Castelnuovo-Mumford regularity for complexes and weakly Koszul modules

Let A be a noetherian AS regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and grA the category of finitely generated graded left A-modules. Following Jorgensen, we define the Castelnuovo-Mumford regularity reg(X) of a complex X \in D^b(grA) in terms of the local cohomologies or the minimal projective resolution of X. Let A^! be the quadratic dual ring of A. For the Koszul duality functor DG : D^b(grA) -> D^b(grA^!), we have reg(X) = max {i | H^i(DG (X)) \ne 0}. Using these concepts, we study weakly Koszul modules (= componentwise linear modules) over A^!. As an application, refining a result of Herzog and Roemer, we show that if J is a monomial ideal of an exterior algebra E= \bigwedge with d \geq 3, then the (d-2)nd syzygy of E/J is weakly Koszul.Comment: 21 pages, to appear in J. Pure Appl. Algebra, the description of the "(non-commutative)canonical module" is correcte

Collapse and Fragmentation of Cylindrical Magnetized Clouds. II. Simulation with Nested Grid Scheme

Fragmentation process in a cylindrical magnetized cloud is studied with the nested grid method. The nested grid scheme use 15 levels of grids with different spatial resolution overlaid subsequently, which enables us to trace the evolution from the molecular cloud density $\sim 100 {\rm cm}^{-3}$ to that of the protostellar disk $\sim 10^{10} {\rm cm} ^{-3}$ or more. Fluctuation with small amplitude grows by the gravita- tional instability. It forms a disk perpendicular to the magnetic fields which runs in the direction parallel to the major axis of the cloud. Matter accrets on to the disk mainly flowing along the magnetic fields and this makes the column density increase. The radial inflow, whose velocity is slower than that perpendicular to the disk, is driven by the increase of the gravity. While the equation of state is isothermal and magnetic fields are perfectly coupled with the matter, which is realized in the density range of $\rho \lesssim 10^{10} {\rm cm}^{-3}$, never stops the contraction. The structure of the contracting disk reaches that of a singular solution as the density and the column density obey $\rho(r)\propto r^{-2}$ and $\sigma(r) \propto r^{-1}$, respectively. The magnetic field strength on the mid-plane is proportional to $\rho(r)^{1/2}$ and further that of the center ($B_c$) evolves as proportional to the square root of the gas density ($\propto \rho_c^{1/2}$). It is shown that isothermal clouds experience ``run-away'' collapses. The evolution after the equation of state becomes hard is also discussed.Comment: 12 pages without figures, AASTEX, submitted to ApJ. Postscript version with figures is available from http://quasar.ed.niigata-u.ac.jp/docs/Papers/mag2.ps.g