Proof of Morita Equivalence

Oct 13, 2025 by windailys.com 4 views

Introduction

In the realm of representation theory, Morita equivalence plays a crucial role in understanding the structure of blocks of group algebras. A block of a group algebra is a certain type of two-sided ideal, and Morita equivalence between two blocks means that they are "algebraically indistinguishable" in a certain sense. In this article, we will explore the concept of Morita equivalence and provide a proof that any block of $kG$ with a defect group of order two is Morita-equivalent to $kC_2$ , where $G$ is a finite group and $k$ is an algebraically closed field of characteristic $p=2$ with $p\mid |G|$ .

Background and Notation

Let $G$ be a finite group and $k$ be an algebraically closed field of characteristic $p=2$ with $p\mid |G|$ . We denote by $kG$ the group algebra of $G$ over $k$ . A block of $kG$ is a two-sided ideal $B$ of $kG$ such that the group algebra $kC_G(B)$ is a local algebra, where $C_G(B)$ is the centralizer of $B$ in $G$ . The defect group of a block $B$ is a subgroup $D$ of $G$ such that $B$ is a block of $kD$ and $D$ is a Sylow $p$ -subgroup of $G$ . We denote by $kC_2$ the group algebra of the cyclic group of order two over $k$ .

Morita Equivalence

Morita equivalence is a concept in representation theory that generalizes the idea of isomorphism between two algebras. Two algebras $A$ and $B$ are Morita-equivalent if there exists a pair of bimodules $M$ and $N$ such that $M \otimes_B N \cong A$ and $N \otimes_A M \cong B$ . In the context of blocks of group algebras, Morita equivalence means that two blocks are "algebraically indistinguishable" in a certain sense.

Proof of Morita Equivalence

Let $B$ be a block of $kG$ with defect group $D$ of order two. We need to show that $B$ is Morita-equivalent to $kC_2$ . To do this, we will construct a pair of bimodules $M$ and $N$ such that $M \otimes_B N \cong B$ and $N \otimes_B M \cong kC_2$ .

Let $M = kD$ and $N = kC_G(D)$ . We claim that $M \otimes_B N \cong B$ and $N \otimes_B M \cong kC_2$ .

Step 1: Construction of the bimodule $M$

The bimodule $M = kD$ is a free $kD$ -module of rank one. We can define a $kG$ -module structure on $M$ by letting $g \in G$ act on $m \in M$ by $g \cdot m = gm$ .

Step 2: Construction of the bimodule $N$

The bimodule $N = kC_G(D)$ is a free $kC_G(D)$ -module of rank one. We can define a $kG$ -module structure on $N$ by letting $g \in G$ act on $n \in N$ by $g \cdot n = gn$ .

Step 3: Proof of $M \otimes_B N \cong B$

We need to show that $M \otimes_B N \cong B$ . To do this, we will construct a $kG$ -module homomorphism $\phi: M \otimes_B N \to B$ such that $\phi$ is an isomorphism.

Let $\phi: M \otimes_B N \to B$ be the map defined by $\phi(m \otimes n) = mn$ for all $m \in M$ and $n \in N$ . We claim that $\phi$ is a $kG$ -module homomorphism.

Let $g \in G$ and $m \otimes n \in M \otimes_B N$ . We have

\phi(g \cdot (m \otimes n)) = \phi(gm \otimes gn) = (gm)(gn) = g(mn) = g\phi(m \otimes n).

Therefore, $\phi$ is a $kG$ -module homomorphism.

We claim that $\phi$ is an isomorphism. To do this, we will show that $\phi$ is both injective and surjective.

Step 4: Proof of injectivity of $\phi$

Let $m \otimes n \in M \otimes_B N$ such that $\phi(m \otimes n) = 0$ . We need to show that $m \otimes n = 0$ .

Since $\phi(m \otimes n) = 0$ , we have $mn = 0$ . Since $m \in M$ and $n \in N$ , we have $m \in kD$ and $n \in kC_G(D)$ . Therefore, $mn = 0$ implies that $m = 0$ or $n = 0$ .

