Beilinson-Parshin adeles via solid algebraic geometry

Christopher Brav; Grigorii Konovalov

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arXiv · 2403.08472

Beilinson-Parshin adeles via solid algebraic geometry

Christopher Brav·Grigorii Konovalov

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Abstract

In this paper, we apply Clausen-Scholze's theory of solid modules to the existence of adelic decompositions for schemes of finite type over $\mathbb{Z}$. Specifically, we use the six-functor formalism for solid modules to define the skeletal filtration of a scheme, and then we show that decomposing a quasi-coherent sheaf with respect to this filtration gives rise to a new construction of the Beilinson-Parshin adelic resolution. As an application of the adelic decomposition combined with some nice completeness properties of the solid tensor product, we prove a version of adelic descent for solid quasi-coherent sheaves.

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Christopher Brav, Grigorii Konovalov. 2024-03-13. Beilinson-Parshin adeles via solid algebraic geometry. https://arxiv.org/abs/2403.08472

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