The Hilbert function of a reduced K-algebra.

*(English)*Zbl 0535.13012Let \(A=\oplus_{i\geq 0}A_ i\) be a graded k-algebra of finite type (where \(A_ 0=k\) is a field and A is generated as a k-algebra by \(A_ 1)\). Such algebras are called standard G-algebras by R. P. Stanley [Adv. Math. 28, 57-83 (1978; Zbl 0384.13012)]. The Hilbert function \(\{b_ i\},i\geq 0\), of A is defined by \(b_ i=\dim_ kA_ i\). In 1927 Macaulay characterized those sequences which could be the Hilbert functions of a standard G-algebra. Such sequences are now called 0- sequences. In this paper we characterize those sequences which can be the Hilbert function of a reduced standard G-algebra. We show that the obvious necessary conditions on the sequence (namely that both the sequence and its first difference be 0-sequences) are also sufficient. We call such 0-sequences differentiable. Our proof gives an algorithm for constructing a reduced standard G-algebra from a differentiable 0- sequence.

We also investigate to what extent the geometry of Proj A is determined by the Hilbert function of A.

We also investigate to what extent the geometry of Proj A is determined by the Hilbert function of A.

##### MSC:

13E15 | Commutative rings and modules of finite generation or presentation; number of generators |

14A05 | Relevant commutative algebra |

16W50 | Graded rings and modules (associative rings and algebras) |

13D03 | (Co)homology of commutative rings and algebras (e.g., Hochschild, AndrĂ©-Quillen, cyclic, dihedral, etc.) |

13C15 | Dimension theory, depth, related commutative rings (catenary, etc.) |

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |