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Extremal Betti numbers and applications to monomial ideals. (English) Zbl 0946.13008
Denote by \(\beta_{i,j}= \beta_{i,j}(M)\) the graded Betti numbers associated to a minimal free resolution of a graded \(S\)-module \(M\), \(S\) being the polynomial ring in \(n\) variables over a field. \(\beta_{i,j}\) is called extremal if \(\beta_{l,r}=0\) for all \(l\geq i\), \(r\geq j+l\) and \(r-l\geq j-l\).
In the first part the authors connect the extremal Betti numbers of a submodule of a free \(S\)-module with those of its generic initial module. In the independent second part they relate the \(\beta_{i,j}\) of the minimal resolution of a square free monomial ideal with those of the monomial ideal corresponding to the Alexander dual simplicial complex.

MSC:
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
13D02 Syzygies, resolutions, complexes and commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
Software:
Macaulay2
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References:
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