“Koszul algebra”的意思、由来-开放百科全书

词条

Koszul algebra

释义

See also
References

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

It is named after the French mathematician Jean-Louis Koszul.

Here, is the graded algebra with grading shifted up by , i.e. . The exponents refer to the -fold direct sum.

We can choose bases for the free modules in the resolution; then the maps can be written as matrices. For a Koszul algebra, the entries in the matrices are 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,

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}}.

{{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}}.{{algebra-stub}}

1 : Algebras

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。

See also

References