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c_xrsvn1dwh3y0 | In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces f: A → B {\displaystyle f:A\to B} . It acts as a homotopy theoretic kernel of a mapping of topological space... | Homotopy fiber |
c_lqsgfda2cuxn | In mathematics, especially homotopy theory, the mapping cone is a construction C f {\displaystyle C_{f}} of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated C f {\displaystyle Cf} . Its dual, a fibration, is called the mapping fibre. The mapping cone can be understood to... | Homotopy cofiber |
c_jhjr7h2h1471 | In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1, 2 and k, with k = n − 4. In some older books and papers, E2 and E4 are used as names for G2 and F4. | En (Lie algebra) |
c_lpeapf1z9unf | In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element properties are optional. A quasigroup with an identity element is called a loop. | Quasigroup |
c_y7qph023gj3r | In mathematics, especially in algebraic K-theory, the algebraic K-group of a field is important to compute. For a finite field, the complete calculation was given by Daniel Quillen. | K-group of a field |
c_bhm3x5b8ngee | In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; s... | Coherent sheaf cohomology |
c_q9hrtoa567qr | In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Cohere... | Quasi-coherent sheaf |
c_3epclct9x61x | Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank. Coherent sheaf cohomology is a powerful technique, in pa... | Quasi-coherent sheaf |
c_1meixipu8qqg | In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedded in well-behaved spaces such as projective space or to prove theorems by... | Adjunction formula (algebraic geometry) |
c_fi823hltdmbd | In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4. More specifically there are two closely related types of quartic surface: affine and projective. An affine quartic surface is the solution set of an equation of the form f ( x , y , z ) = 0 {\displaystyl... | Quartic surface |
c_0ezzquw0d7bo | On the other hand, a projective quartic surface is a surface in projective space P3 of the same form, but now f is a homogeneous polynomial of 4 variables of degree 4, so for example f ( x , y , z , w ) = x 4 + y 4 + x y z w + z 2 w 2 − w 4 {\displaystyle f(x,y,z,w)=x^{4}+y^{4}+xyzw+z^{2}w^{2}-w^{4}} . If the base fiel... | Quartic surface |
c_iunubee1ek07 | In mathematics, especially in algebraic geometry, the Beilinson regulator is the Chern class map from algebraic K-theory to Deligne cohomology: K n ( X ) → ⊕ p ≥ 0 H D 2 p − n ( X , Q ( p ) ) . {\displaystyle K_{n}(X)\rightarrow \oplus _{p\geq 0}H_{D}^{2p-n}(X,\mathbf {Q} (p)).} Here, X is a complex smooth projective v... | Beilinson regulator |
c_c5lp5r9s24sv | The Beilinson regulator features in Beilinson's conjecture on special values of L-functions. The Dirichlet regulator map (used in the proof of Dirichlet's unit theorem) for the ring of integers O F {\displaystyle {\mathcal {O}}_{F}} of a number field F O F × → R r 1 + r 2 , x ↦ ( log | σ ( x ) | ) σ {\displaystyle {\... | Beilinson regulator |
c_smfmxz5d4m2f | In mathematics, especially in algebraic geometry, the v-topology (also known as the universally subtrusive topology) is a Grothendieck topology whose covers are characterized by lifting maps from valuation rings. This topology was introduced by Rydh (2010) and studied further by Bhatt & Scholze (2017), who introduced t... | V-topology |
c_zc1uqh4vo9y8 | In mathematics, especially in algebraic geometry, the étale homotopy type is an analogue of the homotopy type of topological spaces for algebraic varieties. Roughly speaking, for a variety or scheme X, the idea is to consider étale coverings U → X {\displaystyle U\rightarrow X} and to replace each connected component o... | Étale homotopy type |
c_3eiz1ksj5l9q | In mathematics, especially in algebraic number theory, the Hermite–Minkowski theorem states that for any integer N there are only finitely many number fields, i.e., finite field extensions K of the rational numbers Q, such that the discriminant of K/Q is at most N. The theorem is named after Charles Hermite and Hermann... | Hermite–Minkowski theorem |
c_sq7phyl4jf6f | In mathematics, especially in algebraic topology, an induced homomorphism is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space X to a topological space Y induces a group homomorphism from the fundamental group of X to the fundamental group of Y. More gene... | Induced homomorphism |
c_eq71ldwfonwl | A homomorphism induced from a map h {\displaystyle h} is often denoted h ∗ {\displaystyle h_{*}} . Induced homomorphisms often inherit properties of the maps they come from; for example, two maps that are inverse to each other up to homotopy induce homomorphisms that are inverse to each other. | Induced homomorphism |
