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79:), the derived-functor cohomology agrees with this Čech cohomology obtained by direct limits. However, like the derived functor cohomology, this cover-independent Čech cohomology is virtually impossible to calculate from the definition. The Leray condition on an open cover ensures that the cover in question is already "fine enough." The derived functor cohomology agrees with the Čech cohomology with respect to any Leray cover.
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is well-suited to calculation, but of limited usefulness because it depends on the open cover chosen, not only on the sheaves and the space. By taking a direct limit of Čech cohomology over arbitrarily fine covers, we obtain a Čech cohomology theory that does not depend on the open cover chosen. In
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Algebraic geometry. Introduction to schemes. W. A. Benjamin, Inc., New York-Amsterdam 1968 vii+113 pp.
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measures the extent to which a locally exact sequence on a fixed topological space, for instance the
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Taylor, Joseph L. Several complex variables with connections to algebraic geometry and Lie groups.
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380:{\displaystyle H^{k}(U_{i_{1}}\cap \cdots \cap U_{i_{n}},{\mathcal {F}})=0}
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451:by open affine subschemes is a Leray cover.
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118:{\displaystyle {\mathfrak {U}}=\{U_{i}\}}
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262:{\displaystyle \{i_{1},\ldots ,i_{n}\}}
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468:Graduate Studies in Mathematics
186:{\displaystyle {\mathfrak {U}}}
36:. Such covers are named after
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424:{\displaystyle {\mathcal {F}}}
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210:{\displaystyle {\mathcal {F}}}
162:{\displaystyle {\mathcal {F}}}
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508:. You can help Knowledge by
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169:a sheaf on X. We say that
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288:{\displaystyle k>0}
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444:{\displaystyle X}
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138:{\displaystyle X}
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77:paracompact
18:mathematics
549:Categories
455:References
72:open cover
38:Jean Leray
34:cohomology
339:∩
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