Correction to the Paper

“Classical Motivic Polylogarithm

According to Beilinson and Deligne”

(Doc. Math. J. DMV 3 (1998), 27–133)

Annette Huber, Jörg Wildeshaus

Received: December 7, 1998

Communicated by Peter Schneider

Abstract. The Hodge theoretic appendix of our paper contains results (A.2.10, A.2.11) about the explicit shape of the groups of one-extensions in the category of Hodge structures. Regrettably, they are incorrect in general. Here, we give the right formulae. Since they coincide with the old ones for Hodge structures of strictly negative weights (e.g., Tate twists A(n), n ≥ 1), this correction has no effect on the main text of our paper.

In fact, the results from [B1] and [Jn3] quoted in the proof of A.2.10 refer to extensions in the category MHS′_{A} of mixed A-Hodge structures, while the statement of A.2.10 was about extensions in the category MHS_{A} of graded-polarizable Hodge structures. Using the notation of Appendix A, we have:

Theorem 1. For any H ∈ MHS_{A}, there is a canonical isomorphism

W_{−1}H_{ℂ}/W_{−1}H_{ℂ} ∩ (W_{0}H_{A} + W_{0}F^{0}H_{ℂ}) ⟶^{∼} Ext1 ; MHS_{A}(A(0), H)

given by sending the class of h ∈ W_{−1}H_{ℂ} to the extension described by the matrix

( \begin{matrix}1 & 0 \\ −h & id_{H} \end{matrix} ) .

This means that we equip ℂ ⊕ H_{ℂ} with the diagonal weight and Hodge filtrations, and the A–rational structure extending the A–rational structure H_{A} of H_{ℂ} by the vector

1 − h ∈ ℂ ⊕ H_{ℂ} .

Proof. We may assume that H does not have strictly positive weights: look at the Ext· ; MHS_{A}(A(0), ) sequence associated to the short exact sequence

0 ⟶ W_{0}H ⟶ H ⟶ H/W_{0}H ⟶ 0 .

Our claim will be a consequence of a comparison of the Ext_{MHS′_{A}}^{·}(A(0), ) and the Ext· ; MHS_{A}(A(0), ) sequences associated to the short exact sequence

0 ⟶ W_{−1}H ⟶ H ⟶ Gr_{0}^{W} H ⟶ 0 .

We use the formula from [B1], § 1 or [Jn3], 9.2, 9.3 for Ext_{MHS′_{A}}^{·}(A(0), ), together with the following observations: 1) The Ext groups coincide if H has only strictly negative weights (since any extension in MHS′_{A} of A(0) by H will then automatically be graded-polarizable). 2) Ext_{MHS′_{A}}^{1}(A(0), H) is trivial if H is pure of weight 0 (any extension in MHS_{A} of pure objects of the same weight splits). 3) The canonical map from Ext1 ; MHS_{A} to Ext_{MHS′_{A}}^{1} is injective.

It follows that Ext1 ; MHS_{A}(A(0), H) equals the image of

Ext1 ; MHS_{A}(A(0), W_{−1}H) = W_{−1}H_{ℂ}/(W_{−1}H_{A} + W_{−1}F^{0}H_{ℂ})

in

Ext_{MHS′_{A}}^{1}(A(0), H) = W_{0}H_{ℂ}/(W_{0}H_{A} + W_{0}F^{0}H_{ℂ}) .

By replacing [B1], § 1 and [Jn3], 9.2, 9.3 by Theorem 1 in the original proofs of A.2.10 and A.2.11, one sees that the correct statements read as follows:

Theorem A.2.10. For any H ∈ MHS_{A}^{+}, there is a canonical isomorphism

(W_{−1}H_{ℂ}/W_{−1}H_{ℂ} ∩ (W_{0}H_{A} + W_{0}F^{0}H_{ℂ}))^{+} ⟶^{∼} Ext_{MHS_{A}^{+}}^{1}(A(0), H)
= H_{ℌ^{p}}^{1}(Spec(ℝ)/ℝ, H) ,

where the superscript + on the left hand side denotes the fixed part of the de Rham– conjugation

W_{−1}H_{ℂ}/W_{−1}H_{ℂ} ∩ (W_{0}H_{A} + W_{0}F^{0}H_{ℂ}) ⟶^{c_{∞}} W_{−1}H_{ℂ}/W_{−1}H_{ℂ} ∩ (W_{0}H_{A} + W_{0}\overline{F}^{0}H_{ℂ})
= W_{−1} ι^{∗}H_{ℂ}/W_{−1} ι^{∗}H_{ℂ} ∩ (W_{0} ι^{∗}H_{A} + W_{0}F^{0}ι^{∗}H_{ℂ})
⟶^{F_{∞}} W_{−1}H_{ℂ}/W_{−1}H_{ℂ} ∩ (W_{0}H_{A} + W_{0}F^{0}H_{ℂ}) .

The isomorphism is given by sending the class of h ∈ W_{−1}H_{ℂ} to the extension described by the matrix

( \begin{matrix}1 & 0 \\ −h & id_{H} \end{matrix} ) .

This means that we equip ℂ ⊕ H_{ℂ} with the diagonal weight and Hodge filtrations, and the A–rational structure extending the A–rational structure H_{A} of H_{ℂ} by the vector

1 − h ∈ ℂ ⊕ H_{ℂ} ,

thereby obtaining an extension E of A(0) by H in the category MHS_{A}.

The conjugate extension ι^{∗}E ∈ Ext1 ; MHS_{A}(A(0), ι^{∗}H) is given, with the same notation, by the matrix

( \begin{matrix}1 & 0 \\ −F_{∞}(h) & id_{ι^{∗}H} \end{matrix} ) ,

and the extension of F_{∞} to an isomorphism

F_{∞} : E ⟶^{∼} ι^{∗}E

sends 1 − h to 1 − F_{∞}(h). Thus

(F_{∞})_{ℂ} = id ⊕(F_{∞})_{ℂ} : ℂ ⊕ H_{ℂ} ⟶ ℂ ⊕ ι^{∗}H_{ℂ} .

Corollary A.2.11. Let X/ℝ be finite and reduced, and M ∈ MHM_{A}(X/ℝ). Then there is a canonical isomorphism

( ⨁_{x∈X(ℂ)} W_{−1}M_{x,ℂ}/W_{−1}M_{x,ℂ} ∩ (W_{0}M_{x,A} + W_{0}F^{0}M_{x,ℂ}))^{+}

⟶^{∼}_{A.2.10} Ext_{MHS_{A}^{+}}^{1} (A(0), ⨁_{x∈X(ℂ)} M_{x})
= H_{ℌ^{p}}^{1}(X/ℝ, M) .

References

[B1] A.A. Beilinson, “Notes on absolute Hodge cohomology”, in “Applications of Algebraic K–theory to Algebraic Geometry and Number Theory”, Proceedings of a Summer Research Conference held June 12–18, 1983, in Boulder, Colorado, Contemp. Math., vol. 55, Part I, AMS, Providence, pp. 35–68.

[Jn3] U. Jannsen, “Mixed Motives and Algebraic K– Theory”, LNM 1400, Springer–Verlag 1990.

Annette Huber Math. Institut Einsteinstr. 62 48149 Münster Germany huber@math.uni-muenster.de Jörg Wildeshaus Math. Institut Einsteinstr. 62 48149 Münster Germany wildesh@math.uni-muenster.de 300

Documenta Mathematica 3 (1998)