c_rnjpwl1adyce | A common use of induced homomorphisms is the following: by showing that a homomorphism with certain properties cannot exist, one concludes that there cannot exist a continuous map with properties that would induce it. Thanks to this, relations between spaces and continuous maps, often very intricate, can be inferred fr... | Induced homomorphism |
c_9hs8zwvf76kx | In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category Ho ( Top ) {\displaystyle {\text{Ho}}({\textbf {Top}})} . The main idea is this: if we have a diagram F: I → Top {\displaystyle F:I\to {\textbf {Top}}}... | Homotopy colimit |
c_qgwm8cpjm82i | In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ(g). In more abstract language, this means that the group homo... | Faithful representation |
c_cturskhe7vg2 | In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: Group with a partial function replacing the binary operation; Category in which every morphism is invertibl... | Transformation groupoid |
c_mv347vrpqb0r | The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed g: A → B {\displaystyle g:A\rightarrow B} , h: B → C {\displaystyle h:B\rightarrow C} , say. Composition is then a total function: ∘: ( B → C ) → ( A → B ) → A → C {\displaystyle \circ :(B\rightarrow C... | Transformation groupoid |
c_dxk8da4ry8wb | In mathematics, especially in category theory, a closed monoidal category (or a monoidal closed category) is a category that is both a monoidal category and a closed category in such a way that the structures are compatible. A classic example is the category of sets, Set, where the monoidal product of sets A {\displays... | Monoidal closed category |
c_hnzpjce01vrp | The internal language of closed symmetric monoidal categories is linear logic and the type system is the linear type system. Many examples of closed monoidal categories are symmetric. However, this need not always be the case, as non-symmetric monoidal categories can be encountered in category-theoretic formulations of... | Monoidal closed category |
c_224tez0nqfk0 | In mathematics, especially in category theory, the codensity monad is a fundamental construction associating a monad to a wide class of functors. | Codensity monad |
c_97uvp6h7a9zi | In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed points as cycles of length one). The Stirling numbers of the first and second k... | Stirling numbers of the first kind |
c_ghh7odyzi7ge | In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes. | Minimal prime (commutative algebra) |
c_z7yhqma13hrr | In mathematics, especially in differential topology, Thom's first isotopy lemma states: given a smooth map f: M → N {\displaystyle f:M\to N} between smooth manifolds and S ⊂ M {\displaystyle S\subset M} a closed Whitney stratified subset, if f | S {\displaystyle f|_{S}} is proper and f | A {\displaystyle f|_{A}} is a s... | Thom's first isotopy lemma |
c_woblw9lnfbny | The local trivializations that the lemma provide preserve the strata. However, they are generally not smooth (not even C 1 {\displaystyle C^{1}} ). On the other hand, it is possible that local trivializations are semialgebraic if the input data is semialgebraic.The lemma is also valid for a more general stratified spac... | Thom's first isotopy lemma |
c_xfc50enwrmgu | The lemma is also valid for the stratification that satisfies Bekka's condition (C), which is weaker than Whitney's condition (B). (The significance of this is that the consequences of the first isotopy lemma cannot imply Whitney’s condition (B).) Thom's second isotopy lemma is a family version of the first isotopy lem... | Thom's first isotopy lemma |
c_8v1yepg9s9tf | In mathematics, especially in differential topology, Thom's second isotopy lemma is a family version of Thom's first isotopy lemma; i.e., it states a family of maps between Whitney stratified spaces is locally trivial when it is a Thom mapping. Like the first isotopy lemma, the lemma was introduced by René Thom. (Mathe... | Thom mapping |
c_vu029774cp3m | In mathematics, especially in geometry and topology, an ambient space is the space surrounding a mathematical object along with the object itself. For example, a 1-dimensional line ( l ) {\displaystyle (l)} may be studied in isolation —in which case the ambient space of l {\displaystyle l} is l {\displaystyle l} , or i... | Ambient space (mathematics) |
c_tqm3w3mag366 | In mathematics, especially in geometry, a double lattice in ℝn is a discrete subgroup of the group of Euclidean motions that consists only of translations and point reflections and such that the subgroup of translations is a lattice. The orbit of any point under the action of a double lattice is a union of two Bravais ... | Double lattice |
