论文标题
学位一级$ k $ - 各种各样的乘法类型
Degree One Milnor $K$-invariants of Groups of Multiplicative Type
论文作者
论文摘要
令$ g $为field $ f $的交换仿射代数组，让$ h \ colon \ colon \ mathrm {fields} _ {f} \ to \ to \ mathrm {abgrps} $成为函数。 $ g $的（同构）$ h $ invariant是一种自然转换$ \ mathrm {tors}（ - ，g）\ to h $，其中$ \ mathrm {tors}（ - ，g）$是functor $ $ $ \ m mathrm {fields} _ {fields} _ {f} $ abgr $ tho $ tho $ tho $ \ $ \ $ \ l/ $ g_ {l} $的同构类别的组 - 超过$ \ mathrm {spec}（l）$。本文的目的是计算$ \ mathrm {inv} _ {\ mathrm {hom}}}^{1}^{1}（g，h）$ h $ g $ g $ g $ g $ g $ g $时，当$ g $是一组乘法时，而$ h $是fighters Exterment of Field Exterment $ l/f $ l/f $ l/f $ l l^$ l l^} \ otimes _ {\ Mathbb {z}}} \ Mathbb {q}/\ Mathbb {z} $。
Let $G$ be a commutative affine algebraic group over a field $F$, and let $H \colon \mathrm{Fields}_{F} \to \mathrm{AbGrps}$ be a functor. A (homomorphic) $H$-invariant of $G$ is a natural transformation $\mathrm{Tors}(-, G) \to H$, where $\mathrm{Tors}(-, G)$ is the functor $\mathrm{Fields}_{F} \to \mathrm{AbGrps}$ taking a field extension $L/F$ to the group of isomorphism classes of $G_{L}$-torsors over $\mathrm{Spec}(L)$. The goal of this paper is to compute the group $\mathrm{Inv}_{\mathrm{hom}}^{1}(G, H)$ of $H$-invariants of $G$ when $G$ is a group of multiplicative type, and $H$ is the functor taking a field extension $L/F$ to $L^{\times} \otimes_{\mathbb{Z}} \mathbb{Q}/\mathbb{Z}$.
