arXiv subjects

Kohji Yanagawa

Publications and source records attributed to Kohji Yanagawa.

40 records · Page 3Linked to original sources

arXiv2004

BGG correspondence and Roemer's theorem on an exterior algebra

Let E = K< y_1, ..., y_n > be the exterior algebra. The ``(cohomological) distinguished pairs" of a graded E-module M describe the growth of a minimal graded injective resolution of M. Roemer gave a duality theorem between the distinguished pairs of M and those of its dual M^*. In this paper, we show that under Bernstein-Gel'fand-Gel'fand correspondence, his theorem is translated into a natural corollary of local duality for (complexes of) graded S=K[x_1, >..., x_n]-modules. Using this idea, we also give a Z^n-graded version of Roemer's theorem.

math.AC↗

arXiv2003

Stanley-Reisner rings, sheaves, and Poincare-Verdier duality

Recently, I defined a squarefree module over a polynomial ring $S = k[x_1, >..., x_n]$ generalizing the Stanley-Reisner ring $k[Δ] = S/I_Δ$ of a simplicial complex $Δ\subset 2^{1, ..., n}$. In this paper, from a squarefree module $M$, we construct the $k$-sheaf $M^+$ on an $(n-1)$ simplex $B$ which is the geometric realization of $2^{1, ..., n}$. For example, $k[Δ]^+$ is (the direct image to $B$ of) the constant sheaf on the geometric realization $|Δ| \subset B$. We have $H^i(B, M^+) = [H^{i+1}_m(M)]_0$ for all $i > 0$. The Poincare-Verdier duality for sheaves $M^+$ on $B$ corresponds to the local duality for squarefree modules over $S$. For example, if $|Δ|$ is a manifold, then $k[Δ]$ is a Buchsbaum ring whose canonical module is a squarefree module giving the orientation sheaf of $|Δ|$ with the coefficients in $k$.

math.AC↗

arXiv2003

Derived Category of Squarefree Modules and Local Cohomology with Monomial Ideal Support

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 $H_{I_Δ}^i(S)$ at a Stanley-Reisner ideal $I_Δ$ using squarefree modules. Among other things, we see that Hochster's formula on the Hilbert function of $H_m^i(S/I_Δ)$ is also a formula on the characteristic cycle of $H_{I_Δ}^{n-i}(S)$ as a module over the Weyl algebra $S<\partial_1, ..., \partial_n >$ (if $chara(k)=0$).

math.AC↗

arXiv1998

Generic and Cogeneric Monomial Ideals

Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions derived from simplicial complexes. For a generic monomial ideal, the associated primes satisfy a saturated chain condition, and the Cohen-Macaulay property implies shellability for both the Scarf complex and the Stanley-Reisner complex. Reverse lexicographic initial ideals of generic lattice ideals are generic. Cohen-Macaulayness for cogeneric ideals is characterized combinatorially; in the cogeneric case the Cohen-Macaulay type is greater than or equal to the number of irreducible components. Methods of proof include Alexander duality and Stanley's theory of local h-vectors.

math.AC↗