The Hopf algebra and the motivic Galois group of a field of characteristic zero. II (Q2878668)

This is the second of two papers (for part I, cf. [ibid. 693, 1--149 (2014; Zbl 1299.14020)]), where the author develops a theory of motivic Galois groups over a field of characteristic \(0\), in the framework of the triangulated category of motives, as constructed by Morel and Voevodsky. In his first paper the author has constructed motivic Galois groups and their algebras of regular functions, described in terms of differential forms or algebraic cycles.NEWLINENEWLINE In the first part of this article he shows the link between the motivic Galois group \(\mathbb{G}_{\mathrm{mot}}(k,\sigma,\Lambda)\) and the usual Galois group of a subfield \(k\) of \(\mathbb{C}\). Here \(\sigma:K\to\mathbb{C}\) is an embedding, \(\overline k\) the algebraic closure of \(k\) in \(\mathbb{C}\) and \(\Lambda\) is a ring of coefficients. There is a morphism of group pro-schemes NEWLINE\[NEWLINE\mathbb{G}_{\mathrm{mot}}(k,\sigma,\Lambda)\to \text{Gal}(\overline k/k)_\Lambda,\tag{1}NEWLINE\]NEWLINE is the Galois Group of \(\overline k/k\), viewed as a constant group pro-scheme over \(\text{Spec}(\Lambda)\). The morphism in (1) is defined from a morphism between Hopf algebras NEWLINE\[NEWLINE{\mathcal C}^0(\text{Gal}(\overline k/k),\Lambda)\to{\mathcal H}_{\mathrm{mot}}(k,\sigma, \Lambda),\tag{2}NEWLINE\]NEWLINE where \({\mathcal C}^0(\text{Gal}(\overline k/k), \Lambda)\) is the ring of continous functions from \(\text{Gal}(\overline k/k)\), endowed with the profinite topology and the ring \(\Lambda\) with the discrete topology. The author proves that the morphism in (2) is an isomorphism in the case \(\Lambda\) is torsion, hence also (1) is invertible.NEWLINENEWLINE The morphism in (1) is always surjective and it is conjectured that its kernel is connected. If so, the Galois group of \(\overline k\) over \(k\) identifies the group of connected components of the motivic Galois group.NEWLINENEWLINE In the case \(\Lambda=\mathbb{Z}/l^n\), with \(l\) a prime number, one gets natural isomorphisms NEWLINE\[NEWLINE\lim_n\,{\mathcal H}_{\mathrm{mot}}(k,\sigma,\mathbb{Z}/l^n)\simeq\lim_n\,{\mathcal C}^0(\text{Gal}(\overline k/k),\mathbb{Z}/l^n)\simeq{\mathcal C}^0(\text{Gal}(\overline k/k), \mathbb{Z}_l),NEWLINE\]NEWLINE where \({\mathcal C}^0(\text{Gal}(\overline k/k), \mathbb{Z}_l)\) is the ring of continuous functions from the Galois group of \(\overline k/k\) to the ring of \(l\)-adic integers, endowed with the usual topology. It follows that to any representation of \(\mathbb{G}_{\mathrm{mot}}(k,\sigma,\Lambda)\) corresponds a \(l\)-adic representation of the construction, in the case of the canonical representation of \(\mathbb{G}_{\mathrm{mot}}(k,\sigma,\Lambda)\) on \({\mathbf B}{\mathbf t}{\mathbf i}^*(M)\) gives the usual \(l\)-adic representation on \({\mathbf B}{\mathbf t}{\mathbf i}^*(M)\otimes_{\mathbb{Z}}\mathbb{Z}_l\). Here \({\mathbf B}{\mathbf t}{\mathbf i}^*\) is the Betti realization functor from the category of motives \({\mathbf D}{\mathbf A}(k,\mathbb{Z})\) to the derived category \(\mathbb{D}(\mathbb{Z})\) of \(\text{Mod}(\mathbb{Z})\) and \(M\in\mathbb{D}{\mathbf A}(k,\mathbb{Z})\).NEWLINENEWLINE The second part of this article contains a theory of ramification and inertia groups, associated Galois group for a finite extension \(K/k\). Let \(\sigma:K\to\mathbb{C}\) be an immersion. The motivic Galois group of \(K\) relative to \(k\) is the kernel \(\mathbb{G}^{\mathrm{rel}}_{/k}(K,\sigma,\Lambda)\) of the homomorphism \(\mathbb{G}_{\mathrm{mot}}(K,\sigma,\Lambda)\to \mathbb{G}_{\mathrm{mot}}(k,\sigma, \Lambda)\).NEWLINENEWLINE At the level of graded \(f\) \(\Lambda\)-algebras this construction yields a Hopf algebra \({\mathcal H}^{\mathrm{rel}}_{/k}(K,\sigma, \Lambda)_{gr}\) such that \(\mathbb{G}^{\mathrm{rel}}_{/k}(K,\sigma, \Lambda)= \text{Spec}({\mathcal H}^{rel}_{/k}(K,\sigma,\Lambda)_{\mathrm{gr}})\). Then it is proved that the algebra \({\mathcal H}^{\mathrm{rel}}_{/k}(K,\sigma, \Lambda)_{\mathrm{gr}}\) is concentrated in degree \(0\). In other words this Hopf algebra is determined by the group pro-scheme \(\mathbb{G}_{\mathrm{mot}}(K,\sigma,\Lambda)\).