Local Cohomology of Multi-Rees Algebras with Applications to joint Reductions and Complete Ideals

Acta Mathematica Vietnamica - Tập 40 - Trang 479-510 - 2015
Shreedevi K. Masuti1, Tony J. Puthenpurakal2, J. K. Verma2
1Institute of Mathematical Sciences, Chennai, India
2Department of Mathematics, Indian Institute of Technology, Bombay Powai, India

Tóm tắt

Let $(R,\mathfrak {m})$ be a Cohen-Macaulay local ring of dimension d and I=(I 1,…,I d ) be $\mathfrak {m}-$ primary ideals in R. We prove that ${\lambda }_{R}([H^{d}_{(x_{ii}t_{i}:1\leq i \leq d)}({\mathcal {R}}^{\prime }({\mathcal {F}})]_{\mathbf {n}})$ $< \infty $ , for all $\mathbf {n} \in \mathbb {N}^{d}$ , where ${\mathcal {F}}=\{{\mathcal {F}}(\mathbf {n}):\mathbf {n}\in \mathbb {Z}^{d}\}$ is an I−admissible filtration and (x i j ) is a strict complete reduction of ${\mathcal {F}}$ and ${\mathcal {R}}^{\prime }({\mathcal {F}})$ is the extended multi-Rees algebra of ${\mathcal {F}}.$ As a consequence, we prove that the normal joint reduction number of I,J,K is zero in an analytically unramified Cohen-Macaulay local ring of dimension 3 if and only if $\overline {e}_{3}(IJK)-[\overline {e}_{3}(IJ)+\overline {e}_{3}(IK)$ $+\overline {e}_{3}(JK)]+\overline {e}_{3}(I)+ \overline {e}_{3}(J)+\overline {e}_{3}(K)=0$ . This generalizes a theorem of Rees on joint reduction number zero in dimension 2. We apply this theorem to generalize a theorem of M. A. Vitulli in dimension 3.

Tài liệu tham khảo

Brodmann, M.P., Sharp, R.Y.: Local cohomology: an algebraic introduction with geometric applications. Cambridge University Press, Second edition (2013) Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge University Press, Revised Edition (1998) D’Cruz, C., Masuti, S.K.: Local cohomology of bigraded Rees algebras, Bhattacharya Coefficients and Joint Reductions. arXiv: 1405.1550 Herrmann, M., Hyry, E., Ribbe, J.: On the Cohen-Macaulay and Gorenstein properties of multigraded Rees algebras. Manuscripta Math. 79, 343–377 (1993) Hyry, E.: The diagonal subring and the Cohen-Macaulay property of a multigraded ring. Trans. Am. Math. Soc. 351, 2213–2232 (1999) Itoh, S.: Coefficients of normal Hilbert polynomials. J. Algebra 150, 101–117 (1992) Jayanthan, A.V., Verma, J.K.: Grothendieck-Serre formula and bigraded Cohen-Macaulay Rees algebras. J. Algebra 254, 1–20 (2002) Kirby, D., Mehran, H.A.: Hilbert functions and the Koszul complex. J. London Math. Soc. 2(24), 459–466 (1981) Marley, T.: Hilbert functions of ideals in Cohen-Macaulay rings. Ph. D, Thesis, Purdue University (1989) Masuti, S.K.: Normal Hilbert coefficients and bigraded Rees algebras. Ph. D, Thesis, Indian Institute of Technology Bombay (2013) Masuti, S.K., Verma, J.K.: Local cohomology of bigraded Rees algebras and normal Hilbert coefficients. J. Pure Appl. Algebra 218, 904–918 (2014) Masuti, S.K., Sarkar, P., Verma, J.K.: Hilbert polynomials of multigraded filtration of ideals. Preprint Rees, D.: Generalizations of reductions and mixed multiplicities. J. London Math. Soc. 29, 397–414 (1984) Rees, D.: Hilbert functions and pseudo-rational local rings of dimension two. J. London Math. Soc. 2(24), 467–479 (1981) Rotman, J.: An introduction to homological algebra, Second edition. Springer, New York (2009) Vitulli, M.A.: Some normal monomial ideals. Contemp. Math. 324, 205–217 (2003)