Hasse–Schmidt Integrable Derivations

• Hasse–Schmidt integrable derivations are sequences of higher-order operators that extend ordinary derivations by satisfying a generalized Leibniz rule.
• They bridge classical differential operators and deformation theory by revealing structural differences in characteristic zero and positive characteristic.
• The theory employs leaps, filtrations, and formal group laws to analyze singularities, base changes, and sheafification in algebraic contexts.

A Hasse–Schmidt integrable derivation is a derivation admitting an extension to a full or truncated Hasse–Schmidt (HS) derivation—a sequence of higher–order operators satisfying a generalized Leibniz rule. Hasse–Schmidt integrability serves as a bridge between the classical theory of differential operators (via ordinary derivations and integrable connections), deformation theory, the algebraic study of singularities in positive characteristic, and model–theoretic approaches to differential fields. The structure and behavior of HS integrable derivations are closely tied to the base ring's characteristic, formal group laws, and specific algebraic features such as smoothness or singularities.

1. Definitions and Fundamental Properties

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

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

with $D_0 = \mathrm{Id}_A$, satisfying for all $a, b \in A$ and $0 \leq i \leq m$:

$D_i(ab) = \sum_{r+s=i} D_r(a) D_s(b).$

Equivalently, $D$ corresponds to a $k$-algebra homomorphism

$$A \to A[[t]]/(t^{m+1}), \quad a \mapsto \sum_{i=0}^m D_i(a) t^i,$$

with reduction modulo $t$ being the identity. The first component $D_1$ is an ordinary $k$-derivation.

A derivation $\delta \in \mathrm{Der}_k(A)$ is said to be $m$–integrable (HS integrable to length $m$) if there exists such a sequence $(D_0, D_1=\delta, D_2, ..., D_m)$ satisfying the HS conditions. The set of $m$–integrable derivations, denoted $\mathrm{Ider}_k(A; m)$, forms an $A$-submodule of $\mathrm{Der}_k(A)$, yielding a descending chain:

$$\mathrm{Der}_k(A) = \mathrm{Ider}_k(A;1) \supseteq \mathrm{Ider}_k(A;2) \supseteq ... \supseteq \mathrm{Ider}_k(A).$$

For $m=\infty$, the term $\mathrm{Ider}_k(A)$ refers to derivations admitting full (infinite-length) HS integrals (Narváez-Macarro, 2011, Chiu et al., 2020, Bravo et al., 27 Sep 2024).

2. Characteristic-Dependent Structure and Integrability

The structure and ubiquity of HS integrable derivations differ sharply between characteristic 0 and positive characteristic.

• Characteristic 0: Every $k$-derivation is HS integrable to infinite length, with the explicit formula $D_i = \delta^i/i!$ (Narváez-Macarro, 2019, Narváez-Macarro, 2018, Chiu et al., 2020). In this setting, all HS modules correspond exactly to modules with integrable (flat) connections; the two categories are equivalent. In particular, in deformation theory or D-module theory, HS integrable derivations add no extra structure compared to the theory of integrable connections or Lie–Rinehart algebras (Narváez-Macarro, 2019, Narváez-Macarro, 2018).
• Positive Characteristic: HS integrability is a subtle condition. Not every derivation admits a full HS integral; the submodules $\mathrm{Ider}_k(A;m)$ can be strict. The failure loci of integrability encode significant arithmetic and geometric information (e.g., $p$–curvature phenomena) (Narváez-Macarro, 2011, Hernández, 2019).

As a result, the theory in positive characteristic involves detailed analysis of the chain of submodules $\mathrm{Ider}_k(A;m)$, the occurrence of "leaps," and obstructions arising from binomial coefficients and Frobenius phenomena.

3. Leaps, Filtrations, and Classification in Positive Characteristic

A leap at index $s \geq 2$ occurs if $\mathrm{Ider}_k(A;s-1)$ is strictly larger than $\mathrm{Ider}_k(A;s)$. The set of leaps, $\mathrm{Leaps}_k(A)$, marks the values at which new obstructions to HS integration arise. The structure of leaps reflects deep number-theoretic features:

• Powers of $p$: In characteristic $p>0$, $\mathrm{Leaps}_k(A)$ can be nonempty, but leaps only occur at powers of $p$; for all other $n$, $\mathrm{Ider}_k(A;n) = \mathrm{Ider}_k(A;n-1)$ (Hernández, 2019, Hernández, 2019). This follows from the vanishing properties of binomial coefficients modulo $p$ and Lucas' theorem, which are crucial in the obstruction theory to extending partial HS derivations.
• Examples: For the plane curve $A = k[u]/(u^{p^e})$, the derivation $u^{p^e-1} d/du$ is $p^e$–integrable but not integrable at any lower length, so there is a leap at $p^e$ (Hernández, 2019, Hernández, 2018).
• Finiteness: In regular or complete intersection settings, or for reduced $A$ over fields, the set of leaps is finite—bounded by the rank and support of $\mathrm{Der}_k(A)$ and controlled by Fitting ideals of the module of Kähler differentials $\Omega_{A/k}$ (Bravo et al., 27 Sep 2024).

