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Exact sequence used to describe the structure of an object
more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules
Resolution_(algebra)
Topics referred to by the same term
Year's Day Dispute resolution, the settlement of a disagreement Resolution (algebra), an exact sequence in homological algebra Resolution (logic), a rule
Resolution
Concept in algebraic geometry
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, which is a non-singular variety
Resolution_of_singularities
Springer resolution is a resolution of the variety of nilpotent elements in a semisimple Lie algebra, or the unipotent elements of a reductive algebraic group
Springer_resolution
In algebraic K-theory, Quillen's resolution theorem states that if A ⊂ C {\displaystyle {\mathcal {A}}\subset {\mathcal {C}}} is an exact subcategory where
Resolution theorem (algebraic K-theory)
Resolution_theorem_(algebraic_K-theory)
In homological algebra, the Cartan–Eilenberg resolution is in a sense, a resolution of a chain complex. It can be used to construct hyper-derived functors
Cartan–Eilenberg_resolution
Branch of mathematics
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins
Homological_algebra
Process in mathematics
example of a Koszul algebra is a polynomial ring over a field, for which the Koszul complex is the minimal graded free resolution of the ground field
Koszul_algebra
Branch of mathematics
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems
Algebraic_geometry
Technique for constructing resolutions in homological algebra
the bar resolution, bar construction, standard resolution, or standard complex, is a way of constructing resolutions in homological algebra. It was first
Bar_complex
glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic geometry, glossary
Glossary of commutative algebra
Glossary_of_commutative_algebra
Graduate level textbook on algebra
semi-simplicity. The fourth part, Homological Algebra, covers general homology theory and finite free resolutions. The Mathematical Association of America
Algebra_(book)
Algebra, a branch of mathematics
In homological algebra, a branch of mathematics, a matrix factorization is a tool used to study infinitely long resolutions, generally over commutative
Matrix factorization (algebra)
Matrix_factorization_(algebra)
Algebraic structure in ring theory
In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion-free modules. Formally, a module M over
Flat_module
Study of systems of inequalitites
mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with
Real_algebraic_geometry
predicts that the gonality of the algebraic curve C can be calculated by homological algebra means, from a minimal resolution of an invertible sheaf of high
Gonality of an algebraic curve
Gonality_of_an_algebraic_curve
French mathematician (1540–1603)
Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to his innovative use of letters as parameters
François_Viète
Mathematical concept
Let π : Y → X {\displaystyle \pi :Y\to X} be a morphism between complex algebraic varieties, where Y {\displaystyle Y} is smooth and carries a symplectic
Symplectic_resolution
In algebraic geometry, the homogeneous coordinate ring is a certain commutative ring assigned to any projective variety. If V is an algebraic variety given
Homogeneous_coordinate_ring
Homological algebra is the study of homological functors
lemma Extension (algebra) Central extension Splitting lemma Projective module Injective module Projective resolution Injective resolution Koszul complex
List of homological algebra topics
List_of_homological_algebra_topics
In algebraic geometry, the Bott–Samelson resolution of a Schubert variety is a resolution of singularities. It was introduced by Bott & Samelson (1958)
Bott–Samelson_resolution
Concept related to resolving singularities in algebraic geometry
In algebraic geometry, local uniformization is a weak form of resolution of singularities, stating that a variety can be desingularized near any valuation
Local_uniformization
Theorem in Boolean algebra
In Boolean algebra, the consensus theorem or rule of consensus is the identity: x y ∨ x ¯ z ∨ y z = x y ∨ x ¯ z {\displaystyle xy\vee {\bar {x}}z\vee
Consensus_theorem
Auslander (1974). An Artin algebra Γ is called an Auslander algebra if gl dim Γ ≤ 2 and if 0 → Γ → I → J → K → 0 is a minimal injective resolution of Γ then I and
Auslander_algebra
Direct summand of a free module (mathematics)
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over
Projective_module
Describes the structure of some free resolutions of a quotient of a local or graded ring
mathematics, the Hilbert–Burch theorem describes the structure of some free resolutions of a quotient of a local or graded ring in the case that the quotient
Hilbert–Burch_theorem
Concept in algebraic geometry
In algebraic geometry, an algebraic variety or scheme X is normal if it is normal at every point, meaning that the local ring at the point is an integrally
Normal_scheme
