Finite ringIn mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements. Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite group, but the concept of finite rings in their own right has a more recent history. Although rings have more structure than groups, the theory of finite rings is simpler than that of finite groups.
Torsion-free moduleIn algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module is torsion free if its torsion submodule is reduced to its zero element. In integral domains the regular elements of the ring are its nonzero elements, so in this case a torsion-free module is one such that zero is the only element annihilated by some non-zero element of the ring.
Triangulated categoryIn mathematics, a triangulated category is a with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the of an , as well as the . The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology. Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology.
Functor categoryIn , a branch of mathematics, a functor category is a category where the objects are the functors and the morphisms are natural transformations between the functors (here, is another object in the category). Functor categories are of interest for two main reasons: many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable; every category embeds in a functor category (via the Yoneda embedding); the functor category often has nicer properties than the original category, allowing certain operations that were not available in the original setting.
Forgetful functorIn mathematics, in the area of , a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signature, this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the underlying set of a structure.
Functor represented by a schemeIn algebraic geometry, a functor represented by a scheme X is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme S is (up to natural bijections) the set of all morphisms . The scheme X is then said to represent the functor and that classify geometric objects over S given by F. The best known example is the Hilbert scheme of a scheme X (over some fixed base scheme), which, when it exists, represents a functor sending a scheme S to a flat family of closed subschemes of .
Lie algebra cohomologyIn mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to properties of the Lie algebra. It was later extended by to coefficients in an arbitrary Lie module. If is a compact simply connected Lie group, then it is determined by its Lie algebra, so it should be possible to calculate its cohomology from the Lie algebra.
Duality (mathematics)In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry. In mathematical contexts, duality has numerous meanings.
Sheaf (mathematics)In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts).
Yoneda lemmaIn mathematics, the Yoneda lemma is arguably the most important result in . It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms). It allows the of any into a (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category.
