Spectral sequenceIn homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra. Motivated by problems in algebraic topology, Jean Leray introduced the notion of a sheaf and found himself faced with the problem of computing sheaf cohomology.
Category of modulesIn algebra, given a ring R, the category of left modules over R is the whose are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For example, when R is the ring of integers Z, it is the same thing as the . The category of right modules is defined in a similar way. One can also define the category of bimodules over a ring R but that category is equivalent to the category of left (or right) modules over the enveloping algebra of R (or over the opposite of that).
Direct limit of groupsIn mathematics, a direct limit of groups is the direct limit of a of groups. These are central objects of study in algebraic topology, especially stable homotopy theory and homological algebra. They are sometimes called stable groups, though this term normally means something quite different in model theory. Certain examples of stable groups are easier to study than "unstable" groups, the groups occurring in the limit. This is surprising, given that they are generally infinite-dimensional, constructed as limits of groups with finite-dimensional representations.
Direct limitIn mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any . The way they are put together is specified by a system of homomorphisms (group homomorphism, ring homomorphism, or in general morphisms in the category) between those smaller objects. The direct limit of the objects , where ranges over some directed set , is denoted by .
Global dimensionIn ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring A denoted gl dim A, is a non-negative integer or infinity which is a homological invariant of the ring. It is defined to be the supremum of the set of projective dimensions of all A-modules. Global dimension is an important technical notion in the dimension theory of Noetherian rings.
Λ-ringIn algebra, a λ-ring or lambda ring is a commutative ring together with some operations λn on it that behave like the exterior powers of vector spaces. Many rings considered in K-theory carry a natural λ-ring structure. λ-rings also provide a powerful formalism for studying an action of the symmetric functions on the ring of polynomials, recovering and extending many classical results (). λ-rings were introduced by . For more about λ-rings see , , and .
Morita equivalenceIn abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like R, S are Morita equivalent (denoted by ) if their are equivalent (denoted by ). It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958. Rings are commonly studied in terms of their modules, as modules can be viewed as representations of rings.
Hilbert symbolIn mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from K× × K× to the group of nth roots of unity in a local field K such as the fields of reals or p-adic numbers. It is related to reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert symbol was introduced by in his Zahlbericht, with the slight difference that he defined it for elements of global fields rather than for the larger local fields. The Hilbert symbol has been generalized to higher local fields.
Whitehead torsionIn geometric topology, a field within mathematics, the obstruction to a homotopy equivalence of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion which is an element in the Whitehead group . These concepts are named after the mathematician J. H. C. Whitehead. The Whitehead torsion is important in applying surgery theory to non-simply connected manifolds of dimension > 4: for simply-connected manifolds, the Whitehead group vanishes, and thus homotopy equivalences and simple homotopy equivalences are the same.
Algebraic cycleIn mathematics, an algebraic cycle on an algebraic variety V is a formal linear combination of subvarieties of V. These are the part of the algebraic topology of V that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety. The most trivial case is codimension zero cycles, which are linear combinations of the irreducible components of the variety. The first non-trivial case is of codimension one subvarieties, called divisors.