homological conjectures have been a focus of research activity in commutative algebra since the early 1960s. They concern a number of interrelated (sometimes
Homological conjectures in commutative algebra
Homological_conjectures_in_commutative_algebra
Construction in homological algebra
homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie algebras, and
Tor_functor
Homological algebra statement
In homological algebra, the horseshoe lemma, also called the simultaneous resolution theorem, is a statement relating resolutions of two objects A ′ {\displaystyle
Horseshoe_lemma
In algebraic geometry, a crepant resolution of a singularity is a resolution that does not affect the canonical class of the manifold. The term "crepant"
Crepant_resolution
Mathematics award
moduli of algebraic varieties." Karim Adiprasito – "For the development, with Eric Katz, of combinatorial Hodge theory leading to the resolution of the log-concavity
Breakthrough Prize in Mathematics
Breakthrough_Prize_in_Mathematics
On the effects of changing the ring of ''K''-groups
In algebra, the fundamental theorem of algebraic K-theory describes the effects of changing the ring of K-groups from a ring R to R [ t ] {\displaystyle
Fundamental theorem of algebraic K-theory
Fundamental_theorem_of_algebraic_K-theory
Concept in commutative algebra
In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is
Excellent_ring
Japanese mathematician (1931–2026)
least 3. In 1964, Hironaka proved that singularities of algebraic varieties admit resolutions in characteristic zero. Hironaka was able to give a general
Heisuke_Hironaka
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
resolution or Koszul–Tate complex of the quotient ring R/M is a projective resolution of it as an R-module which also has a structure of a dg-algebra
Koszul–Tate_resolution
Sheaf theory concept
The Godement resolution of a sheaf is a construction in homological algebra that allows one to view global, cohomological information about the sheaf in
Godement_resolution
projective resolution of singularities X ′ → X {\displaystyle X'\to X} . Despite Chow's theorem, not every complex analytic variety is a complex algebraic variety
Complex_algebraic_variety
Russian-American mathematician (1899–1986)
Zariski, Oscar (1972), Collected papers. Vol. I: Foundations of algebraic geometry and resolution of singularities, Cambridge, Massachusetts-London: MIT Press
Oscar_Zariski
Four mathematical theorems
Ch. V, Resolution Theorem 3.1. Weibel 2013, Ch. V, Cofinality Theorem 2.3. Weibel, Charles (2013). "The K-book: An introduction to algebraic K-theory"
Basic theorems in algebraic K-theory
Basic_theorems_in_algebraic_K-theory
Mathematical object in abstract algebra
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
Injective_module
Tool in mathematical dimension theory
In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a
Hilbert series and Hilbert polynomial
Hilbert_series_and_Hilbert_polynomial
field of an algebraic variety Ample line bundle Ample vector bundle Linear system of divisors Birational geometry Blowing up Resolution of singularities
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Construction in homological algebra
homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and
Ext_functor
In mathematics, specifically algebraic topology, there is a resolution analogous to free resolutions of spectra yielding a tool for constructing the Adams
Adams_resolution
American mathematician (born 1983)
in 2022. He has been noted for the linkages that he has found between algebraic geometry and combinatorics. Huh was born in Stanford, California while
June_Huh
Computer algebra system
commutative algebra and algebraic geometry. Macaulay2 is built around fast implementations of algorithms useful for computation in commutative algebra and algebraic
Macaulay2
Objects in representation theory of Lie algebras
algebras, a branch of mathematics. Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra
Verma_module
mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Harvey White B&W series of films (30m each) 1957 titles (incomplete): Algebra and Powers of Ten / The Atmosphere / Atomic Accelerators / The Bohr Atom
List of Encyclopædia Britannica Films titles
List_of_Encyclopædia_Britannica_Films_titles
In mathematics, an algebra such as ( R , + , ⋅ ) {\displaystyle (\mathbb {R} ,+,\cdot )} has multiplication ⋅ {\displaystyle \cdot } whose associativity
Homotopy_associative_algebra
although Colin Maclaurin also published the method in his 1748 Treatise of Algebra (and probably knew of the method as early as 1729). Pell's equation. The
List_of_misnamed_theorems
Set of vectors used to define coordinates
program Coordinate system Change of basis – Coordinate change in linear algebra Frame of a vector space – Similar to the basis of a vector space, but not
Basis_(linear_algebra)
Family of scientific calculators by Casio
scientific calculators made by Casio which use Casio's Visually Perfect Algebraic Method (V.P.A.M.), Natural Display or Natural V.P.A.M. input methods.
