spectral sequence
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| spectral sequence | |
|---|---|
| Name | Spectral sequence |
| Field | Algebraic topology, Homological algebra |
| Introduced | 1930s |
| Key people | Jean Leray, Henri Cartan, Jean-Pierre Serre, Jean-Louis Koszul, John Milnor, Jean B. B. Brown |
spectral sequence
A spectral sequence is an algebraic tool used to compute homology and cohomology by organizing complex computations into successive approximations. Originating from work of Jean Leray, Henri Cartan, and Jean-Pierre Serre, spectral sequences provide filtrations and page-by-page differentials that connect structures in algebraic topology, homological algebra, and algebraic geometry.
Introduction
Spectral sequences arose in early 20th-century research involving the Leray spectral sequence in sheaf theory and the Serre spectral sequence for fiber bundles; they quickly became central in work by Henri Cartan, Jean-Pierre Serre, Jean Leray, and Jean-Louis Koszul. Foundational applications appear in the study of fiber bundles such as the Hopf fibration and in calculations tied to the Adams spectral sequence developed by J. F. Adams, influencing research by John Milnor, Serge Lang, and Jean B. B. Brown. Major institutions and events that shaped the theory include seminars at the Institut des Hautes Études Scientifiques, Bourbaki meetings, and conferences honoring Emmy Noether and Alexander Grothendieck.
Definitions and Basic Properties
A spectral sequence is typically defined as a sequence of bigraded abelian groups or modules with differentials satisfying d_r^2 = 0 on each page E_r, culminating in an E_∞ page that is related to a graded object of interest. Definitions by Jean Leray, Henri Cartan, and Jean-Pierre Serre formalize filtrations and exact couples, linking to concepts used by Emmy Noether, David Hilbert, and Alexander Grothendieck. Basic properties include functoriality with respect to maps studied by Grothendieck, convergence types (strong, conditional) explored in work by John Milnor and Jean-Pierre Serre, and edge homomorphisms that relate to long exact sequences from Leray and Cartan.
Constructions and Examples
Standard constructions produce spectral sequences from filtered complexes, exact couples, and filtrations of spaces; classical examples include the Serre spectral sequence for fiber bundles like the Hopf fibration, the Leray spectral sequence for maps of sheaves, and the Hochschild–Serre spectral sequence for group extensions studied by Jean-Pierre Serre and Henri Cartan. The Adams spectral sequence and the Adams–Novikov spectral sequence, influenced by J. F. Adams and Sergei Novikov, are key in stable homotopy theory and calculations by John Milnor and Michael Atiyah. Other examples appear in K-theory computations by Michael Atiyah and Friedrich Hirzebruch, cyclic homology settings by Jean-Louis Loday, and filtered de Rham complexes tied to Alexander Grothendieck and Pierre Deligne.
Convergence and Exact Couples
Convergence issues are central: spectral sequences may converge to a graded object, an associated graded group, or require additional extension problems as in work by Jean-Pierre Serre and Jean Leray. Exact couples, introduced by Jean Leray and used extensively by Henri Cartan and Jean-Pierre Serre, give a systematic way to produce spectral sequences from long exact sequences arising in studies by Emmy Noether and Alexander Grothendieck. Techniques for resolving extension problems draw on methods developed by Jean-Pierre Serre, Jean-Louis Koszul, and John Milnor, and are applied in contexts explored by Michael Atiyah and Raoul Bott.
Applications in Algebraic Topology and Homological Algebra
Spectral sequences are indispensable in calculations of homotopy groups of spheres via the Adams spectral sequence (J. F. Adams, John Milnor), in computations of cohomology rings using the Serre spectral sequence (Jean-Pierre Serre), and in sheaf cohomology via the Leray spectral sequence (Jean Leray, Alexander Grothendieck). They play roles in the study of characteristic classes in work by Shiing-Shen Chern and René Thom, index theory influenced by Michael Atiyah and Isadore Singer, and the development of cobordism theories by René Thom and Sergei Novikov. In homological algebra, spectral sequences assist in Tor and Ext computations central to the work of Pierre Samuel, Henri Cartan, and Jean-Pierre Serre.
Computational Techniques and Spectral Sequence Charts
Practitioners use spectral sequence charts and Adams charts pioneered in computations by J. F. Adams, John Milnor, and Mark Mahowald, often annotating differentials derived from operations studied by Jean-Pierre Serre, Norman Steenrod, and Daniel Quillen. Software implementations and computational approaches inspired by institutions like the Clay Mathematics Institute and collaborative projects led by Michael Hopkins and Jacob Lurie aid large-scale calculations in stable homotopy and arithmetic geometry. Techniques include filtrations from Postnikov towers as in work by J. H. C. Whitehead, comparison theorems attributed to Jean-Pierre Serre, and vanishing lines influenced by Vito Volterra and later authors.
Variants and Generalizations
Many variants exist: the Adams spectral sequence (J. F. Adams), Adams–Novikov spectral sequence (Sergei Novikov), Hochschild–Serre spectral sequence (Jean-Pierre Serre), Bockstein spectral sequence, Grothendieck spectral sequence (Alexander Grothendieck), and Eilenberg–Moore spectral sequence (Samuel Eilenberg, John Coleman Moore). Generalizations connect to motivic homotopy theory by Vladimir Voevodsky, algebraic K-theory by Daniel Quillen and Spencer Bloch, and to derived categories popularized by Jean-Louis Verdier and Alexander Grothendieck. Contemporary developments involve work by Jacob Lurie, Michael Hopkins, and Akhil Mathew in higher category theory and chromatic homotopy theory pioneered by Douglas Ravenel.