In the context of arithmetic geometry and the Weil conjectures, purity refers to a property of a geometric object or a scheme that relates to its cohomological dimensions, specifically regarding its behavior under various morphisms and its local properties. This notion plays a crucial role in understanding how rational points behave over different fields and is linked to the concept of étale cohomology, as well as the way singularities can affect the counting of points over finite fields.
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