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Classical motivic polylogarithm according to Beilinson and Deligne. (English) Zbl 0906.19004
Doc. Math. 3, 27-133 (1998); correction ibid. 3, 297-299 (1998).
The central topic of the paper is an alternative proof of the following theorem, for which a sketch of proof was first given by Beilinson, and a thorough treatment of the ideas related to it as advocated by Beilinson and Deligne: Let \(\mu_d^0\), \(d\geq 2\), denote the primitive \(d\)-th roots of unity in \(F:={\mathbb{Q}}(\mu_d)\). Let \(H_{\mathcal M}^{\bullet}\) denote Beilinson’s motivic cohomology, i.e., a suitable eigenspace of the Adams operations on \(K_{\ldots}(F)\otimes{\mathbb{Q}}\). Then, for \(n\geq 0\), there is a map of sets \[ \varepsilon_{n+1}:\mu_d^0\rightarrow H^1_{\mathcal M}(\text{Spec}(F), {\mathbb{Q}}(n+1)), \] which, combined with the regulator map \[ r_{\mathcal D}:H^1_{\mathcal M}(\text{Spec}(F),{\mathbb{Q}}(n+1))\rightarrow\bigoplus_{\sigma:F\hookrightarrow{\mathbb{C}}} {\mathbb{C}}/(2\pi i)^{n+1}{\mathbb{R}}, \] maps an element \(\omega\in\mu_d^0\) to the polylogarithm values \[ (-Li_{n+1}(\sigma\omega))_{\sigma}=\Biggl(-\sum_{k\geq 1}{\sigma\omega^k\over k^{n+1}} \Biggr)_{\sigma}. \] Using the \(\ell\)-adic regular map \[ r_{\ell}:H^1_{\mathcal M} (\text{Spec}(F),{\mathbb{Q}}(n+1))\rightarrow H^1_{cont} (\text{Spec}(F),{\mathbb{Q}}_{\ell}(n+1)), \] one obtains an \(\ell\)-adic counterpart to the above result. In particular, this \(\ell\)-adic result has interesting consequences for Soulé’s work on cyclotomic elements in \(K_{2n+1}(F)\otimes{\mathbb{Q}}\), \(F\) an abelian number field, and it is a key ingredient in the proof of the Tamagawa number conjecture of Bloch-Kato.
The strategy followed by the authors consists in a number of steps. Firstly, the realization theoretic side is constructed, and then secondly, the (motivic) \(K\)-theory enters. Both are related by the regulator maps. The necessary tools are developed in two large and valuable appendices, the first on mixed Hodge modules with coefficients, and the second on \(K\)-theory of simplicial schemes and regulators.
As to the realizations, it is necessary to work with algebraic mixed Hodge modules (MHM) over \({\mathbb{R}}\) on the one side, and with mixed \(\ell\)-adic perverse sheaves (Perv, with respect to a suitable horizontal stratification) on the other one. The base scheme is \(B=\text{Spec}(A)\) with \(A={\mathbb{R}}\) in the first case and \(A={\mathbb{Z}}[{1/\ell}]\) (for a fixed prime \(\ell\)) in the second case. So, for a scheme \(X\rightarrow B\), one has the corresponding categories \(\text{Sh}(X)= \text{MHM}_{\mathbb{Q}}(X/{\mathbb{R}})\) and \(\text{ Sh}(X)=\text{Perv}(X,{\mathbb{Q}}_{\ell})\) of sheaves. For smooth \(X\) one has the subcategories \(\text{Sh}^s(X):= \text{Var}_{\mathbb{Q}}(X/{\mathbb{R}})\subset\text{Sh}(X)\) of admissible variations of mixed \({\mathbb{Q}}\)-Hodge structure (which is equivalent to the category of smooth algebraic \({\mathbb{Q}}\)-Hodge modules on \(X\)) and \(\text{Sh}^s(X)= \text{Et}^{\ell,m}_{{\mathbb{Q}}_\ell}(X)\subset\text{Sh}(X)\) of lisse mixed \({\mathbb{Q}}_{\ell}\)-sheaves on \(X\). \(U\text{Sh}^s(X)\) will denote the category of unipotent objects of \(\text{Sh}^s(X)\).
