论文标题
拓扑Hochschild同源性和Zeta值
Topological Hochschild homology and Zeta-values
论文作者
论文摘要
使用Antieau和Bhatt-Morrow-Scholze的作品，我们定义了拓扑Hochschild同源性的过滤及其变体$ tp $和$ tc^ - $ tc^ - $ tc^ - $ tci-lci响应有界的扭转，在$ p $ adadic的完成后恢复了BMS滤光。然后，我们根据基本环$ \ mathbb {z} $来计算hodge完成的衍生的de rham同胞衍生的de rham共同体。我们用$ \ mathrm {gr}^{n} tc^ - （ - ）$ to $ \ mathrm {gr}^{n} tp（ - ）$ by $lΩ^{<n} _ { - /\ m athbb {s}} $。令$ \ mathcal {x} $为$ \ mathrm {spec}（\ mathbb {z}）$的尺寸$ d $的常规连接方案，然后让$ n \ in \ mathbb {z} $是任意整数。连同紧凑的支持$rγ_{w，c}（\ Mathcal {x}，\ Mathbb {z}（n））$，复杂$LΩ^{<n} _ {\ natcal { $ \pmζ^*（\ Mathcal {x}，n）$。将结果与最近与Flach联合合作的定理相结合，我们获得了一个公式，与$LΩ^{<n} _ {\ Mathcal {x}/\ Mathbb {s}} $，$LΩ tate tate $ n $和$ d-n $的Archimedean Fiber $ \ MATHCAL {X} _ {\ infty} $，bloch导体$ a（\ Mathcal {x}）$以及Zeta-function $ qunction $ quamclunction $ quallinct $ quallyean Euler euler euler euler euler euter firect的特殊值该公式是zeta功能功能方程的阴影。
Using work of Antieau and Bhatt-Morrow-Scholze, we define a filtration on topological Hochschild homology and its variants $TP$ and $TC^-$ of quasi-lci rings with bounded torsion, which recovers the BMS-filtration after $p$-adic completion. Then we compute the graded pieces of this filtration in terms of Hodge completed derived de Rham cohomology relative to the base ring $\mathbb{Z}$. We denote the cofiber of the canonical map from $\mathrm{gr}^{n}TC^-(-)$ to $\mathrm{gr}^{n}TP(-)$ by $LΩ^{<n}_{-/\mathbb{S}}[2n]$. Let $\mathcal{X}$ be a regular connected scheme of dimension $d$ proper over $\mathrm{Spec}(\mathbb{Z})$ and let $n\in\mathbb{Z}$ be an arbitrary integer. Together with Weil-étale cohomology with compact support $RΓ_{W,c}(\mathcal{X},\mathbb{Z}(n))$, the complex $LΩ^{<n}_{\mathcal{X}/\mathbb{S}}$ is expected to give the Zeta-value $\pmζ^*(\mathcal{X},n)$ on the nose. Combining the results proven here with a theorem recently proven in joint work with Flach, we obtain a formula relating $LΩ^{<n}_{\mathcal{X}/\mathbb{S}}$, $LΩ^{<d-n}_{\mathcal{X}/\mathbb{S}}$, Weil-étale cohomology of the archimedean fiber $\mathcal{X}_{\infty}$ with Tate twists $n$ and $d-n$, the Bloch conductor $A(\mathcal{X})$ and the special values of the archimedean Euler factor of the Zeta-function $ζ(\mathcal{X},s)$ at $s=n$ and $s=d-n$. This formula is a shadow of the functional equation of Zeta-functions.
