In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology)...

syzygy module is free, but not the (n − 1)th one (for a proof, see § Koszul complex, below). The theorem is also true for modules that are not finitely...

In mathematics, a Koszul–Tate resolution or Koszul–Tate complex of the quotient ring R/M is a projective resolution of it as an R-module which also has...

days after his 97th birthday. Koszul algebra Koszul complex Koszul duality Koszul cohomology Koszul connection Koszul–Tate resolution Lie algebra cohomology...

complex Eagon–Northcott complex Gersten complex Graph complex Koszul complex Moore complex Schur complex Differential graded algebra Differential graded Lie...

quotient of a polynomial algebra and taking the Koszul complex associated to this presentation. The Koszul complex acts as a semi-free resolution of the differential...

module Injective module Projective resolution Injective resolution Koszul complex Exact functor Derived functor Ext functor Tor functor Filtration (abstract...

C^{\infty }(M)} of the Hamiltonian action. The resulting Koszul complex is the Koszul complex of the S ( g ) {\displaystyle S({\mathfrak {g}})} -module...

zero or linear forms. An example of a Koszul algebra is a polynomial ring over a field, for which the Koszul complex is the minimal graded free resolution...

and named after Jean-Louis Koszul as they are closely related to the Koszul complex. Green (1989) surveys early work on Koszul cohomology, Eisenbud (2005)...

{\displaystyle A\otimes (A/K)\otimes \cdots \otimes (A/K)\otimes A} . Koszul complex Weibel 1994, p. 283. Cartan, Henri; Eilenberg, Samuel (1956), Homological...

cohomology by means of Koszul complexes (see below). ◻ {\displaystyle \square } Let R be a ring and x an element in it. We form the chain complex K(x) given by...

In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory...

being "well suited for computations." The Čech complex can be defined as a colimit of Koszul complexes K ∙ ( f 1 , … , f m ) {\displaystyle K^{\bullet...

to higher order syzygy modules, and this leads to the concept of the Koszul complex of an ideal, which provides information on the non-trivial relations...

fattened. If r1, ..., rd is a regular sequence in a ring R, then the Koszul complex is an explicit free resolution of R/(r1, ..., rd) as an R-module, of...

{\displaystyle F_{0}=R} , and let K ∙ {\displaystyle K_{\bullet }} denote the Koszul complex of R with respect to x 1 , … , x d {\displaystyle x_{1},\ldots ,x_{d}}...

exterior algebra is the main ingredient in the construction of the Koszul complex, a fundamental object in homological algebra. The exterior algebra was...

M → 0. A classic example of a projective resolution is given by the Koszul complex of a regular sequence, which is a free resolution of the ideal generated...

Regular local ring Regular sequence Cohen–Macaulay ring Gorenstein ring Koszul complex Spectrum of a ring Zariski topology Kähler differential Generic flatness...

the quotient of a commutative ring by any regular sequence, using the Koszul complex. For example, if R is the polynomial ring k[x1,...,xn] over a field...

a module homomorphism is the pre-image of the zero element. Koszul complex Koszul complex Krull–Schmidt The Krull–Schmidt theorem says that (1) a finite-length...

theory both of formal and of analytical structure. This is a kind of Koszul complex theory, taken up by numerous mathematicians during the 1960s. In particular...

the quotient of a commutative ring by any regular sequence, using the Koszul complex. For example, if R is the polynomial ring k[x1, ..., xn] over a field...

lower than r-i. A classic example of a free resolution is given by the Koszul complex of a regular sequence in a local ring or of a homogeneous regular sequence...

complete intersection ring. A key argument in such considerations is the Koszul complex, which provides an explicit free resolution of the residue field k of...

emb dim (R) of R. Define a graded algebra H(R) as the homology of the Koszul complex with respect to a minimal system of generators of m/m2; up to isomorphism...

module Cohen-Macaulay ring Gorenstein ring Complete intersection ring Koszul complex Hilbert's syzygy theorem Quillen–Suslin theorem Height (ring theory)...

connections are also called Koszul connections after Jean-Louis Koszul, who gave an algebraic framework for describing them (Koszul 1950). This article defines...

integers of the imaginary quadratic field of discriminant −7. Koszul complex The Koszul complex is a free resolution constructed from a regular sequence....