For a scheme \(a:X\rightarrow B\) one has the Tate twist \(F(n)_X:= a^*F(n)\in D^b\text{Sh}(X)\), where \(F(n)\) is the usual Tate twist on \(B\), i.e., \(F= {\mathbb{Q}}\) in the Hodge setting and \(F= {\mathbb{Q}}_{\ell}\) in the \(\ell\)-adic setting. One can now define, in a straightforward manner, the absolute cohomology with coefficients in the object \(M^{\bullet}\in D^b\text{Sh}(X)\), \(H_{\text{abs}}^i (X,M^{\bullet})\), and likewise, the absolute and relative absolute cohomology with Tate coefficients, \(H_{\text{abs}}^i (X,n):= H_{\text{abs}}^i (X,F(n)_X)\) and \(H_{\text{abs}}^i (X\text{ rel }Z,n):= H_{\text{abs}}^i (X,j_!F(n)_U)\), where \(Z\) is a closed reduced subscheme of the separated reduced flat \(B\)-scheme \(X\) of finite type with \(j:U\hookrightarrow X\) for the complement. For the polylogarithm one puts \(X= {\mathbb{G}}_m= {\mathbb{G}}_{m,B}\), \({\mathbb{U}}= {\mathbb{P}}^1_B\setminus\{0,1,\infty\}_B\), \(j:{\mathbb{U}}\hookrightarrow{\mathbb{G}}_m\), \(p:{\mathbb{G}}_m\rightarrow B\) and \(\widetilde{p}= p\circ j:{\mathbb{U}}\rightarrow B\). On \({\mathbb{G}}_m\) one has the (universal) logarithmic pro-sheaf \({\mathcal L}og\in\text{pro-}U \text{Sh}^s({\mathbb{G}}_m)\), which in the Hodge setting is just the canonical variation with base point \(1\). It has a weight filtration \(W\) such that \(\kappa:\text{Gr}^W{\mathcal L}og{\overset\sim\rightarrow} \prod_{k\geq 0}F(k)\). The cohomology of \({\mathcal L}og\) over \({\mathbb{U}}\) vanishes in degree \(\neq 0\), and in degree zero it has a weight filtration with \(W_{-1}\) isomorphic to the product of the Tate objects \(F(k)\), \(k\geq 1\). Fixing \(\kappa\), one may define the small polylogarithmic extension \(pol\in\text{Ext}^1_{U\text{Sh}^s({\mathbb{U}})}(\text{Gr}_{-2}^W{\mathcal L}og| _{\mathbb{U}},{\mathcal L}og(1)| _{\mathbb{U}})\).
For \(d\geq 2\), let \(R= A[1/d,T]/\Phi_d(T)\), where \(\Phi_d\) is the \(d\)-th cyclotomic polynomial, and write \(C= \text{Spec}(R)\). Then one has an embedding \(i_b:C{\overset\sim\rightarrow} C\hookrightarrow{\mathbb{G}}_m\otimes_AA[1/d]\), \(\zeta\mapsto\zeta^b\). Pulling back \({\mathcal L}og\) and \(pol\) via \(i_b\) one gets \(pol_b\in\text{Ext}_{\text{Sh}^s(C)}^1(F(1),{\mathcal L}og_b(1))\), where the subscript \(b\) denotes the pullback. One has the following splitting principle: \({\mathcal L}og_b= \prod_{k\geq 0} \text{Gr}^W_{-2k}({\mathcal L}og_b)\).
