Leaps in Integrable Derivation Modules

• The topic defines leaps as strict drops in the chain of Hasse–Schmidt-integrable derivations, typically occurring at p-power indices.
• It demonstrates that the arithmetic of prime characteristic and binomial combinatorics control the integrability and singularity structure in commutative algebras.
• Methodologies using combinatorial identities and Artin–Rees techniques establish finite leaps and provide effective tools for analyzing derivation modules.

A leap of modules of integrable derivations refers to a strict drop in the chain of modules of Hasse–Schmidt-integrable derivations associated to a commutative algebra over a ring of positive characteristic. This phenomenon encodes subtle invariants of the singularity structure of the algebra and is controlled by the arithmetic of the prime characteristic $p$ and the properties of the algebraic structure. The subject lies at the intersection of algebraic geometry, commutative algebra, and the theory of differential operators in positive characteristic, connecting module-theoretic, cohomological, and singularity-theoretic aspects.

1. Hasse–Schmidt Integrable Derivations

Let $k$ be a commutative ring and $A$ a unital commutative $k$-algebra. A Hasse–Schmidt derivation of $A$ over $k$ of length $m\in\mathbb{N}\cup\{\infty\}$ is a sequence of $k$-linear endomorphisms

$D = (D_0, D_1, \ldots, D_m)$

satisfying $D_0 = \operatorname{Id}_A$ and, for all $a, b \in A$ and $n\leq m$,

$D_n(ab) = \sum_{i+j=n} D_i(a) D_j(b).$

The space $\mathrm{HS}_k(A;m)$ denotes all such sequences. The first component $D_1$ is an ordinary $k$-derivation; higher $D_n$ are differential operators of order $\leq n$, vanishing at $1$. A $k$-derivation $\delta \in \operatorname{Der}_k(A)$ is called $n$-integrable ($n\in\mathbb{N}\cup\{\infty\}$) if there exists $D\in \mathrm{HS}_k(A;n)$ with $D_1 = \delta$. This leads to the descending chain of $A$-submodules:

$$\operatorname{IDer}_k(A;1) \supseteq \operatorname{IDer}_k(A;2) \supseteq \cdots \supseteq \operatorname{IDer}_k(A;\infty)$$

where $\operatorname{IDer}_k(A;n)$ consists of derivations that are $n$-integrable in the Hasse–Schmidt sense (Hernández, 2019, Bravo et al., 27 Sep 2024, Miyamoto, 30 Nov 2025, Narváez-Macarro, 2011).

2. Filtration and Leaps: Definition and Characterization

A leap occurs at $n>1$ if

$$\operatorname{IDer}_k(A;n-1) \supsetneq \operatorname{IDer}_k(A;n)$$

i.e., if there exists a derivation integrable up to order $n-1$ but not to order $n$. The set of such $n$ is denoted $\operatorname{Leaps}_k(A)$. These inclusions generally stabilize after finitely many steps. For a scheme $X$ essentially of finite type over $k$, there is a corresponding decreasing chain of quasi-coherent sheaves of integrable derivations:

$$\mathcal{D}er_k(\mathcal{O}_X) \supset \mathcal{D}er^p_k(\mathcal{O}_X) \supset \mathcal{D}er^{p^2}_k(\mathcal{O}_X) \supset \cdots$$

and $X$ “leaps” at $p^i$ if the inclusion at that stage is strict (Miyamoto, 30 Nov 2025, Bravo et al., 27 Sep 2024, Narváez-Macarro, 2011).

3. Restrictions on Possible Leaps: Powers of $p$

A central result establishes that, for $k$ of characteristic $p>0$, strict inclusions—i.e., leaps—can only occur at $n=p^r$ for $r\geq 1$. For all $n$ not a pure $p$–power, the modules remain constant:

$$\operatorname{IDer}_k(A;n-1) = \operatorname{IDer}_k(A;n) \quad\text{if } n\text{ is not a power of } p$$

This result, proved via a sequence of combinatorial and lifting arguments, utilizes intricate binomial identities, properties of the characteristic $p$ residue field, and an induction on the $p$–adic expansion of $n$. The restriction to $p$–power indices is intrinsic to the binomial combinatorics present in positive characteristic (Hernández, 2019, Miyamoto, 30 Nov 2025).