c_4z0of03q60g1 | In mathematics, especially in higher-dimensional algebra and homotopy theory, a double groupoid generalises the notion of groupoid and of category to a higher dimension. | Double groupoid |
c_o0ww2j4f2mf8 | In mathematics, especially in homological algebra and algebraic topology, a Künneth theorem, also called a Künneth formula, is a statement relating the homology of two objects to the homology of their product. The classical statement of the Künneth theorem relates the singular homology of two topological spaces X and Y... | Kunneth formula |
c_kxtqaxdxkhk1 | In mathematics, especially in homotopy theory, a left fibration of simplicial sets is a map that has the right lifting property with respect to the horn inclusions Λ i n ⊂ Δ n , 0 ≤ i < n {\displaystyle \Lambda _{i}^{n}\subset \Delta ^{n},0\leq i | Fibration of simplicial sets |
c_4pbhjqv58us9 | In mathematics, especially in linear algebra and matrix theory, a centrosymmetric matrix is a matrix which is symmetric about its center. More precisely, an n×n matrix A = is centrosymmetric when its entries satisfy Ai,j = An−i + 1,n−j + 1 for i, j ∊{1, ..., n}.If J denotes the n×n exchange matrix with 1 on the antidi... | Centrosymmetric matrix |
c_gb62ifnwuvq1 | In mathematics, especially in linear algebra and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix into the vectorized form of its transpose. Specifically, the commutation matrix K(m,n) is the nm × mn matrix which, for any m × n matrix A, transforms vec(A) into vec(AT): K(m,... | Commutation matrix |
c_zo29zxkzuu8o | In mathematics, especially in linear algebra and matrix theory, the duplication matrix and the elimination matrix are linear transformations used for transforming half-vectorizations of matrices into vectorizations or (respectively) vice versa. | Elimination matrix |
c_91l3y0udjq9j | In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a vector. Specifically, the vectorization of a m × n matrix A, denoted vec(A), is the mn × 1 column vector obtained by stacking the columns of the matrix A on top of on... | Vectorization (mathematics) |
c_hxhf7plareky | In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fundamental theorem of Galois theory about the correspondence between subgrou... | Galois connection |
c_fbh7nycdekvn | A Galois connection can also be defined on preordered sets or classes; this article presents the common case of posets. The literature contains two closely related notions of "Galois connection". In this article, we will refer to them as (monotone) Galois connections and antitone Galois connections. A Galois connection... | Galois connection |
c_w4f9qvqwxhvm | In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames. Although these three categorie... | Complete Heyting algebra |
c_tv9pjxga41kf | In mathematics, especially in order theory, a maximal element of a subset S of some preordered set is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some preordered set is defined dually as an element of S that is not greater than any other element in S. The notions ... | Maximal element |
c_4rqjsebepu6q | As an example, in the collection ordered by containment, the element {d, o} is minimal as it contains no sets in the collection, the element {g, o, a, d} is maximal as there are no sets in the collection which contain it, the element {d, o, g} is neither, and the element {o, a, f} is both minimal and maximal. By contra... | Maximal element |
c_43qm0ft0gd6y | {\displaystyle S.} Zorn's lemma states that every partially ordered set for which every totally ordered subset has an upper bound contains at least one maximal element. This lemma is equivalent to the well-ordering theorem and the axiom of choice and implies major results in other mathematical areas like the Hahn–Banac... | Maximal element |
c_kx7w0wsp1d89 | In mathematics, especially in order theory, a partially ordered set with a unique minimal element 0 has the disjunction property of Wallman when for every pair (a, b) of elements of the poset, either b ≤ a or there exists an element c ≤ b such that c ≠ 0 and c has no nontrivial common predecessor with a. That is, in th... | Disjunction property of Wallman |
c_7cierxkuuo7o | In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cases of a preorder: an antisymmetric (or skeletal) preorder is a partial order, an... | Preordered set |
c_56upjpel1mip | However, because they are not necessarily antisymmetric, some of the ordinary intuition associated to the symbol ≤ {\displaystyle \,\leq \,} may not apply. On the other hand, a preorder can be used, in a straightforward fashion, to define a partial order and an equivalence relation. Doing so, however, is not always use... | Preordered set |
c_btzxiw0xkegb | In words, when a ≤ b , {\displaystyle a\leq b,} one may say that b covers a or that a precedes b, or that b reduces to a. Occasionally, the notation ← or → or ≲ {\displaystyle \,\lesssim \,} is used instead of ≤ . {\displaystyle \,\leq .} To every preorder, there corresponds a directed graph, with elements of the set c... | Preordered set |
