Universal Equivalence Sample Clauses

Universal Equivalence. To study HS(K) for local fields K, we need to quote and explain some results of Manin [10, Chapter II] on universal equivalence. Let S be a smooth cubic variety over a field K (not assumed to be local for now). Let ∼ be an equivalence relation on S(K). We say that ∼ is admissible if it is compatible with collinearity in the following sense: if both triples P1, P2, P3 and P1j , P2j , P3j are collinear, and if P1 ∼ P1j and P2 ∼ P2j then P3 ∼ P3j . Now let U1, U2 be two admissible relations on S(K). We say that U1 is finer than U2 (and so U2 is coarser than U1) if whenever PU1Q then PU2Q. It is clear that every equivalence class of U2 is a union of equivalence classes for U1 and so U2 has fewer or the same number of equivalence classes as U1.
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Related to Universal Equivalence

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