Generalized F-Signature in Commutative Algebra

• Generalized F-signature is a numerical invariant in commutative algebra that extends classical F-signature to pairs and diverse algebraic contexts.
• It employs module theory, Cartier operators, and algorithmic techniques to measure Frobenius splitting properties and detect singularities like strong F-regularity.
• Its applications span toric geometry and invariant theory, providing practical methods for explicit computation and deeper insights into singularity types.

The generalized F-signature is a numerical invariant in commutative algebra and singularity theory, extending the classical F-signature from rings to "pairs" and further to a variety of algebraic contexts. It quantifies the asymptotic proportion of certain types of direct summands under the iteration of the Frobenius endomorphism in characteristic $p > 0$, generalizing both the algebraic and representation-theoretic decompositions associated with the Frobenius functor. Through a combination of module theory, Cartier operators, invariant theory, and toric geometry, it detects subtle properties of singularities—such as strong F-regularity, F-purity, and F-rationality—and admits explicit computations in several key classes.

1. General Definition and Cartier Subalgebras

Let $R$ be a Noetherian, F-finite ring of characteristic $p>0$, equipped with a Cartier subalgebra $$D = \bigoplus D_e \subseteq \mathcal{C}_R = \bigoplus_e \operatorname{Hom}_R(F_*^e R, R)$$. A pair $(R, D)$ encodes both the algebraic structure of $R$ and the allowed $p^{-e}$-linear operators. For each $e > 0$, define the ideal

$$I_e^D = \{ r \in R : \varphi(F_*^e r) \in m \text{ for all } \varphi \in D_e \},$$

where $m$ is the maximal ideal.

The splitting number $R$0 is the maximal number of direct summands isomorphic to $R$1 in a decomposition $R$2, with projection maps in $R$3. Several equivalent formulations hold: $R$4 where $R$5 (the imperfection degree over its residue field).

The generalized F-signature of the pair is

$R$6

with $R$7, the set of levels at which $R$8 has non-nilpotent elements. The existence of this limit is guaranteed for F-finite rings and any Cartier subalgebra (Blickle et al., 2011).

2. Existence, Properties, and Positivity

The generalized F-signature $R$9 exists for any F-finite local ring and Cartier subalgebra $p>0$0. The central dichotomy of F-signature theory remains intact:

• $p>0$1 if and only if the pair $p>0$2 is strongly F-regular; that is, for every $p>0$3, there exists some $p>0$4 and $p>0$5 with $p>0$6 (Blickle et al., 2011).
• For non-strongly F-regular but F-pure $p>0$7, the F-splitting ratio

$p>0$8

measures the “undercurrent of splitting” along the largest $p>0$9-compatible ideal $$D = \bigoplus D_e \subseteq \mathcal{C}_R = \bigoplus_e \operatorname{Hom}_R(F_*^e R, R)$$0 (the splitting prime). This ratio is strictly positive and equals the F-signature of the strongly F-regular quotient $$D = \bigoplus D_e \subseteq \mathcal{C}_R = \bigoplus_e \operatorname{Hom}_R(F_*^e R, R)$$1.

A generalized F-signature function for pairs with exponents, such as $$D = \bigoplus D_e \subseteq \mathcal{C}_R = \bigoplus_e \operatorname{Hom}_R(F_*^e R, R)$$2, tracks regularity thresholds and their order of vanishing. For such settings, the asymptotics of $$D = \bigoplus D_e \subseteq \mathcal{C}_R = \bigoplus_e \operatorname{Hom}_R(F_*^e R, R)$$3 near the F-pure threshold are controlled by the F-splitting ratio and the codimension of the splitting prime (Canton, 2012).

