论文标题
双变量Hermitian $ K $ - 理论和Karoubi的基本定理
Bivariant Hermitian $K$-theory and Karoubi's fundamental theorem
论文作者
论文摘要
令$ \ ell $为一个通勤戒指，其中包含一个元素$λ$，以便$λ+λ^*= 1 $，让$ \ operatatorName {alg}^*_ \ ell $是$ \ ell $ -Algebras的类别，配备了半岛化的参与性和屈服于预先提供的派生。我们构建一个三角类别$ kk^h $和fuction $ j^h：\ peripatorName {alg}^*_ \ ell \ to kk^h $，这是同质的，母亲和赫尔米尼亚稳定和兴奋性，并且具有这些属性的普遍性。我们证明，Karoubi的基本定理版本以$ kk^h $为单位。通过后者的通用属性，这意味着任何函数$ h：\ permatorAtorname {alg}^*_ \ ell \ to \ m athfrak {t} $带有三角形类别中的值，这是同型不变的，术语上，毕生型和遗产稳定和兴奋性可满足基本属性的属性。我们还证明了Karoubi的$ 12 $ - Term精确序列的双变量版本。
Let $\ell$ be a commutative ring with involution $*$ containing an element $λ$ such that $λ+λ^*=1$ and let $\operatorname{Alg}^*_\ell$ be the category of $\ell$-algebras equipped with a semilinear involution and involution preserving homomorphisms. We construct a triangulated category $kk^h$ and a functor $j^h:\operatorname{Alg}^*_\ell\to kk^h$ that is homotopy invariant, matricially and hermitian stable and excisive and is universal initial with these properties. We prove that a version of Karoubi's fundamental theorem holds in $kk^h$. By the universal property of the latter, this implies that any functor $H:\operatorname{Alg}^*_\ell\to\mathfrak{T}$ with values in a triangulated category which is homotopy invariant, matricially and hermitian stable and excisive satisfies the fundamental theorem. We also prove a bivariant version of Karoubi's $12$-term exact sequence.
