Castelnuovo-Mumford regularity of products of ideals

By Aldo Conca, Jürgen Herzog. 2002

179 citations recorded by the source corpus.

The Castelnuovo-Mumford regularity reg$(M)$ is one of the most important invariants of a finitely generated graded module $M$ over a polynomial ring $R$. For instance, it measures the amount of computational resources that working with $M$ requires. In general one knows that the regularity of a module can be doubly exponential in the degrees of the minimal generators and in the number of the variables. On the other hand, in many situations one has or one conjectures a much better behavior. One may ask, for instance, whether the Castelnuovo-Mumford regularity reg($IM$) of the product of an ideal $I$ with a module $M$ is bounded by the sum reg($I$) + reg($M$). In general this is not the case. But we show that it is indeed the case if either dim $R/I\leq 1$ or $I$ is generic (in a very precise sense). Further we show that products of ideals of linear forms have always a linear resolution and that the same is true for products of determinantal ideals of a generic Hankel matrix.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/9d9b650a2051e493127ae84725f997752ee192ce

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Citations in the corpus

The depth of powers of an ideal (2004, background): “The product of polymatroidal ideals is again polymatroidal ([6] and [9]).”
How to compute the Stanley depth of a monomial ideal (2007, background): “It is known [9] thatIn,d has linear quotients since In,d is a polymatroidal ideal.”
Discrete Polymatroids (2002, background): “This fact has been shown directly in [2].”
Regularity jumps for powers of ideals (2003, background): “(I) = 3 and reg(I2) = 7. The only non-linear syzygy for I2 comes at the very end of the resolution. In characteristic 2 the ideal I does not have a linear resolution. The second example is taken from [CH]. It is monomial and characteristic free. The ideal is deﬁned by 5 monomials and very likely the”
Koszul Algebras and Regularity (2012, background): “I / (does not hold in general but it) holds provided dimS=I 1, [23].”