Powers of monomial ideals with characteristic-dependent Betti numbers

• Published in: Research in the mathematical sciences Vol. 9; no. 2
• Main Authors: Bolognini, Davide, Macchia, Antonio, Strazzanti, Francesco, Welker, Volkmar
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-06-2022; Springer Nature B.V
• Subjects: Applications of Mathematics; Computational Mathematics and Numerical Analysis; Developments in Commutative Algebra: In honor of Jürgen Herzog on the occasion of his 80th Birthday; Mathematics; Mathematics and Statistics; Polynomials; Castelnuovo–Mumford regularity; Powers of monomial ideals; Edge ideals; Betti splitting; 13F55; 13D02; Betti numbers; Field dependence; Binomial edge ideals
• ISSN: 2522-0144; 2197-9847
• DOI: 10.1007/s40687-022-00318-2

We explore the dependence of the Betti numbers of monomial ideals on the characteristic of the field. A first observation is that for a fixed prime
p
either the
i
-th Betti number of all high enough powers of a monomial ideal differs in characteristic 0 and in characteristic
p
or it is the same for all high enough powers. In our main results, we provide constructions and explicit examples of monomial ideals all of whose powers have some characteristic-dependent Betti numbers or whose asymptotic regularity depends on the field. We prove that, adding a monomial on new variables to a monomial ideal allows to spread the characteristic dependence to all powers. For any given prime
p
, this produces an edge ideal such that all its powers have some Betti numbers that are different over
Q
and over
Z
p
. Moreover, we show that, for every
r
≥
0
and
i
≥
3
there is a monomial ideal
I
such that some coefficient in a degree
≥
r
of the Kodiyalam polynomials
P
3
(
I
)
,
…
,
P
i
+
r
(
I
)
depends on the characteristic. We also provide a summary of related results and speculate about the behavior of other combinatorially defined ideals.