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Comparison of the regulators of Beilinson and of Borel. (English) Zbl 0667.14005

Beilinson’s conjectures on special values of L-functions, Meet. Oberwolfach/FRG 1986, Perspect. Math. 4, 169-192 (1988).
[For the entire collection see Zbl 0635.00005.]
Using the evaluation map e: \(Spec({\mathbb{C}})\times B.GL_ N({\mathbb{C}})\to B.GL_ N/{\mathbb{C}}\) and the n-th Chern class \(c_ n\) in Deligne-Beilinson cohomology \(H^*_{{\mathcal D}}\) one obtains a map from \(H_{2n-1}(GL_ N({\mathbb{C}}),{\mathbb{Z}})\) to \({\mathbb{R}}(n-1):=(2\pi i)^{n-1}\cdot {\mathbb{R}}\), which, composed with the Hurewicz map, gives a map \(K_{2n- 1}({\mathbb{C}})=\pi_{2n-1}(B.GL({\mathbb{C}})^+)\to {\mathbb{R}}(n-1)=H^ 1_{{\mathcal D}}(Spec({\mathbb{C}}),{\mathbb{R}}(n))\). For a finite number field k this leads to Beilinson’s regulator map \(r_{{\mathcal D}}:\quad H^ 1_{{\mathcal M}}(X,{\mathbb{Q}}(n))\to H^ 1_{{\mathcal D}}(X_{/{\mathbb{R}}},{\mathbb{R}}(n)),\) where \(X=Spec(k)\), \(H^ 1_{{\mathcal M}}\) denotes Beilinson’s motivic cohomology, i.e. the \(k^ n\)-eigensubspace of \(K_{2n-1}(k)\otimes {\mathbb{Q}}\) under the Adams operations \(\psi^ k\) and \(H^ 1_{{\mathcal D}}(X_{/{\mathbb{R}}},{\mathbb{R}}(n))\) is the Gal(\({\mathbb{C}}/{\mathbb{R}})\)-invariant part of \(\oplus_{\sigma:k\to {\mathbb{C}}}H^ 1_{{\mathcal D}}(Spec({\mathbb{C}}),{\mathbb{R}}(n)).\)
On the other hand, A. Borel [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Sér. 4, 613-636 (1977; Zbl 0382.57027)] constructed canonical “regulator elements” \(b_{2n-1}\) in the continuous cohomology groups \(H^{2n-1}_{cont}(GL({\mathbb{C}}),{\mathbb{R}}(n-1))\). These are the images of \((2\pi i)^ n\) times the generators \(u_{2n-1}\) of \(H^*_{Betti}(U_ N,{\mathbb{Q}})\) under the isomorphism \(H^*_{Betti}(U_ N,{\mathbb{R}})\otimes {\mathbb{C}}\simeq H^*_{cont}(GL_ N({\mathbb{C}}),{\mathbb{R}})\otimes {\mathbb{C}}\). Forgetting the topology and using the Hurewicz map one obtains Borel’s regulator map \(r_{Borel}:\quad K_{2n-1}(k)\otimes {\mathbb{Q}}\to H^ 1_{{\mathcal D}}(X_{/{\mathbb{R}}},{\mathbb{R}}(n)).\) It is the purpose of the paper under review to give a fairly detailed proof of the equivalence (up to a non- zero rational number) of both regulator maps, thus clarifying Beilinson’s sketch of a proof [A. A. Bejlinson, J. Sov. Math. 30, 2036-2070 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 24, 181-238 (1984; Zbl 0588.14013); appendix to § 2]. An essential step in the proof is the result that the image \(v_{2n-1}\) of the n-th Chern class \(c_ n\in H^{2n}_{DR}(B.GL_{N/{\mathbb{C}}})\) under the homomorphism \(H^{2n}_{DR}(B.GL_{N/{\mathbb{C}}})\to H^{2n-1}(g)\) is equal to a non-zero rational multiple of the image of \((2\pi i)^ n.u_{2n-1}\in H^{2n-1}_{Betti}(U_ N,{\mathbb{C}}) \) under the comparison isomorphism \(H^*_{DR}(U_ N)\otimes {\mathbb{C}}\simeq H^*_{Betti}(U_ N)\otimes {\mathbb{C}}\). Here g is the Lie algebra of \(GL_ N({\mathbb{C}})\). This homomorphism depends on the fact that \(H^{2n}\) of a suitable part of the Weil algebra equals the g-invariant subspace of the n-fold symmetric power of the dual of g. The construction of the Weil algebra rests on the property of the normalization functor inducing an equivalence of categories between the category of so-called reduced small co-simplicial algebras in a linear tensor category \({\mathcal A}\) and the category of reduced small differential graded algebras in \({\mathcal A}\). These notions were introduced by Beilinson in his paper cited above.
A second main ingredient in the proof of equivalence is Beilinson’s interpretation of the van Est isomorphism in continuous cohomology, necessary in Borel’s formulation of the regulator map, as a restriction map to the cohomology of an infinitesimal version of the classifying space, namely the largest small simplicial subscheme.
Reviewer: W.W.J.Hulsbergen

MSC:

14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
17B56 Cohomology of Lie (super)algebras
14A20 Generalizations (algebraic spaces, stacks)