[Paper Review] Local Cohomology at Monomial Ideals

Mircea Mustaţă|ArXiv.org|Jan 26, 2000

Commutative Algebra and Its Applications8 references4 citations

TL;DR

This paper provides a combinatorial description of local cohomology modules at monomial ideals using simplicial complexes and Betti numbers of Alexander dual ideals. It establishes that the multigraded components of Ext and local cohomology are isomorphic to simplicial cohomology groups, and gives a canonical filtration of Ext(R/B, R) indexed by Betti numbers of the dual ideal, enabling explicit computation and characterization of associated primes.

ABSTRACT

For a reduced monomial ideal B in R=k[X_1,...,X_n], we write H^i_B(R) as the union of {Ext^i(R/B^[d],R)}_d, where {B^[d]}_d are the "Frobenius powers of B". We describe H^i_B(R)_p, for every p in Z^n, in the spirit of the Stanley-Reisner theory. As a first application we give an isomorphism Tor_i(B', k)_p\iso Ext^{|p|-i}(R/B,R)_{-p} for all p in {0,1}^n, where B' is the Alexander dual ideal of B. We deduce a canonical filtration of Ext^i(R/B,R) with succesive quotients of the form R/(X_{j_1},...,X_{j_i}) suitably shifted, the multiplicities being computed from the Betti numbers of B'. As a final application, we give a topological description for the associated primes of Ext^i(R/B,R).

Motivation & Objective

• To provide a combinatorial description of local cohomology modules H^i_B(R) for monomial ideals B in polynomial rings.
• To establish a filtration on Ext^i_R(R/B, R) for squarefree monomial ideals B using the Betti numbers of the Alexander dual B^∨.
• To characterize the homological associated primes of R/B via topological conditions on simplicial complexes associated to B^∨.
• To derive explicit formulas for multigraded components of Ext and local cohomology using simplicial cohomology.
• To connect the structure of Ext^i_R(R/B, R) to the Betti numbers of B^∨, extending results of Hochster and Eisenbud.

Proposed method

• Use of Frobenius powers B_0^{[d]} of the radical of B to stabilize the limit defining local cohomology.
• Description of the degree α component of H^i_B(R) as reduced simplicial cohomology of a complex depending on B and the signs of α.
• Construction of a canonical filtration on Ext^i_R(R/B, R) with graded pieces isomorphic to R/P_α(α) tensored with Betti numbers β_{l−i,α}(B^∨).
• Application of Hochster’s formula to relate Tor^R_{|α|−i}(B^∨, k)_α to the multigraded Ext component Ext^i_R(R/B, R)_{−α}.
• Use of simplicial inclusion maps to model the R-module structure on local cohomology via induced cohomology maps.
• Topological characterization of associated primes via kernels of restriction maps in simplicial cohomology.

Experimental results

Research questions

• RQ1 How can the multigraded components of H^i_B(R) be described combinatorially for monomial ideals B?
• RQ2 What is the structure of Ext^i_R(R/B, R) as a filtered R-module when B is a squarefree monomial ideal?
• RQ3 Which prime ideals P_α arise as associated primes of Ext^i_R(R/B, R), and what conditions determine minimality?
• RQ4 How do the Betti numbers of the Alexander dual B^∨ control the structure of Ext^i_R(R/B, R)?
• RQ5 Can the associated primes of Ext^i_R(R/B, R) be characterized purely in terms of simplicial cohomology of complexes derived from B^∨?

Key findings

• The natural map Ext^i_R(R/B_0^{[d]}, R) → H^i_B(R) is an isomorphism onto the submodule of elements of multidegree α with α_j ≥ −d for all j.
• For a squarefree monomial ideal B, there exists a canonical filtration 0 = M_0 ⊂ ⋯ ⊂ M_n = Ext^i_R(R/B, R) such that M_l / M_{l−1} ≅ ⨁_{|α|=l} (R/P_α(α))^{β_{l−i,α}(B^∨)}.
• The multigraded component Ext^i_R(R/B, R)_{−α} is isomorphic to Tor^R_{|α|−i}(B^∨, k)_α for all α ∈ {0,1}^n.
• The associated primes of Ext^i_R(R/B, R) are contained in {P_α | β_{|α|−i,α}(B^∨) ≠ 0}, and P_α is in Ass(Ext^i_R(R/B, R)) if and only if the kernel intersection ∩_{j∈F} Ker(H^{i−2}(Δ_F; k) → H^{i−2}(Δ_{F∖j}; k)) is nonzero.
• A prime P_α is minimal in Ass(Ext^i_R(R/B, R)) if and only if β_{|α|−i,α}(B^∨) ≠ 0 and β_{|α′|−i,α′}(B^∨) = 0 for all α′ < α, α′ ≠ α.
• The result extends the duality between Betti numbers of B and B^∨, recovering and strengthening inequalities from Bayer, Charalambous, and Popescu (1998), and generalizing results of Eagon–Reiner and Terai.