# 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} }

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…

## 25 Citations

Dualizing complex of the incidence algebra of a finite regular cell complex

- Mathematics
- 2004

Let $\Sigma$ be a finite regular cell complex with $\emptyset \in \Sigma$, and regard it as a partially ordered set (poset) by inclusion. Let $R$ be the incidence algebra of the poset $\Sigma$ over a…

LYUBEZNIK NUMBERS OF LOCAL RINGS AND LINEAR STRANDS OF GRADED IDEALS

- MathematicsNagoya Mathematical Journal
- 2017

In this work, we introduce a new set of invariants associated to the linear strands of a minimal free resolution of a $\mathbb{Z}$ -graded ideal $I\subseteq R=\Bbbk [x_{1},\ldots ,x_{n}]$ . We also…

Lyubeznik numbers of local rings and linear strands of graded ideals

- Mathematics
- 2014

In this work we introduce a new set of invariants associated to the linear strands of a minimal free resolution of a $\mathbb{Z}$-graded ideal $I\subseteq R=\Bbbk[x_1, \ldots, x_n]$. We also prove…

BGG Correspondence and Römer’s Theorem on an Exterior Algebra

- Mathematics
- 2004

Let $E = K{\left\langle {y_{1} ,...,y_{n} } \right\rangle }$ be the exterior algebra. The (cohomological) distinguished pairs of a graded E-module N describe the growth of a minimal graded injective…

Notes on C-Graded Modules Over an Affine Semigroup Ring K[C]

- Mathematics
- 2005

Let C ⊂ ℕ d be an affine semigroup, and R = K[C] its semigroup ring. This article is a collection of various results on “C-graded” R-modules M = ⨁ c∈C M c , especially, monomial ideals of R. For…

On a Group Graded Version of BGG

- Mathematics
- 2007

A major result in Algebraic Geometry is the theorem of Bernstein–Gelfand–Gelfand that states the existence of an equivalence of triangulated categories: gr Λ ≅ 𝒟b(Coh ℙn), where gr Λ denotes the…

## References

SHOWING 1-10 OF 34 REFERENCES

Alexander Duality for Stanley–Reisner Rings and Squarefree Nn-Graded Modules

- Mathematics
- 2000

Abstract Let S = k[x1,…,xn] be a polynomial ring, and let ωS be its canonical module. First, we will define squarefreeness for N n-graded S-modules. A Stanley–Reisner ring k[Δ] = S/IΔ, its syzygy…

Bass numbers of local cohomology modules with supports in monomial ideals

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2001

In this paper, we will study the local cohomology modules HiI(S) of a polynomial ring S = k[x1, …, xn] with supports in a (radical) monomial ideal I. When S/I is a Cohen–Macaulay ring of dimension d…

Local Cohomology of Stanley–Reisner Rings with Supports in General Monomial Ideals☆

- Mathematics
- 2001

We study the local cohomology modules HiIΣ(k[Δ]) of the Stanley–Reisner ring k[Δ] of a simplicial complex Δ with support in the ideal IΣ ⊂ k[Δ] corresponding to a subcomplex Σ ⊂ Δ. We give a…

Sheaf Cohomology and Free Resolutions over Exterior Algebras

- Mathematics
- 2003

In this paper we derive an explicit version of the Bernstein- Gel'fand-Gel'fand (BGG) correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free…

Local cohomology at monomial ideals

- Mathematics
- 2000

Abstract We prove that if B ⊂ R = k [ X 1 ,⋯ , X n ] is a reduced monomial ideal, then H B i ( R ) = ∪ d ≥ 1 Ext R i ( R / B [ d ] , R ), where B [ d ] is the d th Frobenius power of B .…

The Alexander Duality Functors and Local Duality with Monomial Support

- Mathematics
- 2000

Abstract Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated N n-graded module. The functors associated with Alexander duality provide a…

Koszul Duality Patterns in Representation Theory

- Mathematics
- 1996

The aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of Koszul duality to repre- sentation theory. The paper consists of three parts…