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In mathematics, the Adams spectral sequence is a spectral sequence introduced by . Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan and Jean-Pierre Serre. ==Motivation== For everything below, once and for all, we fix a prime ''p''. All spaces are assumed to be CW complexes. The ordinary cohomology groups ''H'' *(''X'') are understood to mean ''H'' *(''X''; Z/''p''Z). The primary goal of algebraic topology is to try to understand the collection of all maps, up to homotopy, between arbitrary spaces ''X'' and ''Y''. This is extraordinarily ambitious: in particular, when ''X'' is ''Sn'', these maps form the ''n''th homotopy group of ''Y''. A more reasonable (but still very difficult!) goal is to understand (''Y'' ), the maps (up to homotopy) that remain after we apply the suspension functor a large number of times. We call this the collection of stable maps from ''X'' to ''Y''. (This is the starting point of stable homotopy theory; more modern treatments of this topic begin with the concept of a spectrum. Adams' original work did not use spectra, and we avoid further mention of them in this section to keep the content here as elementary as possible.) (''Y'' ) turns out to be an abelian group, and if ''X'' and ''Y'' are reasonable spaces this group is finitely generated. To figure out what this group is, we first isolate a prime ''p''. In an attempt to compute the ''p''-torsion of (''Y'' ), we look at cohomology: send (''Y'' ) to Hom(''H'' *(''Y''), ''H'' *(''X'')). This is a good idea because cohomology groups are usually tractable to compute. The key idea is that ''H'' *(''X'') is more than just a graded abelian group, and more still than a graded ring (via the cup product). The representability of the cohomology functor makes ''H'' *(''X'') a module over the algebra of its stable cohomology operations, the Steenrod algebra ''A''. Thinking about ''H'' *(''X'') as an ''A''-module forgets some cup product structure, but the gain is enormous: Hom(''H'' *(''Y''), ''H'' *(''X'')) can now be taken to be ''A''-linear! A priori, the ''A''-module sees no more of (''Y'' ) than it did when we considered it to be a map of vector spaces over F''p''. But we can now consider the derived functors of Hom in the category of ''A''-modules, Ext''A''''r''(''H'' *(''Y''), ''H'' *(''X'')). These acquire a second grading from the grading on ''H'' *(''Y''), and so we obtain a two-dimensional "page" of algebraic data. The Ext groups are designed to measure the failure of Hom's preservation of algebraic structure, so this is a reasonable step. The point of all this is that ''A'' is so large that the above sheet of cohomological data contains all the information we need to recover the ''p''-primary part of (''Y'' ), which is homotopy data. This is a major accomplishment because cohomology was designed to be computable, while homotopy was designed to be powerful. This is the content of the Adams spectral sequence. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Adams spectral sequence」の詳細全文を読む スポンサード リンク
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