Signature invariants of monomial ideals

Source abstract

Abstract Let I be a monomial ideal of a polynomial ring $$R=K[x_1,\ldots ,x_n]$$ R = K [ x 1 , … , x n ] over a field K , and let $$\textrm{sgn}(I)$$ 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 $$R/\textrm{sgn}(I)$$ R / sgn ( I ) , and the regularity of $$R/\textrm{sgn}(I)$$ R / 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 $$\textrm{sgn}(I)$$ sgn ( I ) are the same, and we show that I is Cohen–Macaulay (resp. Gorenstein) if and only if $$\textrm{sgn}(I)$$ sgn ( I ) is Cohen–Macaulay (resp. Gorenstein), and furthermore, we show that the v-number of $$\textrm{sgn}(I)$$ sgn ( I ) is at most the v-number of I and compare the irreducible decompositions of I and $$\textrm{sgn}(I)$$ sgn ( I ) . We give an algorithm to compute the signature of a monomial ideal using Macaulay 2, 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.