Milnor K symbols and the Nisnevich sheaf

PDF p. 16: Milnor K sheaf in Remark 6.8

All rings here are commutative and unital, and ring homomorphisms preserve the unit. For a ring $A$, let $A^{\times }$ be its abelian group of units under multiplication. Regard it as a $\mathbb{Z}$-module, and form its tensor algebra $T_{\mathbb{Z}}(A^{\times })={⨁}_{n\ge 0}(A^{\times }{)}^{\otimes n}$, with degree-zero summand $\mathbb{Z}$. Let $I_A$ be the two-sided homogeneous ideal generated by $a\otimes (1-a)$ for all $a$ such that both $a$ and $1-a$ are units. The naive Milnor K-ring is $K_{\ast }^{M,\mathrm{n}\mathrm{a}\mathrm{i}\mathrm{v}\mathrm{e}}(A)=T_{\mathbb{Z}}(A^{\times })\mathrm{/}{I}_A$; its degree-$d$ abelian group, for an integer $d\ge 0$, is $K_d^{M,\mathrm{n}\mathrm{a}\mathrm{i}\mathrm{v}\mathrm{e}}(A)$. The image of $a_1\otimes \cdots \otimes a_d$ is written $\{a_1,\dots ,a_d\}$ and called a Milnor symbol. Thus symbols are additive in each entry with respect to multiplication of units, and $\{a,1-a\}=0$ whenever both entries are units. In degree zero the group is $\mathbb{Z}$.

Fix a scheme $X$ and the suitably small Nisnevich site of étale $X$-schemes specified in Nisnevich covers and distinguished squares, with the size convention of Scheme sites and sheafification. For an object $U$ put $P_d(U)=K_d^{M,\mathrm{n}\mathrm{a}\mathrm{i}\mathrm{v}\mathrm{e}}(\mathrm{\Gamma }(U,{\mathcal{O}}_U))$; for a morphism $V\to U$ use restriction of global functions. The naive Milnor K-sheaf ${\mathcal{K}}_d^{M,\mathrm{n}\mathrm{a}\mathrm{i}\mathrm{v}\mathrm{e}}$ is the Nisnevich sheafification $a{P}_d=(P_d^{+}{)}^{+}$, with its induced abelian-group structure. Here the plus construction and its sheafification unit are those defined in Scheme sites and sheafification. In particular, degree-zero sheafification is the sheafification of the constant presheaf $\mathbb{Z}$; its sections are not asserted to be $\mathbb{Z}$ on every object.

This specifies the naive convention only. In particular, the name ${\mathcal{K}}_d^M$ in a comparison must be accompanied by its intended convention; no identification with the separately constructed improved Milnor K-theory, including for local rings with finite residue fields, is asserted here. No comparison with motivic cohomology is asserted by this definition.

A unital ring homomorphism sends units to units and sends each generator $a\otimes (1-a)$ of $I_A$ to the corresponding generator for the target ring. It therefore induces a graded ring homomorphism on the quotients. Identities and composition are preserved, so restriction of global functions makes $P_d$ a presheaf of abelian groups. The ideal is homogeneous and has no degree-zero generators, giving the stated degree-zero group.

By Scheme sites and sheafification, applying plus twice produces a sheaf of sets with the stated universal property. Its group operations are induced by the presheaf operations: matching families can be added, negated and given zero componentwise. Families on two covering sieves can be compared on a common covering subsieve, and agreement after restriction is preserved by these operations. Hence the operations are well-defined on plus classes and satisfy the group laws locally, and therefore on the classes. Applying the same argument a second time gives an abelian-group sheaf. The degree-zero assertion follows because sheafification is applied to the presheaf with value $\mathbb{Z}$ and identity restrictions, rather than being evaluated before sheafification.

Remark 6.8 of the article uses $K_d^M$ in a reported comparison; the passage does not specify the naive or improved convention. This item fixes a usable naive convention and does not attribute that choice to the article.