Schemes Essentially of Finite Type

• Schemes essentially of finite type are defined as schemes locally modeled on localizations of finitely generated k-algebras, ensuring Noetherian properties and structural rigor.
• They support a robust sheaf-theoretic framework, linking Kähler differentials and tangent sheaves to practical studies of singularities and smoothness.
• Recent finiteness results in positive characteristic reveal that leaps in integrable derivations occur only at powers of p, stabilizing the descending chain of derivation modules.

A scheme is said to be essentially of finite type over an algebraically closed field $k$ if it is locally modeled on affine rings that are localizations of finitely generated $k$-algebras. Such schemes and their coordinate rings serve as the ambient structures for foundational advances in algebraic geometry, particularly in the paper of singularities, derivations, and higher differential operators in positive characteristic. The theory of integrable derivations and their jumps, or “leaps,” in this context illuminates the interplay between singularities and the arithmetic of the underlying field, culminating in recent finiteness results.

1. Definitions and Examples

A $k$-algebra $R$ is essentially of finite type if there exists a finitely generated polynomial algebra $A = k[x_1, \dotsc, x_n]$ and a multiplicative subset $S \subset A$ such that

$R \cong S^{-1}A / I$

for some ideal $I \subset S^{-1}A$; equivalently, $R$ is a localization of a finitely generated $k$-algebra. Schemes essentially of finite type are those with affine open covers $\Spec R_i$ for such $R_i$. This includes the local rings of points on affine varieties, open subschemes defined by inverting regular functions, and more general constructions obtained by gluing such affine pieces.

Key examples include:

• $k[x, y]_x$: localization of the polynomial ring at the nonzerodivisor $x$.
• The local ring at any closed point of a hypersurface, e.g., $\bigl(k[x, y]/(y^2 - x^3)\bigr)_{\mathfrak{p}}$.
• The complement of a divisor, e.g., $\Spec k[x, y, z]/(xy - z^2) \setminus V(x)$.

2. Fundamental Properties and Sheaf-Theoretic Aspects

Because finitely generated algebras over a field are Noetherian, their localizations and quotients retain the Noetherian property. In the smooth case, localizations of polynomial algebras are formally smooth over $k$. For a scheme $X$ essentially of finite type over $k$, coherent sheaves such as the sheaf of Kähler differentials $\Omega^1_{X/k}$ and the tangent sheaf $T^1_{X/k}$ are obtained by gluing the corresponding modules on affine opens. This categorical and sheaf-theoretic infrastructure enables the construction and paper of differential operators and their modules in local and global settings.

3. Derivations, Hasse–Schmidt Derivations, and Integrability

Given a $k$-algebra $R$, a derivation is a $k$-linear map $d: R \to R$ satisfying the Leibniz rule

$$d(xy) = x\,d(y) + y\,d(x),\quad \forall x, y \in R.$$

The set of all derivations forms the $R$-module $\Der_k(R, R)$. Hasse–Schmidt derivations of length $m$ generalize this notion to sequences $D = (D_0, D_1, \ldots, D_m)$ of $k$-linear endomorphisms with $D_0 = \mathrm{id}$ and

$$D_i(xy) = \sum_{j + k = i} D_j(x) D_k(y),\quad 1 \leq i \leq m.$$

Here, $D_1$ coincides with an ordinary derivation, giving an isomorphism $\HS^1_k(R) \cong \Der_k(R, R)$. The image of the truncation $\HS^m_k(R) \to \HS^1_k(R)$ (and thus to $\Der_k(R, R)$) is the module $\Der^m_k(R)$ of $m$-integrable derivations. This construction yields a descending chain of submodules: $\Der_k(R) \supset \Der^2_k(R) \supset \Der^3_k(R) \supset \cdots \supset \Der^\infty_k(R),$ where $\Der^\infty_k(R)$ denotes the set of derivations arising from infinite-length Hasse–Schmidt derivations.

4. The Structure and Finiteness of Leaps

A leap occurs at $m$ if $\Der^{m-1}_k(R) \supsetneq \Der^m_k(R)$, i.e., there exists a derivation integrable to order $m-1$ but not to $m$. In characteristic $p > 0$, Hernández–Narváez–Macarro established that leaps can only occur at powers of $p$, i.e., $m = p^i$. The set of such exponents where the chain of modules strictly descends is called the set of leaps.

The main theorem of Takuya Miyamoto (Miyamoto, 30 Nov 2025) establishes:

Finiteness Theorem.

Let $k$ be an algebraically closed field of characteristic $p > 0$, and $R$ a $k$-algebra essentially of finite type. Then there exists $M$ such that

$\Der^M_k(R) = \Der^{M+1}_k(R) = \cdots = \Der^\infty_k(R).$

Thus, only finitely many submodules in the descending chain are distinct; the set of leaps is finite. The same holds over any perfect field via a separable base change.

The proof proceeds by:

• Introducing, for each $m = p^i$, an obstruction map $\ob_m : \HS^{m-1}_k(R) \rightarrow T^1_{R/k}$, with image in a finitely generated submodule $\Ob^m_R \subset T^1_{R/k}$.
• Showing the chain $\Ob^{p^i}_R \subset \Ob^{p^{i+1}}_R \subset \cdots$ becomes stationary using coherence and the Artin–Rees lemma.
• Employing Noetherian induction on the support of $\Ob^{p^i}_R/\Ob^{p^{i-1}}_R$ to globalize local results.

5. Significance, Applications, and Further Directions

In characteristic zero, or for smooth $k$-schemes, every derivation is arbitrarily integrable: no nontrivial leaps occur. Finiteness is therefore exclusive to the setting of positive characteristic and is closely linked to singularities and the Frobenius endomorphism. Knowing that only finitely many integrability conditions can arise offers structural control over modules of higher differential operators and Hasse–Schmidt derivations. This has implications for de Rham cohomology mod $p$, singularity classification via differential operators, and understanding the behavior of the tangent sheaf under Frobenius.

Further, the result addresses an open question of L. Narváez-Macarro and L.N. Macarro, confirming that all algebras essentially of finite type have only finitely many leaps. As a corollary, a Hasse–Schmidt derivation of length $n$ extends to infinite length if and only if it extends to length $m$ for every $m \geq n$ (Miyamoto, 30 Nov 2025).

Outstanding questions include determining effective bounds for the stabilization index $i$ in terms of invariants such as dimension, clarifying the relationship between leaps and properties like $F$-purity or $F$-regularity, and pursuing representation-theoretic approaches (e.g., via Cartier theory) to obstruction modules. There are also prospects for extending these finiteness results to contexts of mixed characteristic and log geometry.

Miyamoto’s theorem thus provides a comprehensive classification of the behavior of higher integrable derivations in singular, positive characteristic settings, offering new tools for differentiating and studying singularities through the lens of differential operators (Miyamoto, 30 Nov 2025).