Lifting Frobenius splittings through geometric vertex decomposition

Technical Summary: Lifting Frobenius Splittings Through Geometric Vertex Decomposition

Problem Statement
The paper addresses the problem of lifting Frobenius splittings from the components of a geometric vertex decomposition back to the original ideal. Frobenius splitting is a technique in characteristic $p > 0$ commutative algebra used to control singularities. While Knutson [Knu09] established that Frobenius splittings of a specific form descend through geometric vertex decomposition (a type of Gröbner degeneration), the converse direction—constructing a splitting for an ideal $I$ given splittings for its "link" and "deletion" components—remained an open question. The authors seek to determine under what conditions a splitting compatible with the link and deletion can be lifted to a splitting compatible with the original ideal, and to identify obstructions to such a lift.

Methodology
The authors employ a combination of Frobenius splitting theory and Gorenstein liaison theory.

1. Geometric Vertex Decomposition: They utilize the framework introduced by Knutson, Miller, and Yong [KMY09], where an ideal $I$ is decomposed into a link $C(y, I)$ and a deletion $N(y, I)$ relative to a variable $y$.
2. Isomorphism Construction: Central to their method is the construction of an explicit module isomorphism $\psi: C/N \to I/N$ derived from the geometric vertex decomposition. This isomorphism, originally developed in the context of Gorenstein liaison by Klein and Rajchgot [KR21], is adapted here to the setting of Frobenius splittings.
3. Trace Map Analysis: The authors analyze the trace map $\text{Tr}$ and the specific form of the splitting polynomial $f$. They investigate how the splitting polynomial for the components relates to the splitting polynomial for the original ideal via the isomorphism $\psi$.
4. Counterexample Construction: To demonstrate the necessity of their hypotheses, they construct a specific counterexample using an ideal studied by Fedder and Singh, showing that the lifting fails without a specific divisibility condition.

Key Contributions and Results

• Main Theorem (Theorem 3.3): The paper provides a partial converse to Knutson's result. It states that if an ideal $I$ admits a nondegenerate geometric vertex decomposition $in_y(I) = C \cap (N + (y))$, and if a Frobenius splitting $\text{Tr}((y g)^{p-1} \cdot)$ compatibly splits both $C$ and $N$, then a splitting for $I$ can be constructed provided an additional hypothesis holds: there must exist an element $u \in C$ such that $u$ does not divide zero modulo $N$ and $u$ divides $g$. Under these conditions, there exists an element $r$ such that $\text{Tr}((yg + r)^{p-1} \cdot)$ compatibly splits $I$.
• Obstruction to a Full Converse (Example 3.4): The authors demonstrate that the hypothesis regarding the existence of $u$ is necessary. They present an example (based on Fedder's work) where a splitting exists for the link and deletion, but the ideal $I$ itself is not Frobenius split. In this case, the required element $u$ (a factor of $g$ that is a nonzerodivisor modulo $N$) does not exist.
• Application to Determinantal Ideals (Section 4): The main theorem is applied to show that:
  • Ideals generated by $2$-minors of a $2 \times n$ generic matrix are compatibly split.
  • The defining ideals of lower bound cluster algebras are compatibly split, recovering and extending results from [BMRS15].
• Li's Double Determinantal Varieties (Section 5): The authors show that double determinantal ideals defined by maximal minors are Frobenius split. However, they note that this result cannot be obtained as a direct consequence of Theorem 3.3 applied inductively. In these cases, the standard inductive lifting fails the hypothesis on $u$. Instead, the authors construct the splitting by viewing factors of the splitting polynomial as sums of multiples of elements in the link, applying the isomorphism to these components individually, and recombining them. This reveals a phenomenon related to the main theorem that is not fully explained by it.

Significance and Claims
The paper claims to provide a "partial converse" to Knutson's theorem, establishing a constructive method for lifting Frobenius splittings through geometric vertex decomposition under specific algebraic constraints. The significance lies in:

1. Clarifying the Relationship: It elucidates the relationship between geometric vertex decomposition and Frobenius splitting, mediated by the isomorphisms constituting elementary G-biliaisons.
2. Identifying Limits: It explicitly identifies the obstruction (the non-existence of a nonzerodivisor factor $u$) that prevents a full converse, thereby refining the understanding of when Frobenius splittings deform through degeneration.
3. New Examples: It provides explicit Frobenius splittings for lower bound cluster algebras and Li's double determinantal varieties.
4. Open Problem: The authors modestly note that their construction for double determinantal varieties suggests a need for a refinement of the theory. They leave as an open problem the development of a theory that can explain the splitting construction in Section 5, which operates outside the direct scope of their main theorem.

The work relies heavily on the interplay between the combinatorial structure of geometric vertex decompositions and the algebraic properties of Frobenius splittings, utilizing the isomorphism $\psi$ as the primary technical tool for the lifting process.