Analysis on some infinite modules, inner projection, and applications

Abstract

A projective scheme X X is called ‘quadratic’ if X X is scheme-theoretically cut out by homogeneous equations of degree 2 2 . Furthermore, we say that X X satisfies ‘property N 2 , p \textbf {N}_{2,p} ’ if it is quadratic and the quadratic ideal has only linear syzygies up to the first p p -th steps. In the present paper, we compare the linear syzygies of the inner projections with those of X X and obtain a theorem on ‘embedded linear syzygies’ as one of our main results. This is the natural projection-analogue of ‘restricting linear syzygies’ in the linear section case. As an immediate corollary, we show that the inner projections of X X satisfy property N 2 , p − 1 \textbf {N}_{2,p-1} for any reduced scheme X X with property N 2 , p \textbf {N}_{2,p} . Moreover, we also obtain the neccessary lower bound ( codim ⁡ X ) ⋅ p − p ( p − 1 ) 2 (\operatorname {codim} X)\cdot p -\frac {p(p-1)}{2} , which is sharp, on the number of quadrics vanishing on X X in order to satisfy N 2 , p \textbf {N}_{2,p} and show that the arithmetic depths of inner projections are equal to that of the quadratic scheme X X . These results admit an interesting ‘syzygetic’ rigidity theorem on property N 2 , p \textbf {N}_{2,p} which leads the classifications of extremal and next-to-extremal cases. For these results we develop the elimination mapping cone theorem for infinitely generated graded modules and improve the partial elimination ideal theory initiated by M. Green. This new method allows us to treat a wider class of projective schemes which cannot be covered by the Koszul cohomology techniques because these are not projectively normal in general.