Casio_V.P.A.M._calculators
Kind of partial function between algebraic varieties
particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article
Rational_mapping
Subject area in mathematics
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic
Algebraic_K-theory
Type of logical formula
properties for use in logic programming, formal specification, universal algebra and model theory. Horn clauses are named for the logician Alfred Horn,
Horn_clause
Brieskorn–Grothendieck resolution is a resolution conjectured by Alexander Grothendieck, that in particular gives a resolution of the universal deformation
Brieskorn–Grothendieck resolution
Brieskorn–Grothendieck_resolution
In mathematics, more particularly in the field of algebraic geometry, a scheme X {\displaystyle X} has rational singularities, if it is normal, of finite
Rational_singularity
pair. Canonical singularity Crepant resolution Kollár, János; Mori, Shigefumi (1998), Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics
Discrepancy (algebraic geometry)
Discrepancy_(algebraic_geometry)
Symmetry between bosons and fermions
algebra requires the introduction of a Z2-grading under which the bosons are the even elements and the fermions are the odd elements. Such an algebra
Supersymmetry
Algebraic Logic Functional (ALF) programming language combines functional and logic programming techniques. Its foundation is Horn clause logic with equality
Algebraic Logic Functional programming language
Algebraic_Logic_Functional_programming_language
Algebraic structure
The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific
Commutative_ring
Weil cohomology theory for schemes X over a base field k
Jean-Marc Fontaine, the resolution of which is called p-adic Hodge theory. For a variety X {\displaystyle X} over an algebraically closed field of characteristic
Crystalline_cohomology
Branch of mathematics
{\displaystyle A} . In the case of analytic algebras these resolutions are called the Tjurina resolution for the mathematician who first studied such
Deformation_(mathematics)
Topics referred to by the same term
top-level domain Google Reader, a defunct RSS reader service The low-resolution mode of Apple II graphics GR, METAR reporting code for hail ≥ 5 mm in
GR
Theory for associative algebras over rings
Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors
Hochschild_homology
Generalization of vector bundles
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric
Coherent_sheaf
-differential graded algebra ( B ∙ , d B ) {\displaystyle (B_{\bullet },d_{B})} can be constructed using a semi-free resolution ( R ∙ , d R ) → ( B ∙
Derived_scheme
Branch of mathematics that studies the properties of groups
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known
Group_theory
October 1970). "On the Possibility of a Projective Plane of Order Ten". Algebraic Theory of Codes II (Report). Air Force Cambridge Research Laboratories
Lam's_problem
Type of ring in commutative algebra
In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal
Regular_local_ring
Concept in mathematics
In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called
Morphism of algebraic varieties
Morphism_of_algebraic_varieties
German mathematician (1882–1935)
German mathematician who made many important contributions to abstract algebra. She also proved Noether's first and second theorems, which are fundamental
Emmy_Noether
Various mathematical dualites
of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra) and topology (e.g., equivariant cohomology). The prototypical
Koszul_duality
Tools for studying groups based on techniques from algebraic topology
abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper. As in algebraic topology
Group_cohomology
Well-behaved sequence in a commutative ring
In commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent as possible, in a precise sense. This
Regular_sequence
Mathematical object in sheaf cohomology
mathematics, injective sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as
Injective_sheaf
Prize awarded by the American Mathematical Society
American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to number theory. The prize
