$F$-injectivity does not imply $F$-fullness in normal domains

This paper constructs examples of noetherian local geometrically normal domains in prime characteristic that are $F$-injective but fail to be $F$-full or $F$-anti-nilpotent, highlighting the critical role of $F$-injectivity's behavior under purely inseparable finite base changes.

The Gist

Here is an explanation of the paper "F-Injectivity Does Not Imply F-Fullness in Normal Domains" using simple language, analogies, and metaphors.

The Big Picture: A Family of "Good" Rings

Imagine a vast family of mathematical objects called Rings. In the world of algebra, some rings are "well-behaved" (smooth, predictable), while others are "sick" (full of singularities or cracks).

Mathematicians have developed a set of "health checks" to see how sick a ring is. These checks are based on a special operation called the Frobenius map (think of it as a magical photocopier that copies the ring's structure in a specific way).

• F-Pure: The ring is very healthy. The photocopier works perfectly, and you can always reverse the copy to get the original back.
• F-Injective: The ring is "mostly healthy." The photocopier doesn't lose any information (it's injective), but you might not be able to reverse the process perfectly.
• F-Full: The ring is "robustly healthy." Not only does the photocopier keep information, but it also ensures that every possible pattern in the copy can be traced back to a source in the original.

The Big Question: For a long time, mathematicians wondered: If a ring passes the "F-Injective" test (it doesn't lose information), does it automatically pass the tougher "F-Full" test (it's robust)?

Most people suspected the answer was "Yes," especially if the ring was Normal (a specific type of structural integrity, like a building that is perfectly symmetrical and has no hidden cracks).

The Discovery: The "Normal" Lie

The authors of this paper (De Stefani, Polstra, and Simpson) say: "Nope. Not always."

They constructed specific examples of rings that are:

1. Normal: They look perfect and symmetrical.
2. F-Injective: They pass the basic health check (the photocopier doesn't lose data).
3. NOT F-Full: They fail the robustness check. They have a hidden weakness that only shows up under specific conditions.

The Analogy: The "Invisible Crack"

Imagine you have a beautiful, normal-looking vase (the Normal Domain).

• You shine a standard light on it, and it looks perfect. No cracks are visible. This is F-Injectivity.
• However, the authors discovered that if you shine a very specific, strange light (a purely inseparable base change—think of it as looking at the vase through a special filter that distorts reality in a weird way), a hidden crack appears.
• Because of this hidden crack, the vase isn't as "full" or "robust" as we thought. It fails the F-Full test.

The paper proves that even if a ring looks perfectly normal and passes the basic test, it can still have this "hidden crack" that breaks the stronger property.

The Two Main Examples

The authors built two specific "vases" to prove their point:

1. The 2-Dimensional Example (The "Anti-Nilpotent" Failure)

• What it is: A ring that is 2-dimensional (like a surface).
• The Flaw: It is F-Injective but not F-anti-nilpotent.
• The Metaphor: Imagine a machine that sorts mail. It sorts every letter correctly (Injective). But, if you put a specific type of letter in a special box (a submodule), the machine gets confused and can't sort that specific box correctly. The "anti-nilpotent" property is about the machine handling every possible box of letters perfectly. This ring fails that specific test.

2. The 3-Dimensional Example (The "F-Full" Failure)

• What it is: A ring that is 3-dimensional (like a solid object).
• The Flaw: It is F-Injective and Normal, but NOT F-Full.
• The Metaphor: This is the main headline. The ring is a solid, normal block. It passes the basic "no data loss" test. But, it lacks the "completeness" of F-fullness. It's like a sponge that holds water (F-Injective) but has a tiny, invisible hole that lets water leak out if you squeeze it in a specific, weird way (the base change).

Why Does This Matter? (The "Deformation" Problem)

Mathematicians love to study how shapes change when you tweak them slightly (deformation).

• There is a famous open question: If you have a ring that is "almost" F-Injective (specifically, if you cut a piece off it and the piece is F-Injective), is the whole ring F-Injective?
• To solve this, many hoped that Normal + F-Injective would automatically mean F-Full. If that were true, it would make solving the big question much easier.
• The Punchline: The authors proved this hope is false. You can have a Normal, F-Injective ring that is not F-Full. This means the path to solving the big open question is much harder than we thought; we can't rely on "Normality" to save us.

