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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

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.