A user's guide to spectral sequences by John McCleary

By John McCleary

Spectral sequences are one of the such a lot stylish and robust tools of computation in arithmetic. This booklet describes essentially the most very important examples of spectral sequences and a few in their so much astounding purposes. the 1st half treats the algebraic foundations for this type of homological algebra, ranging from casual calculations. the center of the textual content is an exposition of the classical examples from homotopy thought, with chapters at the Leray-Serre spectral series, the Eilenberg-Moore spectral series, the Adams spectral series, and, during this re-creation, the Bockstein spectral series. The final a part of the ebook treats functions all through arithmetic, together with the speculation of knots and hyperlinks, algebraic geometry, differential geometry and algebra. this is often a great reference for college students and researchers in geometry, topology, and algebra.

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A graded vector space H ∗ is a Γ∗ module if there is a mapping of graded vector spaces ψ : Γ∗ ⊗k H ∗ −→ H ∗ satisfying the usual module axioms, that is, the following diagram commutes with m, the product on Γ∗ : Γ∗ ⊗ Γ∗ ⊗ H ∗ m⊗1 1⊗ψ u Γ∗ ⊗ H ∗ w Γ∗ ⊗ H ∗ u ψ ψ w H ∗. As the reader might expect, we want to consider how a graded algebra, Γ∗ , might act on a bigraded vector space E ∗,∗ . In this instance, we consider only one case of such an action that represents the interesting case in later chapters; more general definitions are possible.

If V ∗ ∼ = Λ(x2n+1 ), then W ∗ ∼ = Q[y2n ]. This example is a simple case of a theorem of [Borel53] that we discuss in detail in later chapters. 22. y x⊗y x2⊗y 1 x x2 Recall that the differential di , applied to an element written as w ⊗ z, satisfies the Leibniz rule di (w ⊗ z) = di (w) ⊗ z + (−1)deg w w ⊗ di (z). ∗,0 Furthermore, the differential restricted to V ∗ ∼ = E2 is null and restricted to 0,∗ ∗ ∼ ∗ ∗ W = E2 it must have its image in V ⊗ W . If di (1 ⊗ u) = j vj ⊗ wj , then di (1 ⊗ uk ) = k j vj ⊗ (wj uk−1 ) .

Thus d∞ is an isomorphism. The boundedness of the filtration on (A, d) implies that the induced filtration on H(A, d) is bounded and so a finite sequence of ∗,∗ to H(A, d). 1 the relation between a spectral sequence associated to a filtered differential graded module and the homology of that differential module is developed further. The role of assumptions like boundedness of the filtration in determining H(A, d) uniquely is explored. Weaker conditions that guarantee convergence and uniqueness of the target are also discussed.

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