Leopoldt's conjectural description of the zero Tate twist over an arithmetic base

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Let SS be the arithmetic scheme in the source, let jj be the relevant open immersion, let ee be the indicated cup-product class, and let A(1)A(1) be the real Tate-twist object. Consider the complex Ql(0)∣Sc{\Bbb Q}_l(0)^c_{|S}. Leopoldt conjecture. There is an exact triangle

j∗Ql[−1]→⋅eQl(0)∣Sc→j∗A(1)⊗Ql[−1]→j∗Ql[0].j_*{\Bbb Q}_l[-1]\xrightarrow{\cdot e}{\Bbb Q}_l(0)^c_{|S}\to j_*A(1)\otimes {\Bbb Q}_l[-1]\to j_*{\Bbb Q}_l[0].

In particular, Hi(Ql(0)∣Sc)=0\mathcal H^i({\Bbb Q}_l(0)^c_{|S})=0 for i≠1i\ne1. The source states that this conjectural description is equivalent to the Leopoldt conjecture.

References

Primary source

Bruno Kahn, “A sheaf-theoretic reformulation of the Tate conjecture”, arXiv:math/9801017 (1998).

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