For the “values” of \(pol_b\) one obtains: (i) in the Hodge setting: \(pol_b= ((-1)^kLi_k(\omega^b))_{\omega,k}\in\prod_{k\geq 1} (\bigoplus_{\omega\in C({\mathbb{C}})}{\mathbb{C}}/(2\pi i)^k{\mathbb{Q}})^+\) and (ii) in the \(\ell\)-adic setting, choosing a geometric point \(\zeta\in C(\overline{\mathbb{Q}})\), \[ pol_b= \Biggl( (-1)^{k-1}\cdot{1\over d^{k-1}}\cdot{1\over(k-1)!}\cdot\sum_{\alpha^{\ell^r}= \zeta^b} [1-\alpha]\otimes(\alpha^d)^{\otimes(k-1)}\Biggr)_{r,k\geq 1} \] in \(H^1_{cont}({\mathbb{Q}}(\zeta),{\mathbb{Q}}_{\ell}(k))\), which may be identified with a \({\mathbb{Q}}_{\ell}\)-subspace of \[ \biggl(\varprojlim_{r\geq 1}\Bigl({\mathbb{Q}}(\mu_{\ell^{\infty}},\zeta)^*/({\mathbb{Q}}(\mu_{\ell^{\infty}},\zeta)^*)^{\ell^r}\otimes\mu_{\ell^r}^ {\otimes(k-1)}\Bigl)\otimes_{{\mathbb{Z}}_{\ell}}{\mathbb{Q}}_{\ell} \biggr)^{ \text{Gal}({\mathbb{Q}}(\mu_{\ell^{\infty}},\zeta)/{\mathbb{Q}}(\zeta))}. \] The whole set-up may be given a more geometric flavour by defining a pro-unipotent sheaf \({\mathcal G}\) on \({\mathbb{P}}^1\setminus\{0,1,\infty\}\), which may be identified with \({\mathcal L}og| _{\mathbb{U}}\). \({\mathcal G}\) is defined as the projective limit of relative cohomology objects of powers of \({\mathbb{G}}_m\) with coefficients in Tate twists over \({\mathbb{U}}\) relative to certain singular subschemes. This leads to the absolute cohomological interpretation of \(pol_b\) as an element of \(\varprojlim_{n\geq 1} H^{n+1}_{\text{abs}}({\mathbb{G}}_{m,C}^{\vee n},n)^{ \text{sgn}}\), where \({\mathbb{G}}_{m,C}^{\vee n}\) means \({\mathbb{G}}_{m,C}^n\) relative to \(Z^{(n)}\) for \(Z= \alpha(C)\coprod\beta(C)\subset{\mathbb{G}}_{m,C}\), where \(\alpha\), \(\beta\) are disjoint sections of \({\mathbb{G}}_m\rightarrow C\). The superscript sgn means invariants under the sign of the permutations induced from the action of \(\mathfrak S_n\) on the factors. The geometric construction of \({\mathcal G}\) also leads to an isomorphism \(H^0_{ \text{abs}}({\mathbb{U}},{\mathcal L}og| _{\mathbb{U}}){\overset\sim\rightarrow}H^0_{ \text{abs}}(B,0)\) such that \(pol\) maps to \(1\). In the other direction one obtains an inverse system \(pol^{(n)}\).
Concerning the motivic \(K\)-theoretic part of the theory, the setup parallels the absolute cohomological one. The material from the second appendix assures that one can work with simplicial schemes with regular components to imitate the absolute cohomological theory with singular schemes. In particular, one may construct a long exact sequence (the residue sequence) that is compatible under the regulator map, with a similar one constructed in absolute cohomology. This so-called motivic residue sequence maps to the absolute cohomological residue sequence. Now let \(B=\text{Spec}({\mathbb{Z}})\). Imitating the construction of the polylogarithm mentioned above, one is led to define the universal motivic polylog as the system \(pol_n=s_n(1)\) with \(s_n:H^0_{\mathcal M}(B,0)\rightarrow H^{n+1}_{\mathcal M}({\mathbb{G}}_{m,{\mathbb{U}}}^{\vee n}, n)^{\text{sgn}}\), which is mapped to the system \(pol^{(n)}\). The final step is the splitting principle in motivic cohomology. It can be stated as follows: There is a canonical isomorphism \[ \varprojlim_{n} H^{n+1}_{\mathcal M}({\mathbb{G}}_{m,C}^{\vee n},n)^{\text{sgn}}{\overset\sim\rightarrow}\prod_{i\geq 1}H^1_{\mathcal M}(C,i). \] Then, under the regulators, the element \(pol_b\in\displaystyle{ \varprojlim_{n}}H^{n+1}_{\mathcal M}({\mathbb{G}}_{m,C}^ {\vee n},n)^{\text{sgn}}\) is mapped to \[ pol_b\in\varprojlim_{n} H^{n+1}_{\text{abs}}({\mathbb{G}}_{m,C}^ {\vee n},n)^{\text{sgn}}=\prod_{i\geq 1}H^1_{\text{abs}}(C,i). \]
See the correction in Zbl 0906.19005.

19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
11R18 Cyclotomic extensions
11R34 Galois cohomology
11R42 Zeta functions and \(L\)-functions of number fields
14D07 Variation of Hodge structures (algebro-geometric aspects)
14F99 (Co)homology theory in algebraic geometry
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