University of Paderborn Department of Mathematics Diploma Thesis ...

8 CHAPTER 1. NOTATIONS AND PREREQUISITESOn the set M we define the lexicographic order >:=> lex . For two monomials m = x a 00 ·. . . · x an0 , n = x b 00 · . . . · x ann ∈ R, we have m > n if and only if the first non-zero entry in thevector (a 0 − b 0 , . . . , a n − b n ) is positive. The lexicographic order is a total order on the setM of all monomials of R.The set M serves as a basis of R viewed as vector space over K. By [R] d for a d ∈ N 0 wedenote all homogeneous elements of R having degree d, i.e.[R] d := {f | f ∈ R, deg f = d, f homogeneous}.For each d, [R] d itself is a vector space over K( with)a basis given by all monomials in Mn + dhaving degree d. Note that there are exactly of these monomials having degree dnand that this implies( ) n + ddim K [R] d = ,nwhere dim K denotes the vector space dimension over K. This result will be useful todetermine the Hilbert function, the Hilbert polynomial and the Hilbert series of R, as wewill see later.In this terminology we can write R as the direct sumR = ⊕ d∈Z[R] dor, if we shift the grading of R by −k,R(−k) = ⊕ k∈Z[R(−k)] d = ⊕ k∈Z[R] d−k .For an ideal I ⊂ R generated by the polynomials f 1 , . . . , f s ∈ R we write I = (f 1 , . . . , f s ).According to the notation mentioned above, we denote by [I] d the set of all homogeneouselements of I having degree d. For each d the set [I] d and the ideal I itself can again beviewed as a vector space over K. All ideals treated in this thesis will be homogeneous,i.e. we can and will consider the elements f 1 , . . . , f s generating I to be homogeneous polynomials.The class of ideals, which is of main interest in our context, is the class of monomial ideals.An ideal I = (m 1 , . . . , m s ) ⊂ R is called a monomial ideal if it can be generated bymonomials m 1 , . . . , m s , i.e. m i = x a i00 · x a i11 · . . . · x a inn , where a i0 , a i1 , . . . , a in ∈ N 0 . The set ofmonomials {m 1 , . . . , m s } is called a set of monomial generators of I. We call {m 1 , . . . , m s }a set of minimal monomial generators of I if it consists of a minimal number of generatorsof I, i.e. if for any other set of monomials {n 1 , . . . , n t } ⊂ R generating I, we have s ≤ t.In this case, µ(I) := s is the well-defined number of minimal generators of I.The class of monomial ideals is closed under the usual operations: If I, J ⊂ R are monomialideals, then I ∩ J, I + J and I : J = {f | f ∈ R, f · m ∈ I for every m ∈ J} aremonomial ideals. Furthermore, if I ⊂ R is a monomial ideal, the primary ideals q i and the