Cole_Prize
Point without a tangent space
In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric
Singular point of an algebraic variety
Singular_point_of_an_algebraic_variety
Mathematical theory
the algebraic variety at the point. Intensive studies of such singularities led in the end to Heisuke Hironaka's fundamental theorem on resolution of singularities
Singularity_theory
Ring whose ideals are projective
path algebra of a quiver. This is a consequence of the existence of the standard resolution (which is of length 1) for modules over a path algebra. The
Hereditary_ring
singular point of an algebraic variety (cf. resolution of singularities). Because of this aspect, valuation theory, Rees algebras and integral closure
Multiplicity_theory
Notation for recording chess games
concise algebraic notation was in use. Since 1981, FIDE no longer recognizes descriptive notation for the purposes of dispute resolution, and algebraic notation
Descriptive_notation
Study of dimension in algebraic geometry
dimension theory is the study in terms of commutative algebra of the notion of dimension of an algebraic variety (and by extension that of a scheme). The need
Dimension_theory_(algebra)
structure on smooth algebraic varieties is recovered when the degree is zero. Spectral invariant Spectral invariants. Springer resolution symplectic action
Glossary of symplectic geometry
Glossary_of_symplectic_geometry
French mathematician
97th birthday. Koszul algebra Koszul complex Koszul duality Koszul cohomology Koszul connection Koszul–Tate resolution Lie algebra cohomology Décès de Jean-Louis
Jean-Louis_Koszul
Type of mathematical expression
used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. The word polynomial joins two
Polynomial
German mathematician (1844–1921)
up in order to prove resolution of singularities for plane curves. Noether made major contributions to the theory of algebraic surfaces. Noether's formula
Max_Noether
Homological construction in category theory
choice of fibrant or cofibrant resolution, etc.) Manin, Yuri Ivanovich; Gelfand, Sergei I. (2003), Methods of Homological Algebra, Berlin, New York: Springer-Verlag
Derived_functor
Singularities of algebraic varieties
theorems on resolution of singularities relate an arbitrary variety to a divisor with simple normal crossings in a smooth variety. Let X be an algebraic variety
Normal_crossing_singularity
French mathematician (1811–1832)
foundations for Galois theory and group theory, two major branches of abstract algebra. Galois was a staunch Republican and was heavily involved in the political
Évariste_Galois
Formula that provides the solutions to a quadratic equation
In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic
Quadratic_formula
In algebra, a module over a ring
In algebra, a free presentation of a module M over a commutative ring R is an exact sequence of R-modules: ⨁ i ∈ I R → f ⨁ j ∈ J R → g M → 0.
Free_presentation
Area of discrete mathematics
where he drew an analogy between "quantic invariants" and "co-variants" of algebra and molecular diagrams. The definition of a graph can vary, but one can
Graph_theory
On polynomial rings over fields
result of homological algebra. It is the starting point of the use of homological methods in commutative algebra and algebraic geometry. The syzygy theorem
Hilbert's_syzygy_theorem
Equations of degree 5 or higher cannot be solved by radicals
of ever arriving at the resolution of the general equation algebraically, it appears more and more likely that this resolution is impossible and contradictory
Abel–Ruffini_theorem
Type of smooth complex surface of kodaira dimension 0
any quartic surface Y with du Val singularities, the minimal resolution of Y is an algebraic K3 surface. The intersection of a quadric and a cubic in P
K3_surface
Scheme used in the numerical solution of partial differential equations
High-resolution schemes are used in the numerical solution of partial differential equations where high accuracy is required in the presence of shocks
High-resolution_scheme
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