2026/01/06 by Jovanny Ibarguen, Carlos E. Valencia, Rafael H. Villarreal
#math.AC #math.CO
Let I be a monomial ideal of a polynomial ring R=K[x1,…,xn] over a field K and let \rm sgn(I) be its signature ideal. If I is not a principal ideal, we show that the depth of R/I is the depth of R/\rm sgn(I), and the regularity of R/\rm sgn(I) is at most the regularity of R/I. For ideals of height at least 2, we show that the associated primes of I and \rm sgn(I) are the same, and we show that I is Cohen--Macaulay (resp. Gorenstein) if and only if \rm sgn(I) is Cohen--Macaulay (resp. Gorenstein), and furthermore we show that the v-number of \rm sgn(I) is at most the v-number of I and compare the irreducible decompositions of I and \rm sgn(I). We give an algorithm to compute the signature of a monomial ideal using Macaulay2, and an algorithm to examine given families of monomial ideals by computing their signature ideals and determining which of these are Cohen--Macaulay or Gorenstein.