The Hopf algebra and the motivic Galois group of a field of characteristic zero. I (Q2878666)

This is the first of two papers (for part II, cf. [ibid. 693, 151--226 (2014; Zbl 1299.14022)]) 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 \textit{F. Morel} and \textit{V. Voevodsky} [Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007)]. Let \(k\) be a field with an immersion \(\sigma: k\to\mathbb{C}\) and let \(\Lambda\) be a ring of coefficients. Let \(\mathbb{D}\mathbb{A}(k,\Lambda)\) (resp. \(\mathbb{D}\mathbb{A}^{\mathrm{eff}}(k,\Lambda)\)) be the category of motives, without transfers (resp. of effective motives) over \(k\) with coefficients in \(\Lambda\).NEWLINENEWLINE The first step is to develop a general formalism that associates to a monoidal functor \(f\) a Hopf algebra in the target category of \(f\). Then the author applies this formalism to the Betti realization functor \(f\) from \(\mathbb{D}\mathbb{A}(k, \Lambda)\) to the derived category \(\mathbb{D}(\Lambda)\) of \(\text{Mod\,}\Lambda\), and to the functor \(g\) from \(\mathbb{D}\mathbb{A}^{\mathrm{eff}}(k,\Lambda)\), to \(\mathbb{D}(\Lambda)\). In order to do so one has to show that \(f\) and \(g\) satisfy certain conditions for applying a sort of Tannakian duality. In is ways he gets two bialgebras in \(\mathbb{D}(\Lambda)\), denoted by \({\mathcal H}_{\mathrm{mot}}(k,\sigma,\Lambda)\) and \({\mathcal H}^{\mathrm{eff}}_{\mathrm{mot}}(k,\sigma,\Lambda)\), where \({\mathcal H}_{\mathrm{mot}}(k,\sigma,\Lambda)\) is a Hopf algebra.NEWLINENEWLINE By using Grothendieck's result on the comparison between singular and de Rham cohomology, a construction is given of a quasi-isomorphism between \({\mathcal H}^{\mathrm{eff}}_{\mathrm{mot}}(k,\sigma,\Lambda)\) and an explicit complex of differential forms, which is \(0\) in strictly negative homological degree. It follows that NEWLINE\[NEWLINEH_n({\mathcal H}_{\mathrm{mot}}(k,\sigma,\Lambda)={\mathcal H}^{\mathrm{eff}}_{\mathrm{mot}}(k,\sigma,\Lambda)= 0NEWLINE\]NEWLINE for \(n<0\). Therefore \(H_0({\mathcal H}_{\mathrm{mot}}(k,\sigma,\Lambda)\) and \(H_0({\mathcal h}_{\mathrm{MOT}}(K,\sigma,\Lambda)\) are naturally \(\Lambda\)-bialgebras. In particular \(H_0({\mathcal H}_{\mathrm{mot}}(k,\sigma,\Lambda)\) is a Hopf \(\Lambda\)-algebra. Then one defines NEWLINE\[NEWLINE\mathbb{G}_{\mathrm{mot}}(k,\sigma,\Lambda) = \text{Spec}(H_0({\mathcal H}_{\mathrm{mot}}(k,\sigma,\Lambda))NEWLINE\]NEWLINE which is an affine pro-group scheme on \(\text{Spec}(\Lambda)\).NEWLINENEWLINE The above construction resembles that of Grothendieck, which is however based on the standard conjectures. Other constructions are due to Y. André, André-Kahn and Nori. In particular Nori's construction of a motivic Galois group \(\mathbb{G}_{\mathrm{Nori}}(k,\sigma,\Lambda)\) is based on his Tannakian theorem. While it is not yet proved that \(\mathbb{G}_{\mathrm{mot}}(k,\sigma, \Lambda)\simeq \mathbb{G}_{\mathrm{Nori}}(k,\sigma, \Lambda)\), the advantage of the construction given in this paper seems to be that it fits better into the theory of motives as developed by Morel and Voevodsky [loc. cit.]. In particular it gives an explicit description of \({\mathcal H}_{\mathrm{mot}}(k,\sigma, \Lambda)\) a cycles complex.NEWLINENEWLINE In the last part of this paper the author suggests two conjectures, in the case \(\Lambda=\mathbb{Q}\), the first of them asserting that the homology of the complex \({\mathcal H}^{\mathrm{eff}}_{\mathrm{mot}}(k, \sigma,\mathbb{Q})\) is concentrated in degree \(0\). Then he shows that these conjectures imply the so= called Beilinson-Soulé conjecture, which predicts that the motivic complexes \(\mathbb{Q}(r)\) have no cohomology in strictly negative degrees. He also asserts that these conjecture should probably imply the existence of a motivic \(t\)-structure on \(\mathbb{D}\mathbb{A}^{\mathrm{et}}(k,\mathbb{Q})\) as well as the equivalence between the subcategory of \(\mathbb{D}\mathbb{A}^{\mathrm{et}}(k,\mathbb{Q})\) generated by compact objects and the derived category of bounded complexes of the category of finite-dimensional representations of the motivic Galois group \(\mathbb{G}_{\mathrm{mot}}(k,\sigma,\mathbb{Q})\).