3. Computation and Algorithmic Approaches

Fedder-type criteria allow the F-signature and generalized F-signature to be computed via module-length formulas involving colon ideals. For $$D = \bigoplus D_e \subseteq \mathcal{C}_R = \bigoplus_e \operatorname{Hom}_R(F_*^e R, R)$$4 in a regular local ring $$D = \bigoplus D_e \subseteq \mathcal{C}_R = \bigoplus_e \operatorname{Hom}_R(F_*^e R, R)$$5, with a suitable Cartier subalgebra, one has: $$D = \bigoplus D_e \subseteq \mathcal{C}_R = \bigoplus_e \operatorname{Hom}_R(F_*^e R, R)$$6 and thus

$$D = \bigoplus D_e \subseteq \mathcal{C}_R = \bigoplus_e \operatorname{Hom}_R(F_*^e R, R)$$7

This framework extends to pairs $$D = \bigoplus D_e \subseteq \mathcal{C}_R = \bigoplus_e \operatorname{Hom}_R(F_*^e R, R)$$8 by considering sequences $$D = \bigoplus D_e \subseteq \mathcal{C}_R = \bigoplus_e \operatorname{Hom}_R(F_*^e R, R)$$9, producing length formulas for $(R, D)$0 and enabling explicit calculations for monomial ideals and toric settings (Blickle et al., 2011).

For toric or Hibi rings (which possess FFRT), the generalized F-signature of any reflexive conic module $(R, D)$1 is

$(R, D)$2

where $(R, D)$3 is an explicit polytope associated to $(R, D)$4 via the poset or the cone defining the ring (Higashitani et al., 2019, Korff, 2011). Enumeration via polytopes and descent-conditions in the symmetric group gives highly effective combinatorial formulas.

In the context of invariant subrings $(R, D)$5 (with $(R, D)$6 a finite group), for each indecomposable reflexive $(R, D)$7-module $(R, D)$8 corresponding to irreducible $(R, D)$9-module $R$0,

$R$1

provided $R$2 acts without pseudo-reflections and $R$3. This recovers the classical McKay correspondence in the context of F-signature theory (Hashimoto et al., 2013, Hashimoto et al., 2023).

4. Geometric and Representation-Theoretic Extensions

For section rings of globally F-regular projective varieties, the F-signature function becomes a locally Lipschitz, positively homogeneous real-valued function on the ample cone, vanishing precisely on nef but non-big divisors (Lee et al., 2022). The generalized F-signature function $R$4 is controlled by the counting of global sections modulo splitting contribution, and is closely estimated by the volume function up to explicit error terms.

In the invariant theory setting of finite group schemes acting on polynomial rings, the generalized F-signature encodes the asymptotic Frobenius splitting data of reflexive modules. The limit decomposes as

$R$5

in the Grothendieck group, revealing how indecomposable modules distribute in asymptotic Frobenius direct images. This refines the correspondence between singularity type and representation structure of $R$6 (Hashimoto et al., 2023).

Quasi-polynomiality arises in the F-signature functions of quotient singularities, with higher coefficients determined by the group-theoretic data of pseudoreflections. Notably, for small groups (no $R$7-pseudoreflections), the coefficient of $R$8 vanishes identically in the generalized F-signature function's expansion (Caminata et al., 2018).

5. Connections to Other Invariants and Theoretical Significance

The generalized F-signature subsumes the classical F-signature and further bridges to Hilbert–Kunz multiplicities, $R$9-splitting ratios, and the spectrum of Frobenius-pullback summands in FFRT rings. It quantifies not only the abundance of free summands, but also the distribution of all indecomposable summands under Frobenius, thereby measuring singularity type in much finer detail.

Generalizations to F-rationality yield invariants such as the dual F-signature and the relative F-rational signature, which are defined via maximal surjective quotient maps from Frobenius-pullbacks of the canonical module or via infima over relative Hilbert–Kunz slopes. The unified invariant $p^{-e}$0 detects F-rationality, always satisfies $p^{-e}$1, and is lower semi-continuous under localization (Smirnov et al., 2019).