c_difhi425yhd0 | The converse is not true: most directed graphs are neither reflexive nor transitive. In general, the corresponding graphs may contain cycles. A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to a directed acyclic graph. A preorder that is symmetric is an equivalence relation... | Preordered set |
c_c31ihjnzyak2 | In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A. This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member. The cofinality of... | Cofinality |
c_iej1b0j55i9c | In mathematics, especially in order theory, the greatest element of a subset S {\displaystyle S} of a partially ordered set (poset) is an element of S {\displaystyle S} that is greater than every other element of S {\displaystyle S} . The term least element is defined dually, that is, it is an element of S {\displaysty... | Bottom element |
c_sqx07w3q6js2 | In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix) is a square matrix X = ( x i j ) {\displaystyle X=(x_{ij})} of nonnegative real numbers, each of whose rows and columns sums to 1, i.e., ∑ i x i j = ∑ j x i j = 1 , {\displaystyle \sum _{i}x_{ij}=\s... | Bistochastic matrix |
c_1qz2n54wm355 | In mathematics, especially in real algebraic geometry, a semialgebraic space is a space which is locally isomorphic to a semialgebraic set. | Semialgebraic space |
c_3kvdqvuat3b5 | In mathematics, especially in representation theory and algebraic geometry, the Beilinson–Bernstein localization theorem relates D-modules on flag varieties G/B to representations of the Lie algebra g {\displaystyle {\mathfrak {g}}} attached to a reductive group G. It was introduced by Beilinson & Bernstein (1981). Ext... | Beilinson–Bernstein localization |
c_9fyti33jxxbm | In mathematics, especially in set theory, two ordered sets X and Y are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f: X → Y {\displaystyle f\colon X\to Y} such that both f and its inverse are monotonic (preser... | Order type |
c_hbi5mwfk33fv | In mathematics, especially in sheaf theory—a domain applied in areas such as topology, logic and algebraic geometry—there are four image functors for sheaves that belong together in various senses. Given a continuous mapping f: X → Y of topological spaces, and the category Sh(–) of sheaves of abelian groups on a topolo... | Image functors for sheaves |
c_etz88uvp6bua | : D(Sh(Y)) → D(Sh(X)).The exclamation mark is often pronounced "shriek" (slang for exclamation mark), and the maps called "f shriek" or "f lower shriek" and "f upper shriek"—see also shriek map. The exceptional inverse image is in general defined on the level of derived categories only. Similar considerations apply to ... | Image functors for sheaves |
c_u653wtntv42m | In mathematics, especially in singularity theory, the splitting lemma is a useful result due to René Thom which provides a way of simplifying the local expression of a function usually applied in a neighbourhood of a degenerate critical point. | Splitting lemma (functions) |
c_cjt9ml5zpuca | In mathematics, especially in the area of abstract algebra dealing with ordered structures on abelian groups, the Hahn embedding theorem gives a simple description of all linearly ordered abelian groups. It is named after Hans Hahn. | Hahn embedding theorem |
c_wzlkv0glgi5y | In mathematics, especially in the area of abstract algebra known as combinatorial group theory, Nielsen transformations, named after Jakob Nielsen, are certain automorphisms of a free group which are a non-commutative analogue of row reduction and one of the main tools used in studying free groups, (Fine, Rosenberger &... | Nielsen transformation |
c_45erdm0t20z2 | In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element. More precisely, if A is a finite set of generators for G then the word ... | Kuznetsov's theorem |
c_9xivamtfr5hc | In mathematics, especially in the area of abstract algebra known as module theory, a principal indecomposable module has many important relations to the study of a ring's modules, especially its simple modules, projective modules, and indecomposable modules. | Principal indecomposable module |
c_qlkpmacdi0th | In mathematics, especially in the area of abstract algebra known as module theory, a ring R is called hereditary if all submodules of projective modules over R are again projective. If this is required only for finitely generated submodules, it is called semihereditary. For a noncommutative ring R, the terms left hered... | Hereditary ring |
c_ktq1tt32jj6v | In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as gr... | Semisimple module |
