Coherent sheaves on \({\mathbb{P}}^n\) and problems of linear algebra.

*(English)*Zbl 0424.14003In this classic paper the author studies the bounded derived category \(D^b ({\mathbb P}^n)\) of coherent sheaves on a projective space \({\mathbb P}^n\). The main result shows that this category is equivalent (as a triangulated category) to a certain subcategory of the category of graded modules over a symmetric algebra of \(V=H^0({\mathbb P}^n,{\mathcal O}(1))\). One can also replace the symmetric algebra by the exterior algebra of \(V\). This result was motivated by earlier results of Horrocks who introduced monads and it was the first example of computation of the derived category of coherent sheaves on an algebraic variety.

Two other, although similar, descriptions of \(D^b ({\mathbb P}^n)\) were obtained at the same time by I. N. Bernstein, I. M. Gel’fand and S. I. Gel’fand [see Funkts. Anal. Prilozh. 12, 66–67 (1978; Zbl 0402.14005)] using a more algebraic approach.

However, Beilinson’s paper is very remarkable not only because of this result but also because it introduces a very useful method of describing the derived category of coherent sheaves on a variety \(X\) using a resolution of the structure sheaf of the diagonal \(\Delta \subset X\times X\). Implicitly, it also contains the Beilinson spectral sequences that proved to be extremely useful, e.g., when studying sheaves whose cohomology groups are known.

In a nutshell, this small paper contains the striking concept that relevant geometric information is encoded in a seemingly highly abstract object, which has lead to a whole industry today, along the results of A. Bondal and D. Orlov [Compos. Math. 125, No.3, 327–344 (2001; Zbl 0994.18007)] that \(D^b_{coh}(X)\) indeed determines \(X\) for varieties with ample anticanonical (e.g., the Fano case) or ample canonical divisor, the theory of spherical objects and the associated spherical twists introduced by P. Seidel and R. Thomas [Duke Math. J. 108, No. 1, 37–108 (2001; Zbl 1092.14025)] or the homological formulation of mirror symmetry following M. Kontsevich [Proceedings of the international congress of mathematicians, ICM ’94, August 3–11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120–139 (1995; Zbl 0846.53021)].

For recent advances in the study of derived categories of sheaves for algebraic varieties we refer to the books of D. Huybrechts [Fourier-Mukai transforms in algebraic geometry. Oxford Mathematical Monographs, Oxford: Clarendon Press (2006; Zbl 1095.14002)] and C. Bartocci, U. Bruzzo and D. Hernandez-Ruiperez [Fourier-Mukai and Nahm transforms in geometry and mathematical physics. Progress in Mathematics 276, Birkhäuser (2009; Zbl 1186.14001)] and the references within.

Two other, although similar, descriptions of \(D^b ({\mathbb P}^n)\) were obtained at the same time by I. N. Bernstein, I. M. Gel’fand and S. I. Gel’fand [see Funkts. Anal. Prilozh. 12, 66–67 (1978; Zbl 0402.14005)] using a more algebraic approach.

However, Beilinson’s paper is very remarkable not only because of this result but also because it introduces a very useful method of describing the derived category of coherent sheaves on a variety \(X\) using a resolution of the structure sheaf of the diagonal \(\Delta \subset X\times X\). Implicitly, it also contains the Beilinson spectral sequences that proved to be extremely useful, e.g., when studying sheaves whose cohomology groups are known.

In a nutshell, this small paper contains the striking concept that relevant geometric information is encoded in a seemingly highly abstract object, which has lead to a whole industry today, along the results of A. Bondal and D. Orlov [Compos. Math. 125, No.3, 327–344 (2001; Zbl 0994.18007)] that \(D^b_{coh}(X)\) indeed determines \(X\) for varieties with ample anticanonical (e.g., the Fano case) or ample canonical divisor, the theory of spherical objects and the associated spherical twists introduced by P. Seidel and R. Thomas [Duke Math. J. 108, No. 1, 37–108 (2001; Zbl 1092.14025)] or the homological formulation of mirror symmetry following M. Kontsevich [Proceedings of the international congress of mathematicians, ICM ’94, August 3–11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120–139 (1995; Zbl 0846.53021)].

For recent advances in the study of derived categories of sheaves for algebraic varieties we refer to the books of D. Huybrechts [Fourier-Mukai transforms in algebraic geometry. Oxford Mathematical Monographs, Oxford: Clarendon Press (2006; Zbl 1095.14002)] and C. Bartocci, U. Bruzzo and D. Hernandez-Ruiperez [Fourier-Mukai and Nahm transforms in geometry and mathematical physics. Progress in Mathematics 276, Birkhäuser (2009; Zbl 1186.14001)] and the references within.

Reviewer: Adrian Langer (Warszawa)

##### MSC:

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

15A75 | Exterior algebra, Grassmann algebras |

18E30 | Derived categories, triangulated categories (MSC2010) |

##### Keywords:

coherent sheaves; classification of vector bundle; exterior algebra; derived category; projective space
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\textit{A. A. Beilinson}, Funct. Anal. Appl. 12, 214--216 (1979; Zbl 0424.14003)

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##### References:

[1] | W. Barth, Invent. Math.,42, 63–92 (1977). · Zbl 0386.14005 · doi:10.1007/BF01389784 |

[2] | V. G. Drinfel’d and Yu. I. Manin, Usp. Mat. Nauk,33, No. 3, 165–166 (1978). |

[3] | J.-L. Verdier, Lecture Notes Math.,569, 262–311 (1977). · doi:10.1007/BFb0091525 |

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