1,721,054 research outputs found
Reduction numbers and initial ideals
No full textThe reduction number r(A) of a standard graded algebra A is the
least integer k such that there exists a minimal reduction J of the homogeneous maximal ideal m of A such that Jm^k = m^{k+1}. Vasconcelos conjectured that r(R/I)=r(R/in(I)) where in(I) is the initial ideal of an ideal I in a polynomial ring R with respect to a term order. The goal of this note is to prove the conjecture
Divisor class group and canonical class of determinantal rings defined by ideals of minors of a symmetric matrix
No full text
Koszul homology and extremal properties of Gin and Lex
No full textFor every homogeneous ideal I in a polynomial ring R and for
every p\leq dim R we consider the Koszul homology H_i(p;R/I) with respect to a sequence of p of generic linear forms. The Koszul-Betti number \beta_{ijp}(R?I) is, by definition, the dimension of the degree j part of H_i(p;R/I). In characteristic 0, we show that the Koszul-Betti numbers of any ideal I are bounded above by those of the gin-revlex Gin(I) of I and also by those of the Lex-segment
Lex(I) of I. We show that \beta_{ijp}(R/I) = \beta_{ijp}(R/Gin(I)) iff I is componentwise linear and that and \beta_{ijp}(R/I) = \beta_{ijp}(R/Lex(I)) iff I is Gotzmann. We also investigate the set Gins(I) of all the gin of I and show that the Koszul-Betti numbers of any ideal in Gins(I) are bounded below by those of the gin-revlex
of I. On the other hand, we present examples showing that in general there is no J is Gins(I) such that the Koszul-Betti numbers of any ideal in Gins(I) are bounded above by those of J
Linear spaces, transversal polymatroids and ASL domains
No full textWe study a class of algebras associated with linear spaces and its relations with polymatroids and integral posets, i.e. posets supporting homogeneous ASL. We prove that the base ring of a transversal polymatroid is Koszul and describe a new class of integral posets. As a corollary we obtain that every Veronese subring of a polynomial ring is an ASL
Gröbner bases for spaces of quadrics of codimension 3
No full textAbstractLet R=⊕i≥0Ri be an Artinian standard graded K-algebra defined by quadrics. Assume that dimR2≤3 and that K is algebraically closed, of characteristic ≠2. We show that R is defined by a Gröbner basis of quadrics with, essentially, one exception. The exception is given by K[x,y,z]/I where I is a complete intersection of three quadrics not containing a square of a linear form
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