c_5e5h3u05bcrt | An Artinian ring is initially understood via its largest semisimple quotient. The structure of Artinian semisimple rings is well understood by the Artin–Wedderburn theorem, which exhibits these rings as finite direct products of matrix rings. For a group-theory analog of the same notion, see Semisimple representation. | Semisimple module |
c_ljbtz2831pdx | In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, giv... | Baer's criterion |
c_0wcnjq0ppv8n | Injective resolutions measure how far from injective a module is in terms of the injective dimension and represent modules in the derived category. Injective hulls are maximal essential extensions, and turn out to be minimal injective extensions. | Baer's criterion |
c_301iqihxu237 | Over a Noetherian ring, every injective module is uniquely a direct sum of indecomposable modules, and their structure is well understood. An injective module over one ring, may not be injective over another, but there are well-understood methods of changing rings which handle special cases. Rings which are themselves ... | Baer's criterion |
c_4c6m0vqvq73i | In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the polynomial ring may be regarded as a free commutative algebra. | Noncommutative polynomial ring |
c_wpvjoimp151i | In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group G is said to be virtually P if there is a finite index subgroup H ≤ G {\displaystyle H\leq G}... | Virtually nilpotent |
c_1ywp6j0p9xdi | This terminology is also used when P is just another group. That is, if G and H are groups then G is virtually H if G has a subgroup K of finite index in G such that K is isomorphic to H. In particular, a group is virtually trivial if and only if it is finite. Two groups are virtually equal if and only if they are comm... | Virtually nilpotent |
c_pdfqmxc83ugk | In mathematics, especially in the area of abstract algebra, every module has an associated character module. Using the associated character module it is possible to investigate the properties of the original module. One of the main results discovered by Joachim Lambek shows that a module is flat if and only if the asso... | Character module |
c_coj3meg6npzj | In mathematics, especially in the area of algebra known as group theory, a complement of a subgroup H in a group G is a subgroup K of G such that G = H K = { h k: h ∈ H , k ∈ K } and H ∩ K = { e } . {\displaystyle G=HK=\{hk:h\in H,k\in K\}{\text{ and }}H\cap K=\{e\}.} Equivalently, every element of G has a unique expre... | Complement (group theory) |
c_8vu0m4dn6yfr | In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it represents the smallest subgroup which "controls" the structure of G when G is solvable. When G is not solv... | Generalized Fitting subgroup |
c_476be8t9tr8e | In mathematics, especially in the area of algebra known as group theory, the holomorph of a group is a group that simultaneously contains (copies of) the group and its automorphism group. The holomorph provides interesting examples of groups, and allows one to treat group elements and group automorphisms in a uniform c... | Holomorph (mathematics) |
c_uqy18src2dle | In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups: in the study of finite groups, a Z-group is a finite group whose Sylow subgroups are all cyclic. in the study of infinite groups, a Z-group is a group which possesses a very general ... | Z-group |
c_1w00ezbic6d6 | Such groups are elementarily equivalent to the integers ( Z , + , < ) {\displaystyle (\mathbb {Z} ,+,<)} . Z-groups are an alternative presentation of Presburger arithmetic. occasionally, (Z)-group is used to mean a Zassenhaus group, a special type of permutation group. | Z-group |
c_cqoy9cuo1sew | In mathematics, especially in the area of algebra known as module theory, Schanuel's lemma, named after Stephen Schanuel, allows one to compare how far modules depart from being projective. It is useful in defining the Heller operator in the stable category, and in giving elementary descriptions of dimension shifting. | Schanuel's lemma |
c_sdl5azse1aoe | In mathematics, especially in the area of algebra known as representation theory, the representation ring (or Green ring after J. A. Green) of a group is a ring formed from all the (isomorphism classes of the) finite-dimensional linear representations of the group. Elements of the representation ring are sometimes call... | Representation ring |
c_dgve8iv6vtwe | In mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Øystein Ore, is a special type of a ring extension whose properties are relatively well understood. Elements of a Ore extension are called Ore polynomials. Ore extensions appear in several natural contexts, including s... | Ore polynomial |
c_7cd9ggqvaei2 | In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring. The right Ore condition for a mult... | Ore ring |
