[Paper Review] The arithmetical rank of a special class of monomial ideals

Pietro Mongelli|arXiv (Cornell University)|May 14, 2010

Commutative Algebra and Its Applications3 references3 citations

TL;DR

This paper establishes that the arithmetical rank of a special class of square-free monomial ideals, generated by consecutive products of variables $ I_t(L_n) $, equals their projective dimension. Using a key lemma on ideal radicals and a recursive replacement technique based on Barile's result, the authors derive exact formulas for the arithmetical rank depending on the modular residue of $ n $ modulo $ t+1 $, resolving a conjecture in the literature.

ABSTRACT

We give an affirmative answer to a question due to J. He and A. Van Tuyl, proving that the arithmetical rank of a special monomial ideal equals to the projective dimension of corresponding quotient module.

Motivation & Objective

• To generalize previous results on arithmetical ranks of monomial ideals, particularly extending Theorem 4.5 from [2] to a broader class of ideals.
• To establish the equality $ \operatorname{ara}(I_t(L_n)) = \operatorname{pd}(R/I_t(L_n)) $ for all positive integers $ t $ and $ n $.
• To derive explicit closed-form expressions for the arithmetical rank of $ I_t(L_n) $, depending on the value of $ n \mod (t+1) $.
• To provide a constructive method for reducing the number of generators needed to define the radical of $ I_t(L_n) $, using a lemma on ideal extension and variable separation.

Proposed method

• Leveraging a key lemma showing that $ \operatorname{ara}I \leq \operatorname{ara}(I+J) $ when generators of $ J $ are divisible by a variable not dividing any generator of $ I $.
• Applying Barile's result that the ideal generated by $ n $-fold consecutive products in $ 2n $ variables has arithmetical rank 2.
• Partitioning the generators of $ I_t(L_n) $ into blocks of size $ t+1 $, and replacing each block with two polynomials that generate the same radical.
• Using induction and modular arithmetic to handle different cases based on $ n \mod (t+1) $, distinguishing between $ d = t $ and $ d < t $.
• Computing the arithmetical rank via the radical-preserving reduction of generators, relying on the fact that radical ideals are preserved under such substitutions.
• Combining the upper bounds from the reduction process with the known projective dimension from [2, Theorem 4.1] to conclude equality.

Experimental results

Research questions

• RQ1 Does the arithmetical rank of the ideal $ I_t(L_n) $, generated by consecutive products of $ t $ variables, equal its projective dimension for all $ t,n \geq 1 $?
• RQ2 Can the arithmetical rank of $ I_t(L_n) $ be expressed as a closed-form function of $ n $ and $ t $?
• RQ3 What is the precise dependence of $ \operatorname{ara}(I_t(L_n)) $ on the residue of $ n $ modulo $ t+1 $?
• RQ4 Is there a constructive method to reduce the number of generators of $ I_t(L_n) $ while preserving the radical, using variable separation and ideal extension?

Key findings

• The arithmetical rank of $ I_t(L_n) $ is exactly $ \frac{2(n-d)}{t+1} $ when $ n \equiv d \pmod{t+1} $ and $ 0 \leq d \leq t-1 $.
• When $ n \equiv t \pmod{t+1} $, the arithmetical rank is $ \frac{2n - (t-1)}{t+1} $.
• The equality $ \operatorname{ara}(I_t(L_n)) = \operatorname{pd}(R/I_t(L_n)) $ holds for all positive integers $ t $ and $ n $, confirming a conjecture in the literature.
• The proof relies on a reduction technique that replaces blocks of $ t+1 $ consecutive monomial generators with two polynomials, preserving the radical.
• The bound $ \operatorname{ara}I \leq \operatorname{ara}(I+J) $ is not generally true, but holds under the specific variable separation condition used in Lemma 1.
• The result extends Barile’s result on $ 2n $-variable ideals with $ n $-fold consecutive products to the general case of $ n $-variable ideals with $ t $-fold consecutive products.