Notes on symmetric and exterior depth and annihilator numbers

  • Gesa Kampf Universitat Osnabruck
  • Martina Kubitzke Philipps-Universitat Marburg
Keywords: Exterior depth, Annihilator numbers, Stanley-Reiner ring


We survey and compare invariants of modules over the polynomial ring and the exterior algebra. In our considerations, we focus on the depth. The exterior analogue of depth was first introduced by Aramova, Avramov and Herzog. We state similarities between the two notion of depth and exhibit their relation in the case of squarefree modules. Work of Conca, Herzog and Hibi and Trung, respectively, shows that annihilator numbers are a meaningful generalization of depth over the polynomial ring. We introduce and study annihilator numbers over the exterior algebra. Despite some minor differences in the definition, those invariants show common behavior. In both situations a positive linear combination of the annihilator numbers can be used to bound the symmetric and exterior graded Betti numbers, respectively, from above.

Author Biographies

Gesa Kampf, Universitat Osnabruck

Fachbereich Mathematik und Informatik
49069 Osnabruck, Germany

Martina Kubitzke, Philipps-Universitat Marburg
Fachbereich Mathematik und Informatik
35032 Marburg, Germany