4. Finiteness of Leaps and Structural Theorems

For algebras essentially of finite type over an algebraically closed field of characteristic $p>0$, the set of leaps is always finite. Specifically, there is some $M\gg 0$ such that

$$\operatorname{Der}^m_k(A) = \operatorname{Der}^{m+1}_k(A) = \cdots = \operatorname{Der}^\infty_k(A) \quad\text{for } m\geq M$$

and

$$\operatorname{Leaps}_k(A) \subset \{p, p^2, ..., p^{M-1}\}$$

The proof involves constructing and analyzing obstruction modules $\operatorname{Ob}_R^{p^i} \subset T^1_{R/k}$ (where $T^1_{R/k}$ is the module of first cotangent cohomology), and using Artin–Rees techniques to show that, after a bounded stage, all further extension problems for integrable derivations are unobstructed (Miyamoto, 30 Nov 2025, Bravo et al., 27 Sep 2024).

Locally, the number of leaps is bounded by lengths of finite modules or dimensions of appropriate quotients; for instance,

$$|\operatorname{Leaps}_k(A_\mathfrak{p})| \le \dim_{\kappa(\mathfrak{p})} \left( \operatorname{Der}_k(A_\mathfrak{p}) / \mathfrak{p}^M \operatorname{Der}_k(A_\mathfrak{p}) \right)$$

for minimal primes $\mathfrak{p}$ of the critical Fitting ideal (Bravo et al., 27 Sep 2024).

5. Role of Fitting Ideals and Singularities

Let $\Omega_{A/k}$ denote the module of Kähler differentials; its Fitting ideals, specifically $J_r(A)$—the ideal of $(n-r)$-minors of a presenting Jacobian—play a decisive role. For (reduced) complete intersection algebras $A=R/(f_1,\ldots,f_r)$, any derivation vanishing on $J_r(A)$ is $\infty$-integrable; thus, the critical Fitting ideal detects loci where nontrivial leaps may occur. Singularities where the Jacobian drops rank control higher-order integrability, and the support of the successive quotients $$\operatorname{IDer}_k(A;m)/\operatorname{IDer}_k(A;m+1)$$ is contained in the singular locus (Bravo et al., 27 Sep 2024).

These conclusions extend to the context of schemes and formal power series. Localizations at minimal primes of $J_r(A)$ reduce leap detection to the paper of finite-length vector space quotients. This mechanism provides effective criteria for the finiteness of leaps and explicit bounds in concrete settings.

6. Examples, Applications, and Algorithmic Aspects

Explicit examples:

• For $A = k[x]/(x^p)$ in characteristic $p$, $\delta = d/dx$ is $(p-1)$-integrable but not $p$-integrable, so a leap occurs at $p$.
• For the plane cuspidal curve $k[x,y]/(y^2 - x^3)$ in $p=2$, there is a single leap at $2$, with all higher modules stabilizing.
• For reduced one-dimensional $k$-algebras over perfect fields, leaps are always finite and concentrated at finitely many $p$-powers determined by the conductor.

Computational algorithm: For a finitely presented algebra $A=k[x_1,\ldots,x_n]/I$ and integer $N\geq2$, an explicit finite-step algorithm determines the sets $\operatorname{IDer}_k(A;m)$ for all $m\leq N$ and thus locates the possible leaps (Narváez-Macarro, 2011).

7. Theoretical Implications and Open Problems

Finiteness and $p$–power restriction for leaps sharpen previous results of Narváez-Macarro, Molinelli, and others. These results provide a discrete invariant of singularities in positive characteristic, encoding subtle information about wild ramification and the structure of higher-order differential operators. The theory establishes a bridge between integrable-derivation modules and classical algebraic invariants such as the module of Kähler differentials and cotangent cohomology.

Open questions include explicit descriptions of the modules $\operatorname{IDer}_k(A;p^r)$ in terms of singularity data, analogues for non-commutative or graded algebras, and the behavior of the gap $$\operatorname{Der}_k(A)/\operatorname{IDer}_k(A;p^r)$$ as an arithmetic measure of singularity. Another direction concerns the relation of this theory to stratification problems in arithmetic and geometric D-module theory in positive characteristic (Hernández, 2019, Miyamoto, 30 Nov 2025, Bravo et al., 27 Sep 2024).