abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...

, a Koszul algebra

R

is a graded

k

algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...

over which the ground field

k

has a linear minimal graded free resolution, ''i.e.'', there exists an

exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the conte ...

: :

\cdots \rightarrow R(-i)^ \rightarrow \cdots \rightarrow R(-2)^ \rightarrow R(-1)^ \rightarrow R \rightarrow k \rightarrow 0.

Here,

R(-j)

is the graded algebra

R

with grading shifted up by

j

, ''i.e.''

R(-j)_i = R_

. The exponents

b_i

refer to the

b_i

-fold direct sum. Choosing bases for the free modules in the resolution, the chain maps are given by matrices, and the definition requires the matrix entries to be zero or linear forms. An example of a Koszul algebra is a

polynomial ring 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 variable ...

over a field, for which the Koszul complex is the minimal graded free resolution of the ground field. There are Koszul algebras whose ground fields have infinite minimal graded free resolutions, ''e.g'',

R = k,y (xy)

. The concept is named after the French mathematician Jean-Louis Koszul.

References

*. *. *. *. Algebras {{algebra-stub

See also

References