AI 中文总结
该研究建立亨塞尔赋值域的有限系数K-理论等价关系，证明诺特亨塞尔正则局部环的有限系数格斯滕内射性，推广到普吕弗环上的亨塞尔局部 ind-光滑代数的K-理论内射性。
AI 中文摘要
设W为具有分式域L、剩余域k和值群Γ_W的亨塞尔赋值环，N=ℓ^ν在W中可逆。选取Γ_W/NΓ_W的有序Z/N基B，合适类的乘积定义了完全过滤谱的等价：对所有有限子集J⊆B，求和项为Σ^{|J|}Fil_{mot}^{•-|J|}K(k;Z/N)，该求和项等价于Fil_{mot}^{•}K(L;Z/N)。由刚性等价K(W;Z/N)≃K(k;Z/N)，由空集索引的求和项为通用限制映射，因此是分裂内射的。独立于该分裂，切除定理给出正则亨塞尔对的提升定理，特别地，这给出诺特亨塞尔正则局部环的有限系数格斯滕内射性。对正则素数的完备化应用得到相对且有时非亨塞尔的例子。更一般地，若P为普吕弗环，R为亨塞尔局部 ind-光滑P-代数，则R是整环，且对每个n，当N∈R^×时，K_n(R;Z/N)→K_n(Frac(R);Z/N)是内射的，这些内射性结论通过 primary decomposition 从素幂系数扩展到任意有限可逆系数。
英文摘要
Let $W$ be a henselian valuation ring with fraction field $L$, residue field $k$, and value group $Γ_W$. Let $N=\ell^ν$ be invertible in $W$. Choose an ordered $\mathbf Z/N$-basis $B$ of $Γ_W/NΓ_W$. Products of suitable classes define an equivalence of complete filtered spectra $$ \bigoplus_{\substack{J\subseteq B\\J\text{ finite}}} Σ^{|J|}\operatorname{Fil}_{\mathrm{mot}}^{\bullet-|J|} K(k;\mathbf Z/N) \xrightarrow{\simeq} \operatorname{Fil}_{\mathrm{mot}}^{\bullet} K(L;\mathbf Z/N) $$ The summand indexed by $\varnothing$ is the generic restriction map, after the rigidity equivalence $K(W;\mathbf Z/N)\simeq K(k;\mathbf Z/N)$, and is therefore split injective. Independently of this splitting, excision yields a lifting theorem for regular henselian pairs. In particular, this yields finite-coefficient Gersten injectivity for noetherian henselian regular local rings. Applications to completions along regular primes give relative and sometimes nonhenselian examples. More generally, if $P$ is a Prüfer ring and $R$ is a henselian local ind-smooth $P$-algebra, then $R$ is a domain and $K_n(R;\mathbf Z/N)\to K_n(\operatorname{Frac}(R);\mathbf Z/N)$ is injective for every $n$, provided $N\in R^\times$. These injectivity consequences extend from prime-power to arbitrary finite invertible coefficients by primary decomposition.
Comments42 pages
