Note on Morita equivalence in ring extensions

This paper establishes the Morita invariance of several classes of ring extensions and provides a counterexample of a class that is not Morita invariant, building upon previous results by Miyashita and Ikehata.

The Gist

Imagine you are an architect looking at different buildings. Some buildings are made of brick, some of wood, and some of glass. In the world of mathematics, specifically Ring Theory, these "buildings" are called Rings, and the way one ring sits inside another (like a foundation inside a skyscraper) is called a Ring Extension.

This paper by Satoshi Yamanaka is essentially a study of structural integrity. It asks a very specific question: If I take a building and completely remodel it using a specific set of blueprints (called "Morita Equivalence"), do the special features of the original building survive the renovation?

Here is a breakdown of the paper's journey, using simple analogies.

1. The Concept: "Morita Equivalence" (The Renovation)

Imagine you have a house (Ring $A$) built on a specific foundation (Ring $B$). Now, imagine a magical architect who can take that house and rebuild it into a completely different-looking structure (Ring $A'$) on a different foundation (Ring $B'$).

However, this isn't just a random rebuild. The architect uses a special rule called Morita Equivalence. This rule ensures that while the look of the house changes, the internal logic and the relationships between the rooms remain exactly the same. It's like translating a book from English to French; the words change, but the story remains identical.

The paper asks: If the original house had a special feature (like a "trick door" or a "secret vault"), will the new, translated house also have that feature?

2. The Features: Types of Ring Extensions

The paper looks at several specific "architectural styles" or features of these ring extensions. The author wants to know if these features are Morita Invariant (meaning they survive the renovation).

Here are the features the paper investigates:

• Trivial Extensions (The "Add-on" Room): Imagine a house where you just tack on a small shed that doesn't really change the main structure. The paper proves that if you have this simple "add-on" style, the new house will also have a simple "add-on" style. Verdict: It survives.
• Liberal Extensions (The "Key" System): Imagine a house where you can open any door using a specific set of keys. The paper shows that if the original house has this key system, the new house will have a corresponding key system. Verdict: It survives.
• Depth Two Extensions (The "Two-Step" Rule): This is a bit like a rule saying, "To get from the kitchen to the bedroom, you must pass through exactly two specific hallways." The paper proves that if the original house follows this "two-step" rule, the new house will too. Verdict: It survives.
• Strongly Separable & Weakly Separable Extensions (The "Smooth Flow"): Think of these as houses where water (mathematical operations) flows perfectly without getting stuck or leaking. The paper confirms that if the original house has smooth plumbing, the renovated house will also have smooth plumbing. Verdict: They survive.

3. The Big Discovery: What Doesn't Survive?

The most exciting part of the paper is the "Gotcha!" moment. The author proves that not everything survives the renovation.

He gives an example of a specific type of house where a rule exists: "Every time you walk around the block $n$ times, you end up back at the starting point." (In math terms, this is about elements raised to a power $n$).

• The Original House: Has this rule. If you walk around the block enough times, you are back at the start.
• The Renovated House: The author builds a counter-example where, even though the two houses are "Morita Equivalent" (they have the same internal logic), the new house does not have this rule. You can walk around the block forever and never return to the start.

The Lesson: Just because two buildings are structurally equivalent in a deep mathematical sense, they don't necessarily share every single property. Some features are fragile and get lost in translation.

4. The "I Don't Know" Box

At the end, the author admits there are still some mysteries. There are other types of "architectural styles" (like quasi-separable or finite normalizing extensions) where he isn't sure if they survive the renovation or not. He leaves these as open questions for future mathematicians to solve.

Summary

In plain English, this paper is a checklist for mathematicians. It says:

1. We have a way to translate ring structures (Morita Equivalence).
2. We checked a list of popular features (Trivial, Liberal, Depth Two, Separable).
3. Good News: Most of these features are robust; they survive the translation.
4. Bad News: Not all features survive. There is at least one type of rule that breaks during translation.
5. Mystery: We still don't know about a few other features.

The paper helps mathematicians understand which properties of their mathematical "buildings" are fundamental and which are just superficial details that might disappear when you change your perspective.

Technical Summary

Here is a detailed technical summary of the paper "Note on Morita equivalence in ring extensions" by Satoshi Yamanaka.

1. Problem Statement

The paper addresses the fundamental question of Morita invariance within the theory of ring extensions. While Morita equivalence is a well-established concept for rings (preserving module categories), its application to classes of ring extensions ($A/B$) is less explored.

The central problem is to determine which specific classes of ring extensions are Morita invariant. A class $\mathcal{C}$ of ring extensions is Morita invariant if, for any two Morita equivalent extensions $A/B \sim A'/B'$, the property of belonging to $\mathcal{C}$ is preserved (i.e., if $A/B \in \mathcal{C}$, then $A'/B' \in \mathcal{C}$).

The author builds upon previous work by Y. Miyashita (who established invariance for Galois and Frobenius extensions) and S. Ikehata (who proved invariance for separable, Hirata separable, symmetric, and QF extensions). The goal is to extend these results to other significant classes of extensions and to identify classes that are not Morita invariant.

2. Methodology

The paper employs a structural approach based on the definition of Morita equivalence for ring extensions provided by Miyashita.

