2025/09/15 by Ficarra, Antonino, Lu, Dancheng
#Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2509.11977
Let I be a polymatroidal ideal. In this paper, we study the asymptotic behavior of the homological shift ideals of powers of polymatroidal ideals. We prove that the first homological shift algebra HS1(R(I)) of I is generated in degree one as a module over the Rees algebra R(I) of I. We conjecture that the ith homological shift algebra HSi(R(I)) of I is generated in degrees ≤ i, and we confirm it in many significant cases. We show that I has the 1st homological strong persistence property, and we conjecture that the sequence \Ass HSi(Ik)\k>0 of associated primes of HSi(Ik) becomes an increasing chain for k≥ i. This conjecture is established when i=1 and for many families of polymatroidal ideals. Finally, we explore componentwise polymatroidal ideals, and we prove that HS1(I) is again componentwise polymatroidal, if I is componentwise polymatroidal.