Let $R$ be a non-Archimedean Banach ring, satisfying some mild technical hypothesis that we will specify later on. We prove that it is possible to associate to $R$ a homotopical Huber spectrum ${\mathrm{Spa}}^h(R)$ via the introduction of the notion of derived rational localization. The spectrum so obtained is endowed with a derived structural sheaf ${\mathcal{O}}_{{\mathrm{Spa}}^h(R)}$ of simplicial Banach algebras for which the derived C̆ech–Tate complex is strictly exact. Under some hypothesis, we can prove that there is a canonical morphism of underlying topological spaces $|\mathrm{Spa}(R)|\to |{\mathrm{Spa}}^h(R)|$ that is a homeomorphism in some well-known examples of non-sheafy Banach rings, where $\mathrm{Spa}(R)$ is the usual Huber spectrum of $R$. This permits the use of the tools from derived geometry to understand the geometry of $\mathrm{Spa}(R)$ in cases when the classical structure sheaf $H^0({\mathcal{O}}_{\mathrm{Spa}(R)})$ is not a sheaf.

中文翻译：

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Banach环光谱的弹性性质

让$右$是一个非阿基米德巴纳赫环，满足我们稍后将详细说明的一些温和的技术假设。我们证明可以关联到$右$同伦 Huber 谱${\mathrm{温泉}}^H（右）$通过引入派生理性本地化的概念。如此获得的频谱被赋予了衍生的结构束${\mathcal{氧}}_{{\mathrm{温泉}}^H（右）}$单纯 Banach 代数，其派生的 C̆ech-Tate 复形是严格精确的。在某些假设下，我们可以证明基础拓扑空间存在正则态射$|\mathrm{温泉}（右）|\to |{\mathrm{温泉}}^H（右）|$这是一些著名的非束巴纳赫环例子中的同胚，其中$\mathrm{温泉}（右）$是通常的 Huber 谱$右$。这允许使用衍生几何的工具来理解几何$\mathrm{温泉}（右）$在经典结构束的情况下$H^0（{\mathcal{氧}}_{\mathrm{温泉}（右）}）$不是一捆。