Expanding \v{C}ech cohomology for quantales
| Autor: | Tenório, Ana Luiza, Arndt, Peter, Mariano, Hugo Luiz |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We expand \v{C}ech cohomology of a topological space $X$ with values in a presheaf on $X$ to \v{C}ech cohomology of a commutative ring with unity $R$ with values in a presheaf on $R$. The strategy is to observe that both the set of open subsets of $X$ and the set of ideals of $R$ provide examples of a (semicartesian) quantale. We study a particular pair of (adjoint) functors $(\theta, \tau)$ between the quantale of open subsets of $X$ and the quantale of ideals of $C(X)$, the ring of real-valued continuous functions on $X$. This leads to the main result of this paper: the $q$th \v{C}ech cohomology groups of $X$ with values on the constant sheaf $F$ on $X$ is isomorphic to the $q$th \v{C}ech cohomology groups of the ring $C(X)$ with values on a sheaf $F \circ \tau$ on $C(X)$. Comment: 20 pages. We have removed one of the examples and the notion of a "basis" for a quantale; we will discuss them elsewhere with more details |
| Databáze: | arXiv |
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