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Algebraic Geometry Seminar - Markus Brodmann(Zurich Uni.,Switzerland)

posted Mar 25, 2012, 6:32 PM by Jihye Jung   [ updated Mar 25, 2012, 6:45 PM ]
 - Speaker : Markus Brodmann (University of Zurich,Switzerland)

 - Date & Time : Mar 16(Fri),2012 3:00~4:00 PM

 - Place : Math Science building room 404
 - Abstracts
    : We report on joint work with C.H.Linh (Hu.e) and my former student M.H.Seiler (Zurich).
Initiation to the work we present was given by a question which arose in the research group of Mathematical Physics at the Department of Mathematics of the University of Zurich.
In algebraic terms, this question may be formulated as follows:
Let V(N) . A2nK be the characteristic variety of a .nitely generated module N over then-th Weyl algebra
 An(K) := K[X1; : : : ;Xn;D1; : : : ;Dn] (K a .eld of characteristic 0), furnished with an admissible degree-compatible .ltration N. = (Nn)n2N0 . Does the induced Hilbert function n 7! dimK(Nn=Nn􀀀1)
bound the degree of the polynomials inG := K[X1; : : : :Xn; Y1; : : : ; Yn] which are needed to the set V(N) ?
The answer to this question is indeed a.rmative, and it follows from the stronger and more general result (obtained in the Master thesis of M.H. Seiler) that the Hilbert polynomial, the generating degree and the postulation number of a .nitely generated graded G-module M bound the Castelnuovo-Mumford regularity
reg(0 :G M) of the annihilator (0 :G M) ofthe module M.
It turns out, that the techniques which are used in the proof of this latter bounding result, in conjunction with an earlier bounding result proved jointly with my former student T. Gotsch, allow to establish bounds on the Castelnuovo-Mumford regularities reg(ExtiR(M;N)) and reg(TorRi (M;N)) where M and N are .nitely generated graded modules over the Noetherian homogeneous K-algebra R.
In particular we generalize a bounding result of Eisenbud-Huneke-Ulrich for the Castelnuovo- Mumford regularities reg(TorRi (M;N)) which holds under the additional assumption that dimR(TorR1 (M;N)) . 1 and which was originally proved in the special case where R =K[X1; : : : ;Xr] is a polynomial ring over a .eld K.