c_sojtoxjva1o5 | In mathematics, especially in the area of algebra studying the theory of abelian groups, a pure subgroup is a generalization of direct summand. It has found many uses in abelian group theory and related areas. | Pure subgroup |
c_6s3fu0056dzx | In mathematics, especially in the area of algebra studying the theory of abelian groups, an essential subgroup is a subgroup that determines much of the structure of its containing group. The concept was generalized to essential submodules. | Essential subgroup |
c_8wfczireonkz | In mathematics, especially in the area of algebraic topology known as stable homotopy theory, the Adams filtration and the Adams–Novikov filtration allow a stable homotopy group to be understood as built from layers, the nth layer containing just those maps which require at most n auxiliary spaces in order to be a comp... | Adams filtration |
c_pig303su4mhv | In mathematics, especially in the area of mathematical analysis known as dynamical systems theory, a linear flow on the torus is a flow on the n-dimensional torus which is represented by the following differential equations with respect to the standard angular coordinates ( θ 1 , θ 2 , … , θ n ): {\displaystyle \left(\... | Irrational cable on a torus |
c_h2w58pctt0pe | In mathematics, especially in the area of topology known as knot theory, an invertible knot is a knot that can be continuously deformed to itself, but with its orientation reversed. A non-invertible knot is any knot which does not have this property. The invertibility of a knot is a knot invariant. An invertible link i... | Invertible knot |
c_1zskjfz7uq15 | In mathematics, especially in the areas of abstract algebra dealing with group cohomology or relative homological algebra, Shapiro's lemma, also known as the Eckmann–Shapiro lemma, relates extensions of modules over one ring to extensions over another, especially the group ring of a group and of a subgroup. It thus rel... | Shapiro's lemma |
c_f8zv30ulesxs | In mathematics, especially in the areas of abstract algebra known as universal algebra, group theory, ring theory, and module theory, a subdirect product is a subalgebra of a direct product that depends fully on all its factors without however necessarily being the whole direct product. The notion was introduced by Bir... | Subdirect product |
c_pwf0f74xivxh | In mathematics, especially in the areas of numerical analysis called numerical partial differential equations, a compact stencil is a type of stencil that uses only nine nodes for its discretization method in two dimensions. It uses only the center node and the adjacent nodes. For any structured grid utilizing a compac... | Compact stencil |
c_n1lgjy1r9pqo | In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial... | Polynomial expression |
c_kad6d6xu9tsp | Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have bee... | Polynomial expression |
c_l9tjftdna3qb | In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. The dual notion is that of a projective object. | Injective object |
c_fiiben8tyjq1 | In mathematics, especially in the field of commutative algebra, a connected ring is a commutative ring A that satisfies one of the following equivalent conditions: A possesses no non-trivial (that is, not equal to 1 or 0) idempotent elements; the spectrum of A with the Zariski topology is a connected space. | Connected ring |
c_sfttk7s9sz99 | In mathematics, especially in the field of group theory, a Carter subgroup of a finite group G is a self-normalizing subgroup of G that is nilpotent. These subgroups were introduced by Roger Carter, and marked the beginning of the post 1960 theory of solvable groups (Wehrfritz 1999). Carter (1961) proved that any finit... | Carter subgroup |
c_yyf4lt6px9u6 | Vdovin (2006, 2007) showed that even if a finite group is not solvable then any two Carter subgroups are conjugate. A Carter subgroup is a maximal nilpotent subgroup, because of the normalizer condition for nilpotent groups, but not all maximal nilpotent subgroups are Carter subgroups (Ballester-Bolinches & Ezquerro 20... | Carter subgroup |
c_e6lb88okjuoq | For example, any non-identity proper subgroup of the nonabelian group of order six is a maximal nilpotent subgroup, but only those of order two are Carter subgroups. Every subgroup containing a Carter subgroup of a soluble group is also self-normalizing, and a soluble group is generated by any Carter subgroup and its n... | Carter subgroup |
c_wsbuowebgl2z | In the language of formations, a Sylow p-subgroup is a covering group for the formation of p-groups, a Hall π-subgroup is a covering group for the formation of π-groups, and a Carter subgroup is a covering group for the formation of nilpotent groups (Ballester-Bolinches & Ezquerro 2006, p. 100). | Carter subgroup |
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