Leray Spectral Sequence
In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectral sequence. Definition Let f:X\to Y be a continuous map of topological spaces, which in particular gives a functor f_* from sheaves of abelian groups on X to sheaves of abelian groups on Y. Composing this with the functor \Gamma of taking sections on \text_\text(Y) is the same as taking sections on \text_\text(X), by the definition of the direct image functor f_*: :\mathrm (X) \xrightarrow \mathrm(Y) \xrightarrow \mathrm. Thus the derived functors of \Gamma \circ f_* compute the sheaf cohomology for X: : R^i (\Gamma \cdot f_*)(\mathcal)=H^i(X,\mathcal). But because f_* and \Gamma send injective objects in \text_\text(X) to \Gamma- acyclic objects in \text_\text(Y), there is a spectral sequencepg 33,19 whose second page is : E^_2=(R^p\Gamma \cdot R^q f_*)(\mathcal)=H^p(Y,R^qf_ ...
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of p