• DOI:10.2969/JMSJ/1191418707
• Corpus ID: 16908180

Derived category of squarefree modules and local cohomology with monomial ideal support

@article{Yanagawa2003DerivedCO,
  title={Derived category of squarefree modules and local cohomology with monomial ideal support},
  author={Kohji Yanagawa},
  journal={Journal of The Mathematical Society of Japan},
  year={2003},
  volume={56},
  pages={289-308}
}

• Kohji Yanagawa
• Published 10 March 2003
• Mathematics
• Journal of The Mathematical Society of Japan

A "squarefree module" over a polynomial ring $S = k[x_1, .., x_n]$ is a generalization of a Stanley-Reisner ring, and allows us to apply homological methods to the study of monomial ideals systematically. Let $Sq$ be the category of squarefree modules. Then the derived category $D^b(Sq)$ of $Sq$ has three duality functors which act on $D^b(Sq)$ just like three transpositions of the symmetric group $S_3$ (up to translation). This phenomenon is closely related to the Koszul dulaity (in particular, the Bernstein-Gel'fand-Gel'fand correspondence). We also study the local cohomology module $H_{I_\Delta}^i(S)$ at a Stanley-Reisner ideal $I_\Delta$ using squarefree modules. Among other things, we see that Hochster's formula on the Hilbert function of $H_m^i(S/I_\Delta)$ is also a formula on the characteristic cycle of $H_{I_\Delta}^{n-i}(S)$ as a module over the Weyl algebra $S $ (if $chara(k)=0$).