Throughout this book, \(k = \bar k\) shall always denote an algebraically closed field. We will occasionally also require that it have characteristic zero, but we will always make it clear when we are making this assumption. All varieties and subschemes will be assumed to be projective. We shall denote by S the homogeneous polynomial ring k[X0,… X n ,] and we let ℙ n = ℙ n k = Proj S. Since S is a graded ring, it is the direct sum of its homogeneous components: S = ⊗d≥0 S d ,where S d is the vector space of homogeneous polynomials of degree d. We denote by m the maximal ideal, m=(X0,…,Xn,)⊂S.
KeywordsExact Sequence Complete Intersection Short Exact Sequence Regular Sequence Hilbert Function
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