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Analysis on some infinite modules, inner projection, and applications


Authors: Kangjin Han and Sijong Kwak
Journal: Trans. Amer. Math. Soc. 364 (2012), 5791-5812
MSC (2010): Primary 14N05, 13D02, 14N25; Secondary 51N35
DOI: https://doi.org/10.1090/S0002-9947-2012-05755-2
Published electronically: June 8, 2012
MathSciNet review: 2946932
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Abstract:

A projective scheme $ X$ is called `quadratic' if $ X$ is scheme-theoretically cut out by homogeneous equations of degree $ 2$. Furthermore, we say that $ X$ satisfies `property $ \textbf {N}_{2,p}$' if it is quadratic and the quadratic ideal has only linear syzygies up to the first $ p$-th steps. In the present paper, we compare the linear syzygies of the inner projections with those of $ 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$ satisfy property $ \textbf {N}_{2,p-1}$ for any reduced scheme $ X$ with property $ \textbf {N}_{2,p}$.

Moreover, we also obtain the neccessary lower bound $ (\operatorname {codim} X)\cdot p -\frac {p(p-1)}{2}$, which is sharp, on the number of quadrics vanishing on $ X$ in order to satisfy $ \textbf {N}_{2,p}$ and show that the arithmetic depths of inner projections are equal to that of the quadratic scheme $ X$. These results admit an interesting `syzygetic' rigidity theorem on property $ \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.


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Additional Information

Kangjin Han
Affiliation: Department of Mathematics, Korea Advanced Institute of Science and Technology, 373-1 Gusung-dong, Yusung-Gu, Daejeon, Korea
Address at time of publication: School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, Korea
Email: han.kangjin@kaist.ac.kr, kangjin.han@kias.re.kr

Sijong Kwak
Affiliation: Department of Mathematics, Korea Advanced Institute of Science and Technology, 373-1 Gusung-dong, Yusung-Gu, Daejeon, Korea
Email: skwak@kaist.ac.kr

DOI: https://doi.org/10.1090/S0002-9947-2012-05755-2
Keywords: Linear syzygies, the mapping cone theorem, partial elimination ideals, inner projection, arithmetic depth, Castelnuovo-Mumford regularity
Received by editor(s): October 14, 2010
Published electronically: June 8, 2012
Additional Notes: The authors were supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (grant No. 2009-0063180)
Dedicated: Dedicated to the memory of Hyo Chul Myung (June, 1937–February, 2010)
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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