For every regular scheme X, every integer q, and every Tate twist r, the Cousin (Gersten) complex for p-adic étale Tate-twist cohomology is exact: 0→ H^q_Nis( Z_p(r))→bigoplus_x∈ X^(0)H^q_x(X, Z_p(r))→bigoplus_x∈ X^(1)H^q+1_x(X, Z_p(r))→bigoplus_x∈ X^(2)H^q+2_x(X, Z_p(r))→…, where X^(c) denotes the set of points of codimension c and H^_x denotes cohomology with support at x.

Gersten conjecture

For every regular scheme $X$ , every integer $q$ , and every Tate twist $r$ , the Cousin (Gersten) complex for $p$ -adic étale Tate-twist cohomology is exact: $0\to\mathcal H^q_{\mathrm{Nis}}(\mathbb Z_p(r))\to\bigoplus_{x\in X^{(0)}}H^{q}_{x}(X,\mathbb Z_p(r))\to\bigoplus_{x\in X^{(1)}}H^{q+1}_{x}(X,\mathbb Z_p(r))\to\bigoplus_{x\in X^{(2)}}H^{q+2}_{x}(X,\mathbb Z_p(r))\to\cdots$ , where $X^{(c)}$ denotes the set of points of codimension $c$ and $H^*_{x}$ denotes cohomology with support at $x$ .

References

The Gersten conjecture for p-adic étale Tate twists and the p-adic cycle class map — manuscripta mathematica — Morten Lüders

Source: manuscripta mathematica

Posted 2 hours ago · edited 2 hours ago

Progress summary

Refreshed 2 hours ago

Open · progress reported

A recent paper proves an important mixed-characteristic special case, but the full conjecture remains open.

The Gersten conjecture concerns exactness phenomena connecting algebraic cycles, étale cohomology, and motivic methods. No proposer or original date is given in the retrieved sources.

2026 mixed-characteristic case

Morten Lüders proves the Gersten property for $p$ -adic étale Tate twists of smooth schemes over suitable mixed-characteristic henselian discrete valuation rings. The proof uses the $\mathbb{P}^{1}$ -bundle formula, identifies the twists with syntomic complexes, and extends the relevant $p$ -adic cycle class map to arbitrary finite residue fields. This is a substantial scoped advance, not a proof of the general conjecture; the mathematical claim remains unverified in the retrieved evidence.

Current status (as of October 2026): A substantial mixed-characteristic case for $p$ -adic étale Tate twists is claimed in Lüders's paper, while the Gersten conjecture in full generality remains open.

Sources

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