In characteristic zero, analogues such as the differential signature replace the Frobenius with modules of principal parts and differential powers, preserving many of the regularity-detection and volume formula properties. In characteristic $p^{-e}$2, the differential signature coincides with the classical F-signature for $p^{-e}$3-pure rings (Brenner et al., 2018).

6. Illustrative Examples

• Simple normal crossings: $p^{-e}$4, $p^{-e}$5, $p^{-e}$6, then $p^{-e}$7 (Blickle et al., 2011).
• Invariant subrings: For $p^{-e}$8 cyclic of order $p^{-e}$9 acting on $e > 0$0, all indecomposable reflexive modules have $e > 0$1, and the ring $e > 0$2 itself has $e > 0$3 (Hashimoto et al., 2013, Hashimoto et al., 2023).
• Monomial ideals: $e > 0$4, $e > 0$5 monomial ideal with Newton polyhedron $e > 0$6, then for any $e > 0$7, $e > 0$8 (Blickle et al., 2011, Lee et al., 2022).
• Hibi rings: For conic divisorial ideal $e > 0$9, $$I_e^D = \{ r \in R : \varphi(F_*^e r) \in m \text{ for all } \varphi \in D_e \},$$0 for an explicit polytope $$I_e^D = \{ r \in R : \varphi(F_*^e r) \in m \text{ for all } \varphi \in D_e \},$$1. For Segre products, $$I_e^D = \{ r \in R : \varphi(F_*^e r) \in m \text{ for all } \varphi \in D_e \},$$2 admits symmetric group enumeration formulas (Higashitani et al., 2019).
• Quotient singularities: $$I_e^D = \{ r \in R : \varphi(F_*^e r) \in m \text{ for all } \varphi \in D_e \},$$3, the F-signature function $$I_e^D = \{ r \in R : \varphi(F_*^e r) \in m \text{ for all } \varphi \in D_e \},$$4 is a quasi-polynomial in $$I_e^D = \{ r \in R : \varphi(F_*^e r) \in m \text{ for all } \varphi \in D_e \},$$5, with coefficients determined by the group and vanishing of certain reflection types (Caminata et al., 2018).

7. Impact and Open Directions

Generalized F-signature theory unifies and extends the role of Frobenius-semistable summands in singularity detection, providing effective tools for explicit computation and deepening the structural understanding of singularities in rational, toric, and equivariant settings. It resolves questions on positivity, existence, and behavior under families, and connects with broader themes such as differential operators, Hilbert–Kunz theory, and invariant theory.

The continued development of algorithmic and combinatorial formulas, especially in the context of toric and FFRT rings, as well as systematic comparisons with characteristic-zero invariants, are active areas of research. Quantitative bounds and geometric interpretations (e.g., in terms of polytopes and volumes) suggest further analogies to birational geometry, motivic integration, and representation theory. Open problems include a full extension of F-signature concepts to mixed characteristic, higher-dimensional families, and deeper connections with derived categories and D-module theory.

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References:

(Blickle et al., 2011): "F-signature of pairs and the asymptotic behavior of Frobenius splittings," (Lee et al., 2022): "The F-signature Function on the Ample Cone," (Higashitani et al., 2019): "Generalized F-signatures of Hibi rings," (Hashimoto et al., 2023): "Generalized $$I_e^D = \{ r \in R : \varphi(F_*^e r) \in m \text{ for all } \varphi \in D_e \},$$6-signatures of the rings of invariants of finite group schemes," (Hashimoto et al., 2013): "Generalized F-signature of invariant subrings," (Caminata et al., 2018): "F-signature function of quotient singularities," (Korff, 2011): "F-Signature of Affine Toric Varieties," (Canton, 2012): "Relating F-Signature and F-Splitting Ratio of Pairs Using Left-Derivatives," (Smirnov et al., 2019): "The theory of F-rational signature," (Brenner et al., 2018): "Quantifying singularities with differential operators."