• Definition of Equivalence: Two extensions $A/B$ and $A'/B'$ are Morita equivalent ($A/B \sim A'/B'$) if there exist Morita modules ${}_A M_{A'}$ and ${}_B N_{B'}$ such that ${}_A A \otimes_B {}_B N_{B'} \cong {}_A M_{B'}$.
• Construction of Isomorphisms: The core methodology involves constructing explicit isomorphisms between the algebraic structures of the equivalent extensions. The author utilizes the Morita context to define:
  • A ring isomorphism $\phi: A \to A'$ (specifically mapping centralizers and endomorphism rings).
  • A map $\psi: A \otimes_B A \to A' \otimes_{B'} A'$.
  • A map $\phi$ acting on endomorphism rings $\text{End}_\ell(BAB) \to \text{End}_\ell(B'A'B')$.
• Transfer of Properties: The author proves that specific structural properties defining various extension classes are preserved under these isomorphisms. This is done by:
  1. Taking the defining elements (e.g., separability idempotents, centralizers, derivations) of the extension $A/B$.
  2. Mapping them via $\phi$ and $\psi$ to $A'/B'$.
  3. Verifying that the images satisfy the defining axioms of the target class in $A'/B'$.
• Counter-Example Construction: To demonstrate non-invariance, the author constructs a specific counter-example using matrix rings over polynomial rings in characteristic $p$, showing that a property holding for $A/B$ fails for the Morita equivalent $A'/B'$.

3. Key Contributions and Results

The paper establishes the Morita invariance for five distinct classes of ring extensions and provides a counter-example for a specific property-based class.

A. Proven Morita Invariant Classes

The author proves that the following classes are Morita invariant:

1. Trivial Extensions:

   • Definition: $A = B \oplus S$ where multiplication is $(b, s)(c, t) = (bc, bt + sc)$.
   • Result: If $A/B$ is trivial, $A'/B'$ is trivial. The proof relies on showing the direct sum decomposition and the specific multiplication rule are preserved under the Morita isomorphism.

2. Liberal Extensions:

   • Definition: There exists a finite set $\{v_i\} \subseteq V_A(B)$ (centralizer of $B$ in $A$) such that $A = \sum v_i B$.
   • Result: The generating set $\{v_i\}$ maps to a corresponding set $\{v'_i\}$ in $V_{A'}(B')$ that generates $A'$ over $B'$.

3. Depth Two Extensions (Left and Right):

   • Definition: Characterized by the existence of elements $t_i \in (A \otimes_B A)_B$ and $\beta_i \in \text{End}_\ell(BAB)$ satisfying $\sum t_i \beta_i(x)y = x \otimes y$.
   • Result: Using the isomorphism $\psi$ on the tensor product and $\phi$ on the endomorphism ring, the author shows the defining relation holds for $A'/B'$.

4. Strongly Separable Extensions:

   • Definition: A generalization of Hirata separable extensions where $V_A(B)$ is finitely generated projective over the center, and a specific splitting map exists. Equivalently characterized by the existence of $v_i \in V_A(B)$ and $\sum x_{ir} \otimes y_{ir} \in (A \otimes_B A)_A$ such that $u = \sum v_i x_{ir} u y_{ir}$ for all $u \in V_A(B)$.
   • Result: The author explicitly constructs the corresponding elements in $A'$ and $A' \otimes_{B'} A'$ to satisfy the characterization theorem (Proposition 1.2).

5. Weakly Separable Extensions:

   • Definition: Every $B$-derivation $D: A \to A$ is inner (i.e., $D(x) = vx - xv$).
   • Result: The paper proves that the isomorphism $\phi$ induces a bijection between the derivation spaces $\text{Der}_B(A, A)$ and $\text{Der}_{B'}(A', A')$. If every derivation in $A$ is inner, the mapped derivation in $A'$ is also inner.

B. Non-Invariant Class (Counter-Example)

The paper provides a specific example of a class that is not Morita invariant:

• Property: The class of extensions where there exists a positive integer $n$ such that $\{x^n \mid x \in A\} \subseteq B$.
• Example: Let $k$ be a field of characteristic $p$. Let $A = k[t]$ and $B = k[t^p]$. Here, $x^p \in B$ for all $x \in A$.
• Morita Equivalent Pair: Construct $A' = M_2(A)$ and $B' = M_2(B)$.
• Failure: While $A/B$ satisfies the property, $A'/B'$ does not. Specifically, the matrix $\begin{pmatrix} 1 & t \\ 0 & 0 \end{pmatrix} \in A'$ raised to any power $n$ does not necessarily land in $B'$. This demonstrates that the property of "power containment" is not preserved under Morita equivalence.

4. Significance

• Unification of Theory: The paper unifies the understanding of Morita invariance across a broad spectrum of ring extension types, moving beyond the previously known Galois and Frobenius cases.
• Methodological Framework: It establishes a robust framework (using the maps $\phi, \psi, \theta$) for testing Morita invariance of new classes. This allows researchers to verify invariance by checking if the defining structural elements (idempotents, derivations, generators) map correctly.
• Clarification of Limits: By providing the counter-example, the author clarifies the boundaries of Morita invariance, showing that not all algebraic properties related to powers or specific subring inclusions are preserved.
• Open Questions: The paper highlights open problems, specifically regarding the invariance of quasi-separable, weakly quasi-separable, and finite normalizing extensions, inviting further research in these areas.

In summary, Yamanaka's work significantly advances the structural theory of ring extensions by rigorously proving that several important generalizations of separability and depth are intrinsic to the Morita equivalence class, while simultaneously delineating the limits of this invariance.