, which we determine using…

规范与希伯特K-理论

Norms and Hermitian $\mathrm{K}$-Theory

本文研究了动机谱中规范代数结构，展示了ko谱在希伯特K-理论中的应用，并证明了导向映射与规范的相容性。

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…

On rings and categories of general representations

In this note we show that similar to the classical case the ring of representations of symmetric groups in a tensor derived category is certain ring of symmetric functions. We also show that in the general setting considered here, the Adams operations compute the characteristic series associated to powers of endomorphisms.

重新审视 β₁ 在球体稳定同调群中的作用

Revisiting the $β_1$-action on the $3$-primary stable homotopy groups of spheres

本文通过BP合成谱和Adams-Novikov谱序列证明了β₁的5重积非零而6重积为零，并推广至其他144周期家族。

代数叠K-理论上的Lambda环结构

Lambda-ring structures on the K-theory of algebraic stacks

该研究构建光滑代数叠K-理论的lambda、gamma运算，证明其高阶K-理论是预lambda环，满足凝聚层为向量丛商条件时为lambda环，进而定义相关运算与上同调，还得到特定情形下绝对上同调与等变高阶Chow群的比较结果。

微分K理论中的λ-环结构

$λ$-ring structure in differential K-theory

本文研究了微分K理论中闭合光滑流形的λ-环结构，并扩展到等变情形。