scholarly journals Stability of associated primes of integral closures of monomial ideals

2009 ◽  
Vol 116 (1) ◽  
pp. 44-54 ◽  
Author(s):  
Tran Nam Trung
Keyword(s):  
Monomial Ideals ◽  
Associated Primes ◽  
Integral Closures

scholarly journals Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals

2011 ◽  
Vol 331 (1) ◽  
pp. 224-242 ◽  
Author(s):  
Christopher A. Francisco ◽  
Huy Tài Hà ◽  
Adam Van Tuyl
Keyword(s):  
Monomial Ideals ◽  
Perfect Graphs ◽  
Associated Primes

Partial Castelnuovo–Mumford regularities of sums and intersections of powers of monomial ideals

2010 ◽  
Vol 149 (2) ◽  
pp. 229-246 ◽  
Author(s):  
LÊ TUÂN HOA ◽  
TRÂN NAM TRUNG
Keyword(s):  
Asymptotic Behavior ◽  
Linear Function ◽  
Polynomial Ring ◽  
Monomial Ideal ◽  
Monomial Ideals ◽  
Symbolic Powers ◽  
Integral Closures

AbstractLet I, I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp be monomial ideals of a polynomial ring R = K[X1,. . ., Xr] and Ln = I+∩jIn1j + ⋅ ⋅ ⋅ + ∩jIpjn. It is shown that the ai-invariant ai(R/Ln) is asymptotically a quasi-linear function of n for all n ≫ 0, and the limit limn→∞ad(R/Ln)/n exists, where d = dim(R/L1). A similar result holds if I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp are replaced by their integral closures. Moreover all limits $\lim_{n\to\infty} a_i(R/(\cap_j \overline{I_{1j}^n} + \cdots + \cap_j \overline{I_{pj}^n}))/n $ also exist.As a consequence, it is shown that there are integers p > 0 and 0 ≤ e ≤ d = dim R/I such that reg(In) = pn + e for all n ≫ 0 and pn ≤ reg(In) ≤ pn + d for all n > 0 and that the asymptotic behavior of the Castelnuovo–Mumford regularity of ordinary symbolic powers of a square-free monomial ideal is very close to a linear function.

scholarly journals Associated primes of monomial ideals and odd holes in graphs

2010 ◽  
Vol 32 (2) ◽  
pp. 287-301 ◽  
Author(s):  
Christopher A. Francisco ◽  
Huy Tài Hà ◽  
Adam Van Tuyl
Keyword(s):  
Monomial Ideals ◽  
Associated Primes

scholarly journals Limit behavior of the rational powers of monomial ideals

2021 ◽  
Author(s):  
James Lewis
Keyword(s):  
Local Cohomology ◽  
Integral Closure ◽  
Monomial Ideals ◽  
Limit Behavior ◽  
Associated Primes ◽  
Powers Of Ideals ◽  
Symbolic Powers ◽  
Local Cohomology Modules ◽  
Cohomology Modules

We investigate the rational powers of ideals. We find that in the case of monomial ideals, the canonical indexing leads to a characterization of the rational powers yielding that symbolic powers of squarefree monomial ideals are indeed rational powers themselves. Using the connection with symbolic powers techniques, we use splittings to show the convergence of depths and normalized Castelnuovo–Mumford regularities. We show the convergence of Stanley depths for rational powers, and as a consequence of this, we show the before-now unknown convergence of Stanley depths of integral closure powers. Additionally, we show the finiteness of asymptotic associated primes, and we find that the normalized lengths of local cohomology modules converge for rational powers, and hence for symbolic powers of squarefree monomial ideals.

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