The Secret Weapon: The "Weird Filter"

The key to their construction was using a purely inseparable base change.

• Imagine you have a picture (the ring).
• You take a photo of it (the base change).
• Usually, the photo looks the same.
• But in these specific examples, the "photo" (the ring after the change) suddenly develops a crack that wasn't there before.
• The authors used this "cracking photo" to prove that the original ring, while looking perfect, wasn't actually as robust as it seemed.

Summary in One Sentence

The authors built mathematical "vases" that look perfectly symmetrical and pass basic health checks, but they secretly fail a deeper test of robustness, proving that being "Normal" and "F-Injective" is not enough to guarantee the stronger property of being "F-Full."

Technical Summary

Here is a detailed technical summary of the paper "F-INJECTIVITY DOES NOT IMPLY F-FULLNESS IN NORMAL DOMAINS" by De Stefani, Polstra, and Simpson.

1. Problem Statement and Context

Background:
In commutative algebra, specifically within the study of singularities in prime characteristic $p > 0$, several classes of rings are defined via the behavior of the Frobenius endomorphism on local cohomology modules. Key classes include:

• F-pure: The inclusion $R \hookrightarrow R^{1/p}$ splits.
• F-injective: The Frobenius map $F: H^i_{\mathfrak{m}}(R) \to H^i_{\mathfrak{m}}(R)$ is injective for all $i$.
• F-full: A property related to the surjectivity of the Peskine–Szpiro Frobenius base-change functor, linked to "liftable local cohomology."
• F-anti-nilpotent: A stronger condition where the Frobenius action is injective on quotients of local cohomology by Frobenius-stable submodules.

The Gap:
It is well-established that F-purity implies F-injectivity. However, F-injectivity is a much weaker condition; F-injective rings need not be normal or Cohen–Macaulay. A central open question in the field is whether F-injectivity behaves well under geometric base change (specifically, whether $R \otimes_k k'$ remains F-injective for finite extensions $k \subseteq k'$) and whether it implies stronger structural properties like F-fullness or F-anti-nilpotence when restricted to normal or geometrically normal domains.

The Specific Problem:
The authors investigate whether the "pathologies" of F-injectivity (failure of base change, failure of deformation properties) persist even when the ring is assumed to be geometrically normal. Specifically, they aim to construct counterexamples to the following potential implications:

1. Does geometric normality + F-injectivity $\implies$ F-anti-nilpotence?
2. Does geometric normality + F-injectivity $\implies$ F-fullness?
3. Does geometric normality + F-injectivity $\implies$ geometric F-injectivity (stability under purely inseparable base change)?

2. Methodology

The authors employ a constructive approach using graded hypersurfaces and Veronese subalgebras. Their strategy involves three main technical components:

A. Construction of Base Hypersurfaces

They construct specific two-dimensional graded hypersurfaces $R = S/(f)$ over the field $k = \mathbb{F}_p(t)$ (where $t$ is an indeterminate).

• Case $p=2$: $f = x^3 + tyz^7 + y^2z^5 + y^3z^3 + y^4z + z^9$.
• Case $p>2$: $f = x^{p-1} - (ty^{2p-1} + y^{p-1}z^{p^2-p} + z^{2p^2-3p+1})$.
  These polynomials are carefully chosen to ensure:

1. Geometric Normality: The Jacobian ideal has finite support (isolated singularity).
2. F-injectivity in negative degrees: Using a criterion (Proposition 2.1), they ensure the Frobenius map is injective on local cohomology $H^2_{\mathfrak{m}}(R)$ for sufficiently negative degrees.
3. Specific Degree 0 Behavior: They engineer the action of Frobenius on the degree 0 component $[H^2_{\mathfrak{m}}(R)]_0$ to be injective over $k$, but to fail injectivity after a purely inseparable base change (e.g., $k \to k^{1/p}$).

B. The Veronese Subalgebra Trick

The constructed hypersurfaces $R$ themselves have positive $a$-invariants and are not F-injective. To fix this, the authors take the $n$-th Veronese subalgebra $T = R^{(n)}$ for a sufficiently large $n$.

