Toward the theory on local cohomologies at the ideals given by simplicial posets

Abstract: For a simplicial poset $P$, Stanley assigned the face ring $A_P$, which is the quotient of the polynomial ring $S:=K[t_x \mid x \in P \setminus {\widehat{0} }]$ by the ideal $I_P$. This is a generalization of Stanley-Reisner rings, but $S$ and $A_P$ are not standard graded, and $I_P$ is not a monomial ideal. To develop the theory on the local cohomology $H_{I_p}^i(S)$ and its injective resolution, this paper establishes the foundation. Specifically, we give an explicit description of the graded injective envelope ${}^*! E_S(S/\mathfrak{p}_x)$, where $\mathfrak{p}_x$ is the prime ideal associated with $x \in P$. We also analyze morphisms between them.

• The paper establishes a general theory of local cohomology modules at ideals from simplicial posets using explicit constructions of graded injective envelopes.
• The methodology leverages combinatorial decompositions and a novel notion of clean morphisms to manage the complexities of nonstandard grading.
• The work provides a characteristic-free formula for Lyubeznik numbers, expanding applications to non-monomial face rings and advanced homological invariants.

Foundations for Local Cohomology at Ideals from Simplicial Posets

Introduction and Context

This paper establishes a general theory of local cohomology modules $H_{I_P}^i(S)$, where $I_P$ is an ideal associated to a simplicial poset $P$. The context extends the extensively studied case of Stanley–Reisner rings—arising from simplicial complexes—to the more general and structurally complex setting of face rings assigned to simplicial posets, as introduced by Stanley [St91]. The crucial complication in this generalized setting is that the face ring $A_P = S/I_P$ (with $S = K[t_x \mid x \in P^*]$) is $\mathbb{Z}^n$-graded, but not standard graded, and $I_P$ is generally non-monomial. Consequently, standard cohomological techniques applicable to monomial ideals or squarefree settings are insufficient, necessitating fundamentally new approaches for understanding the structure of graded injective resolutions and their morphisms.

Main Results

Explicit Construction of Graded Injective Envelopes

The paper presents a complete, explicit description of the $\mathbb{Z}^n$-graded injective envelope $^*_S(S/\mathfrak{p}_x)$ where $\mathfrak{p}_x$ is a prime ideal corresponding to $I_P$0. The construction leverages the combinatorial decomposition of $I_P$1 into rank 1 elements and the higher ranks, inducing a decomposition of the injective envelope as a tensor product of Laurent polynomial rings and inverse polynomial rings depending on the ranks.

For instance, if $I_P$2 has rank $I_P$3, $I_P$4 is described as

$I_P$5

with $I_P$6-action given through a comultiplication map motivated by the Hopf algebra structure of $I_P$7. This explicitness enables computation and manipulation of these injective envelopes, a necessity for constructive homological arguments.

Analysis of Morphisms: Cleanness and Uniqueness

The morphisms between these injective envelopes are inherently more intricate than in the Stanley–Reisner case. The paper defines the notion of cleanness for graded $I_P$8-homomorphisms between injective envelopes, characterized relative to a chosen basis associated to the construction above. Clean homomorphisms are those which respect a certain submodule filtration by degree, ensuring compatibility with the module decomposition intrinsic to the face ring and the poset.

Key results include:

• Existence and uniqueness up to scalar: For $I_P$9 in $P$0, there exists a clean map $P$1 which is unique up to scalar.
• Compositional stability: Clean maps compose to clean maps, facilitating the construction of clean differentials in complexes.
• Characterization: Any non-scalar morphism can be “clean-ized” via precomposition with an automorphism.

Differential Maps in Dualizing Complexes

The authors show the differential maps in the minimal $P$2-graded injective (dualizing) resolution of $P$3 (and thereby in local cohomology complexes with support in $P$4) can be chosen, via appropriate bases, so that all are clean. This reduces the analysis of local cohomology and Lyubeznik numbers to the combinatorics of $P$5 and the behavior of clean maps.

Calculation of Lyubeznik Numbers

A crucial outcome is a formula expressing the Lyubeznik numbers of $P$6,

$P$7

which generalizes prior work in the squarefree monomial ideal (Stanley–Reisner) setting [Y01a]. Remarkably, this approach is characteristic free and does not rely on $P$8 being $P$9-pure, nor on standard gradings.

Technical and Structural Insights

The paper relies heavily on multigraded commutative algebra frameworks, injective module theory, and the combinatorics of posets. Among the notable technical aspects:

• Comultiplication: The use of the comultiplication map in the Hopf algebra structure of $A_P = S/I_P$0 is essential for defining the $A_P = S/I_P$1-module structure on the injective envelopes.
• Basis dependence: Definitions such as cleanness are dependent on the choice of bases, reflecting the noncanonical nature of these constructs outside the monomial ideal situation.
• Induction on rank and filtration: Many results exploit induction on the rank in $A_P = S/I_P$2 and the filtration submodules indexed by multidegree.

Implications and Future Directions

The work significantly advances the understanding of local cohomology modules and Lyubeznik numbers in the generality of face rings of simplicial posets, paving the way for combinatorial-homological calculations in contexts far beyond simplicial complexes. The explicit algebraic models for injective envelopes and clean morphisms offer potential future applications to:

• Gorenstein and Cohen–Macaulay criteria for face rings from generalized cell complexes.
• Algorithmic computation of Lyubeznik numbers and other homological invariants for complex combinatorial topologies.
• Study of dualizing complexes and their cohomological invariants in nonstandard gradings, potentially involving new types of spectral sequences.

A forthcoming sequel [SY2] promises to further develop these results and extend them to a full theory encompassing generalized face rings and deeper invariants.

Conclusion

This paper establishes foundational tools for the study of local cohomology modules at ideals defined by simplicial posets, significantly generalizing classic results for Stanley–Reisner rings. Through explicit models for graded injective envelopes and a systematic theory of morphisms between them, it enables constructive and combinatorial approaches to homological invariants in richly structured face rings. The implications promise to deepen the interface between commutative algebra, combinatorics, and topological invariants within algebraic systems far from the monomial and standard graded settings.