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| Autores principales: | , |
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| Formato: | Recurso digital |
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Zenodo
2025
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| Acceso en línea: | https://doi.org/10.5281/zenodo.17834974 |
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- This paper explores the intricate relationship between étale topology and motivic spectra, two fundamental concepts in modern algebraic geometry and algebraic topology. Motivic homotopy theory, initiated by Voevodsky and Morel, provides a framework to apply homotopy-theoretic methods to algebraic varieties and schemes, extending classical invariants to the algebraic setting. The introduction of the étale topology, a Grothendieck topology that generalizes the Zariski topology, offers a finer probe into the local structure of schemes. We delve into how the stable étale motivic homotopy category incorporates both A¹-homotopy invariance and étale localization, leading to significant theoretical advancements. Key areas of investigation include the nilpotence properties of the motivic Hopf map $eta$ within this category, demonstrating a departure from the classical A¹-invariant stable motivic homotopy category where $eta$ is never nilpotent. Furthermore, we examine finiteness results in real étale cohomology and their implications for the constructible rational stable motivic homotopy category, including computations of Grothendieck groups and generic base change properties. The paper outlines the theoretical foundations, surveys relevant literature, and discusses the profound implications of these interactions for understanding the structure of motives and their associated cohomology theories.