# Mathematical Sciences Research Institute

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# Seminar

Lower Bounds for the Arithmetical Rank of a Homogeneous Ideal in a Polynomial Ring October 19, 2012 (01:00 PM PDT - 01:45 PM PDT)
Location: MSRI: Simons Auditorium
Speaker(s) Matteo Varbaro
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Abstract/Media

Let I = (f_1,...,f_r) be a homogeneous ideal in the polynomial ring S = K[x_1,...,x_n]. The arithmetical rank of I, ara(I), is the smallest number s such that I has the same radical of the ideal J = (g_1,...,g_s) for some homogeneous polynomials g_1,...,g_s in S. One way to get lower bounds for this number is to look at the non-vanishing of local cohomology with support in I: ara(I) >= t provided that H_I^t(S) is not zero. A similar lower bound comes considering étale topology instead of Zariski's one, and basically these have been, so far, the only two successful methods to compute the arithmetical rank of certain families of ideals. Moreover, in many instances, étale topology was successful where Zariski's one failed. This fact led Lyubeznik to conjecture, in 2002, that the lower bound obtained using étale cohomology is never worse than the one obtained using Zariski's.
In the talk I am going to prove that the conjecture is true if the variety defined by I is smooth and K has characteristic 0.