• Mechanism: The Veronese subalgebra shifts the grading. By choosing $n$ large enough, the "bad" behavior (non-injectivity) is pushed into degrees that are no longer relevant for the local cohomology of the Veronese ring, while the "good" behavior (injectivity in negative degrees) is preserved.
• Result: The localization $T_{\mathfrak{n}}$ becomes F-injective and geometrically normal.

C. Failure of Base Change and F-fullness

• Base Change Failure: They demonstrate that while $T$ is F-injective, the base change $T \otimes_k k^{1/p}$ is not F-injective. This is achieved because the Frobenius action on the degree 0 part of the cohomology of the base-changed ring becomes non-injective (due to the specific coefficients involving $t$).
• F-fullness Failure: To construct a ring that is F-injective but not F-full, they utilize the Segre product. They define $A = T \# k[u,v]$, where $k[u,v]$ is a polynomial ring.
  • Using the Künneth formula for local cohomology, they analyze $H^i_{\mathfrak{m}_A}(A)$.
  • They show that $A$ inherits F-injectivity from $T$.
  • However, because $T \otimes_k k^{1/p}$ fails F-injectivity, the ring $A \otimes_k k^{1/p}$ fails to satisfy the conditions required for F-fullness (via Theorem 2.3 and 2.4).

3. Key Contributions and Results

The paper establishes two main theorems that resolve open questions regarding the hierarchy of F-singularities in normal domains.

Theorem A (Theorem 3.4)

Statement: For any prime $p$, there exists a 2-dimensional local domain $(R, \mathfrak{m}, k)$ that is:

• Geometrically normal.
• F-injective.
• Not F-anti-nilpotent.
• Not geometrically F-injective (fails under purely inseparable base change).

Significance: This provides the first examples of geometrically normal F-injective rings that are not F-anti-nilpotent. It shows that the "weak normality" properties of F-pure rings (specifically, stability under base change) do not extend to F-injective rings, even in the Cohen–Macaulay case (since dim 2 normal implies CM).

Theorem B (Theorem 3.5)

Statement: For any prime $p$, there exists a 3-dimensional local domain $(R, \mathfrak{m}, k)$ that is:

• Geometrically normal.
• F-injective.
• Not F-full.
• Not Cohen–Macaulay.

Significance: This answers the question of whether F-injectivity implies F-fullness in normal domains. The answer is no. The dimension 3 is optimal because in dimension 2, normality implies Cohen–Macaulayness, which in turn implies F-fullness for F-injective rings.

Additional Findings

• Minimal Dimensions: The examples are minimal in dimension. The 2D example is minimal for the failure of F-anti-nilpotence; the 3D example is minimal for the failure of F-fullness.
• Imperfect Residue Fields: The authors note that these examples necessarily have imperfect residue fields ($k = \mathbb{F}_p(t)$). In the Gorenstein setting or with perfect residue fields, F-injectivity, F-purity, and F-anti-nilpotence often coincide.
• Enveloping Algebras: They show that the enveloping algebra $T \otimes_k T$ and the Segre product $T \# T$ of their constructed rings are not F-injective, highlighting the instability of F-injectivity under tensor products.

4. Significance and Implications

1. Refining the Singularity Hierarchy: The paper clarifies the relationships between F-singularities. It proves that F-injectivity is an "extremal" class that does not inherit the robust structural properties (like base change stability or F-fullness) of F-pure rings, even when the ring is geometrically normal.
2. Deformation Theory: The property of being F-full is crucial for deformation problems (e.g., if $R/fR$ is F-full and F-injective, is $R$ F-injective?). By showing that normal F-injective rings are not necessarily F-full, the authors indicate that the deformation question for F-injectivity remains difficult and cannot be resolved simply by assuming normality.
3. Counterexamples to Conjectures: The results serve as definitive counterexamples to the hope that normality "fixes" the pathologies of F-injectivity. Specifically, they disprove the conjecture that geometric normality implies geometric F-injectivity or F-anti-nilpotence.
4. Methodological Innovation: The use of Veronese subalgebras combined with carefully engineered hypersurfaces over function fields ($\mathbb{F}_p(t)$) provides a powerful template for constructing counterexamples in positive characteristic commutative algebra.

In summary, the paper demonstrates that F-injectivity is a fragile property that does not imply F-fullness or F-anti-nilpotence, even in the presence of geometric normality, thereby delineating the strict boundaries of F-singularity theory in prime characteristic.