Spectral sequences

Spectral sequences are a powerful mathematical tool used primarily in algebraic topology, homological algebra, and algebraic geometry. They provide a systematic method for computing homology groups, cohomology groups, or other related invariants of topological spaces or algebraic objects. ### Definition and Construction A spectral sequence consists of a sequence of pages (or terms), each represented as a collection of abelian groups or modules, along with differentials that relate these groups across the pages.

The Bockstein spectral sequence is a mathematical tool in the field of homological algebra and algebraic topology, particularly in the study of spectral sequences. It arises in the context of computing homology and cohomology groups with coefficients in a group or ring, especially when the coefficients can be viewed as a module over a more complex ring.

The Chromatic Spectral Sequence is a tool in stable homotopy theory, particularly in the study of stable homotopy groups of spheres and related objects. It is mainly concerned with the chromatic filtration, which categorizes stable homotopy groups based on their interactions with complex oriented theories, such as complex cobordism and various versions of K-theory.

The EHP spectral sequence is a tool in homotopy theory and stable homotopy theory, particularly involving the study of the stable homotopy groups of spheres. It is named after the mathematicians Eilenberg, Henriques, and Priddy—hence EHP. The EHP spectral sequence arises from the framework of stable homotopy types and is associated with the "suspension" of spaces and the mapping spaces between them.

The term "Exact couple" can refer to different concepts depending on the context. 1. **Mathematics**: In mathematics, particularly in the field of algebra and topology, an "exact couple" refers to a specific type of diagram used in homological algebra that combines two different chain complexes. Exact couples are used to construct spectral sequences, which are tools that allow mathematicians to compute homology groups by breaking complex problems into simpler parts.

The Five-Term Exact Sequence is a concept in algebraic topology and homological algebra, particularly in the context of derived functors and spectral sequences. It often arises in the study of homology and cohomology theories. In general, an exact sequence is a sequence of algebraic objects (like groups, modules, or vector spaces) linked by homomorphisms where the image of one homomorphism equals the kernel of the next.

The Frölicher spectral sequence is a tool in the field of differential geometry, particularly useful in the study of differentiable manifolds and their associated sheaf-theoretic or cohomological structures. It provides a way to compute the sheaf cohomology associated with the global sections of a sheaf of differential forms on a smooth manifold.

The Grothendieck spectral sequence is a powerful tool in algebraic geometry and homological algebra, providing a method for computing the derived functors of a functor that is defined in terms of a different functor. It is commonly used in the context of sheaf cohomology. The context in which the Grothendieck spectral sequence typically arises is in the cohomology of sheaves on a topological space (often a variety or scheme).

The May spectral sequence is a mathematical tool used in algebraic topology, particularly in the study of stable homotopy theory and the homotopy theory of spectra. Named after M. M. May, it is particularly useful for computing homotopy groups of spectra and understanding stable homotopy categories. The May spectral sequence arises in the context of a type of cohomology theory called stable cohomology.

A spectral sequence is a mathematical tool used in algebraic topology, homological algebra, and related fields to compute homology or cohomology groups that may be difficult to compute directly. It provides a method to systematically approximate these groups through a sequence of pages (typically indexed by integers) and associated differentials.

The Čech-to-derived functor spectral sequence is a tool in homological algebra and sheaf theory that relates Čech cohomology to derived functors, particularly sheaf cohomology. This spectral sequence emerges in contexts where one is interested in understanding the relationship between local properties, codified by Čech cohomology, and global properties captured by derived functors like the derived functors of sheaf cohomology. ### Overview of the Components Involved 1.