Koszul algebra

In abstract algebra, a Koszul algebra $R$ is a graded $k$ -algebra over which the ground field $k$ has a linear minimal graded free resolution, i.e., there exists an exact sequence:

: $\cdots \rightarrow (R(-i))^{b_i} \rightarrow \cdots \rightarrow (R(-2))^{b_2} \rightarrow (R(-1))^{b_1} \rightarrow R \rightarrow k \rightarrow 0.$

for some nonnegative integers $b_i$ . Here $R(-j)$ is the graded algebra $R$ with grading shifted up by $j$ , i.e. $R(-j)_i = R_{i-j}$ , and the exponent $b_i$ refers 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 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, $R = k[x,y]/(xy)$ .

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

References

{{citation

| last = Fröberg | first = R.

| contribution = Koszul algebras

| location = New York

| mr = 1767430

| pages = 337–350

| publisher = Marcel Dekker

| series = Lecture Notes in Pure and Applied Mathematics

| title = Advances in commutative ring theory (Fez, 1997)

| volume = 205

| year = 1999}}.

{{citation

| last1 = Loday | first1 = Jean-Louis

| last2 = Vallette | first2 = Bruno

| doi = 10.1007/978-3-642-30362-3

| isbn = 978-3-642-30361-6

| location = Heidelberg

| mr = 2954392

| publisher = Springer

| series = Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

| title = Algebraic operads

| url = https://www.math.univ-paris13.fr/~vallette/Operads.pdf

| volume = 346

| year = 2012}}.

{{citation

| last1 = Beilinson | first1 = Alexander

| last2 = Ginzburg | first2 = Victor

| last3 = Soergel | first3 = Wolfgang

| doi = 10.1090/S0894-0347-96-00192-0

| issue = 2

| journal = Journal of the American Mathematical Society

| mr = 1322847

| pages = 473–527

| title = Koszul duality patterns in representation theory

| volume = 9

| year = 1996| doi-access = free

}}.

{{citation

| last1 = Mazorchuk | first1 = Volodymyr

| last2 = Ovsienko | first2 = Serge

| last3 = Stroppel | first3 = Catharina | author3-link = Catharina Stroppel

| doi = 10.1090/S0002-9947-08-04539-X

| issue = 3

| journal = Transactions of the American Mathematical Society

| mr = 2457393

| pages = 1129–1172

| title = Quadratic duals, Koszul dual functors, and applications

| volume = 361

| year = 2009| arxiv = math/0603475}}.

Category:Algebras

Koszul algebra

See also

References