4. Functoriality, Sheafification, and Base Change

The modules of HS integrable derivations localize well and admit canonical base-change isomorphisms in key situations.

• Sheaf Structure: For a scheme $X$ over $S$, the sheaf of $m$–integrable relative derivations $\mathrm{Ider}_S(\mathcal{O}_X;m)$ forms a quasi-coherent (coherent if $S$ is locally noetherian) subsheaf of the derivation sheaf (Narváez-Macarro, 2011, Bravo et al., 27 Sep 2024).
• Base Change: Under polynomial ring extensions or separable algebra extensions in positive characteristic, the module $\mathrm{Ider}_k(A;m)$ behaves well:

$$\Theta_{L,A} : L \otimes_k \mathrm{Ider}_k(A;m) \xrightarrow{\sim} \mathrm{Ider}_L(A_L;m).$$

Consequently, the set of leaps is invariant under such base changes (Hernández, 2019).

5. Multi-derivation Structure, Formal Group Laws, and Iterativity

The concept of HS integrability generalizes to multivariate and formal group law settings.

• Multivariate HS Derivations: Consider a family $D = (D_{\alpha})_{\alpha\in \Delta}$ for a co-ideal $\Delta \subset \mathbb{N}^p$ and $D_0=\mathrm{Id}_A$ with HS–Leibniz identity. Decomposition theorems express any multivariate HS derivation as compositions of monomial substitution maps and univariate HS derivations, and provide explicit factorization structure (Narváez-Macarro et al., 2019, Narváez-Macarro, 2018).
• Iterativity and Formal Groups: An HS derivation is said to be $F$–iterative if it satisfies a specific integrability law prescribed by a formal group law $F(X,Y)$. Notably, for additive and multiplicative group laws, explicit extension and integration theorems are known (Hoffmann et al., 2012, Hoffmann et al., 2014, Hoffmann, 2015). This connects HS theory to group scheme actions and underlies the model theory of "fields with $G$-derivations."
• Uniqueness and Obstructions: In characteristic zero, HS integrals are unique and every derivation is integrable (Narváez-Macarro, 2018). In positive characteristic, uniqueness may fail, and explicit obstruction criteria involve divided-powers, $p$–curvature restrictions, and binomial congruences (Hoffmann et al., 2012, Hoffmann, 2015).

6. Algebraic and Geometric Implications

HS integrable derivations link to several major directions:

• Algebra of Differential Operators: In characteristic zero and under smoothness—i.e., $A$ is HS–smooth—the canonical map from the enveloping algebra of HS derivations to the ring of $k$–linear differential operators,

$U_{HS}(A/k) \to D_{A/k},$

is an isomorphism. Thus, $D_{A/k}$ can be viewed as the enveloping algebra of all HS integrable derivations (Narváez-Macarro, 2018, Narváez-Macarro, 2019).

• D-module and Deformation Theory: The equivalence of HS module structures and flat connections in characteristic $0$ identifies the classical D-module category with that of HS–modules. In deformation theory, the ability to package higher derivations as formal series plays a role, but gives no additional structure over integrable connections in characteristic zero (Narváez-Macarro, 2019).
• Kähler Differentials of HS Algebras: The higher differential structure of $A$ is captured by the Hasse–Schmidt algebra $A_n$ and its module of Kähler differentials, which can be functorially related to HS modules over the original algebra. Commutation theorems guarantee coherence and naturality for iterated applications of higher derivation functors (Chiu et al., 2020).

7. Model-Theoretic and Formal Geometry Applications

• Model Companions and Existential Closedness: The model-theoretic study of fields with HS derivations (notably, iterative for a fixed formal group law) yields complete model companions, quantifier elimination, and precise geometric axiomatizations. For instance, fields with $W_e$–iterative HS derivations (i.e., Witt vector group law) are completely characterized, and the leap structure reflects the underlying arithmetic (Hoffmann et al., 2014, Hoffmann, 2015).
• Prolongations and Canonical Hasse–Schmidt Tuples: In the algebraic–geometric setting, prolongations (jet schemes) and universal vector bundle structures are naturally functorial in HS algebra terms, further connecting the theory to arc spaces and motivic integration (Chiu et al., 2020).

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References:

(Narváez-Macarro, 2011) On the modules of m-integrable derivations in non-zero characteristic (Narváez-Macarro, 2018) Hasse–Schmidt derivations versus classical derivations (Hernández, 2019) Leaps of modules of integrable derivations in the sense of Hasse–Schmidt (Narváez-Macarro, 2019) Hasse–Schmidt modules versus integrable connections (Narváez-Macarro et al., 2019) On the bracket of integrable derivations (Chiu et al., 2020) Higher derivations of modules and the Hasse–Schmidt module (Hoffmann et al., 2014) Existentially closed fields with G-derivations (Hoffmann, 2015) Witt vectors and separably closed fields with higher derivations (Narváez-Macarro, 2018) Rings of differential operators as enveloping algebras of Hasse–Schmidt derivations (Hoffmann et al., 2012) Integrating Hasse–Schmidt derivations (Bravo et al., 27 Sep 2024) Finiteness of Leaps in the sense of Hasse–Schmidt of reduced rings