47 research outputs found

    Koszul hypersurfaces over the exterior algebras

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    We prove that if EE is an exterior algebra over a field, hh is a quadratic form, then E/(h)E/(h) is Koszul if and only if hh is a product of two linear forms.Comment: 4 page

    Regularity bounds for complexes and their homology

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    Let RR be a standard graded algebra over a field kk. We prove an Auslander-Buchsbaum formula for the absolute Castelnuovo-Mumford regularity, extending important cases of previous works of Chardin and R\"omer. For a bounded complex of finitely generated graded RR-modules LL, we prove the equality reg L=maxiZ{reg Hi(L)i}\text{reg}~ L=\max_{i\in \mathbb Z} \{\text{reg}~ H_i(L)-i\} given the condition depth Hi(L)dimHi+1(L)1\text{depth}~ H_i(L)\ge \dim H_{i+1}(L)-1 for all i<supLi<\sup L. As applications, we recover previous bounds on regularity of Tor due to Caviglia, Eisenbud-Huneke-Ulrich, among others. We also obtain strengthened results on regularity bounds for Ext and for the quotient by a linear form of a module.Comment: Final versio