### Abstract

A major result in Algebraic Geometry is the theorem of Bernstein-Gelfand- Gelfand that states the existence of an equivalence of triangulated categories: gr_{Λ} ≅ Db(Coh ℙn), where gr_{Λ} denotes the stable category of finitely generated graded modules over the n + 1 exterior algebra and Db(Coh ℙn) is the derived category of bounded complexes of coherent sheaves on projective space ℙn. Generalizations of this result were obtained in Martínez-Villa and Saorín (2004) and from a different point of view, the theorem has been extended by Yanagawa (2004) to ℤn-graded modules over the polynomial algebra. This generalization has important applications in combinatorial commutative algebra. The aim of the article is to extend the results of Martínez-Villa and Saorín (2004) to group graded algebras in order to obtain a generalization of Yanagawa's results having in mind the application to other settings (Geigle and Lenzing, 1987).

Original language | English |
---|---|

Pages (from-to) | 3145-3163 |

Number of pages | 19 |

Journal | Communications in Algebra |

Volume | 35 |

Issue number | 10 |

DOIs | |

Publication status | Published - Oct 2007 |

### Fingerprint

### Keywords

- Covering
- G-graded
- Koszul

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Communications in Algebra*,

*35*(10), 3145-3163. https://doi.org/10.1080/00914030701409825

**On a group graded version of BGG.** / Green, E. L.; Martínez-Villa, R.; Yoshino, Yuji.

Research output: Contribution to journal › Article

*Communications in Algebra*, vol. 35, no. 10, pp. 3145-3163. https://doi.org/10.1080/00914030701409825

}

TY - JOUR

T1 - On a group graded version of BGG

AU - Green, E. L.

AU - Martínez-Villa, R.

AU - Yoshino, Yuji

PY - 2007/10

Y1 - 2007/10

N2 - A major result in Algebraic Geometry is the theorem of Bernstein-Gelfand- Gelfand that states the existence of an equivalence of triangulated categories: grΛ ≅ Db(Coh ℙn), where grΛ denotes the stable category of finitely generated graded modules over the n + 1 exterior algebra and Db(Coh ℙn) is the derived category of bounded complexes of coherent sheaves on projective space ℙn. Generalizations of this result were obtained in Martínez-Villa and Saorín (2004) and from a different point of view, the theorem has been extended by Yanagawa (2004) to ℤn-graded modules over the polynomial algebra. This generalization has important applications in combinatorial commutative algebra. The aim of the article is to extend the results of Martínez-Villa and Saorín (2004) to group graded algebras in order to obtain a generalization of Yanagawa's results having in mind the application to other settings (Geigle and Lenzing, 1987).

AB - A major result in Algebraic Geometry is the theorem of Bernstein-Gelfand- Gelfand that states the existence of an equivalence of triangulated categories: grΛ ≅ Db(Coh ℙn), where grΛ denotes the stable category of finitely generated graded modules over the n + 1 exterior algebra and Db(Coh ℙn) is the derived category of bounded complexes of coherent sheaves on projective space ℙn. Generalizations of this result were obtained in Martínez-Villa and Saorín (2004) and from a different point of view, the theorem has been extended by Yanagawa (2004) to ℤn-graded modules over the polynomial algebra. This generalization has important applications in combinatorial commutative algebra. The aim of the article is to extend the results of Martínez-Villa and Saorín (2004) to group graded algebras in order to obtain a generalization of Yanagawa's results having in mind the application to other settings (Geigle and Lenzing, 1987).

KW - Covering

KW - G-graded

KW - Koszul

UR - http://www.scopus.com/inward/record.url?scp=34848827908&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34848827908&partnerID=8YFLogxK

U2 - 10.1080/00914030701409825

DO - 10.1080/00914030701409825

M3 - Article

AN - SCOPUS:34848827908

VL - 35

SP - 3145

EP - 3163

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 10

ER -