Manoj Kummini

Publications (3)0 Total impact

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  Available from: ArXiv

  Article: \'Etale cohomological dimension, a conjecture of Lyubeznik and bounds for arithmetic rank

  Manoj Kummini, Uli Walther
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  ABSTRACT: We produce a criterion for open sets in projective $n$-space over a separably closed field to have \'etale cohomological dimension bounded by $2n-3$. We use the criterion to exhibit a scheme for which \'etale cohomological dimension is smaller than what a conjecture of G.~Lyubeznik predicts; the discrepancy is of arithmetic nature. For a monomial ideal, we relate extremal graded Betti numbers and \'etale cohomological dimension of the complement of the corresponding subspace arrangement. Moreover, we derive upper bounds for its arithmetic rank in terms of invariants distilled from the lcm-lattice. Comment: 12pp
  11/2010;
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  Available from: ArXiv

  Article: Regularity, Depth and Arithmetic Rank of Bipartite Edge Ideals

  Manoj Kummini
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  ABSTRACT: We study minimal free resolutions of edge ideals of bipartite graphs. We associate a directed graph to a bipartite graph whose edge ideal is unmixed, and give expressions for the regularity and the depth of the edge ideal in terms of invariants of the directed graph. For some classes of unmixed edge ideals, we show that the arithmetic rank of the ideal equals projective dimension of its quotient.
  02/2009;
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  Available from: ArXiv

  Article: Multiplicity Bounds for Quadratic Monomial Ideals

  Manoj Kummini
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  ABSTRACT: We prove the multiplicity bounds conjectured by Herzog-Huneke-Srinivasan and Herzog-Srinivasan in the following cases: the strong conjecture for edge ideals of bipartite graphs, and the weaker Taylor bound conjecture for all quadratic monomial ideals. We attach a directed graph to a bipartite graph with perfect matching, and describe operations on the directed graph that would reduce the problem to a Cohen-Macaulay bipartite graph. We determine when equality holds in the conjectured bound for edge ideals of bipartite graphs, and verify that when equality holds, the resolution is pure. We characterize bipartite graphs that have Cohen-Macaulay edge ideals and quasi-pure resolutions.
  08/2007;