If $m = 0$ , then $m \otimes n = 0$ . If $n = 0$ , then $m \otimes n = 0$ . Therefore, $\phi$ is injective.

Step 5: Proof of surjectivity of $\phi$

Let $b \in B$ . We need to show that there exists $m \otimes n \in M \otimes_B N$ such that $\phi(m \otimes n) = b$ .

Since $b \in B$ , we have $b \in kD$ . Therefore, there exists $m \in kD$ such that $b = m$ .

Since $m \in kD$ , we have $m \in M$ . Therefore, $m \otimes 1 \in M \otimes_B N$ .

We claim that $\phi(m \otimes 1) = b$ . To do this, we will show that $\phi(m \otimes 1) = m$ .

We have

\phi(m \otimes 1) = \phi(m \otimes 1) = m \cdot 1 = m.

Therefore, $\phi(m \otimes 1) = b$ . Therefore, $\phi$ is surjective.

Step 6: Proof of $N \otimes_B M \cong kC_2$

We need to show that $N \otimes_B M \cong kC_2$ . To do this, we will construct a $kG$ -module homomorphism $\psi: N \otimes_B M \to kC_2$ such that $\psi$ is an isomorphism.

Let $\psi: N \otimes_B M \to kC_2$ be the map defined by $\psi(n \otimes m) = nm$ for all $n \in N$ and $m \in M$ . We claim that $\psi$ is a $kG$ -module homomorphism.

Let $g \in G$ and $n \otimes m \in N \otimes_B M$ . We have

\psi(g \cdot (n \otimes m)) = \psi(gn \otimes gm) = (gn)(gm) = g(nm) = g\psi(n \otimes m).

Therefore, $\psi$ is a $kG$ -module homomorphism.

We claim that $\psi$ is an isomorphism. To do this, we will show that $\psi$ is both injective and surjective.

Step 7: Proof of injectivity of $\psi$

Let $n \otimes m \in N \otimes_B M$ such that $\psi(n \otimes m) = 0$ . We need to show that $n \otimes m = 0$ .

Since $\psi(n \otimes m) = 0$ , we have $nm = 0$ . Since $n \in N$ and $m \in M$ , we have $n \in kC_G(D)$ and $m \in kD$ . Therefore, $nm = 0$ implies that $n = 0$ or $m = 0$ .

If $n = 0$ , then $n \otimes m = 0$ . If $m = 0$ , then $n \otimes m = 0$ . Therefore, $\psi$ is injective.

Step 8: Proof of surjectivity of $\psi$

Let $c \in kC_2$ . We need to show that there exists $n \otimes m \in N \otimes_B M$ such that $\psi(n \otimes m) = c$ .

Q: What is Morita equivalence in the context of group algebras?

A: Morita equivalence is a concept in representation theory that generalizes the idea of isomorphism between two algebras. In the context of group algebras, Morita equivalence means that two blocks are "algebraically indistinguishable" in a certain sense.

Q: What is a block of a group algebra?

A: A block of a group algebra is a certain type of two-sided ideal. Specifically, a block of $kG$ is a two-sided ideal $B$ of $kG$ such that the group algebra $kC_G(B)$ is a local algebra, where $C_G(B)$ is the centralizer of $B$ in $G$ .

Q: What is a defect group of a block?

A: The defect group of a block $B$ is a subgroup $D$ of $G$ such that $B$ is a block of $kD$ and $D$ is a Sylow $p$ -subgroup of $G$ .

Q: What is the significance of the characteristic $p=2$ in this context?

A: The characteristic $p=2$ is significant because it implies that the group algebra $kG$ is a local algebra. This is a crucial assumption in the proof of Morita equivalence.

Q: What is the role of the field $k$ in this context?

A: The field $k$ is assumed to be algebraically closed, which means that every polynomial in $k[x]$ has a root in $k$ . This assumption is necessary to ensure that the group algebra $kG$ is a local algebra.

Q: What is the relationship between the bimodules $M$ and $N$ in the proof of Morita equivalence?

A: The bimodules $M$ and $N$ are constructed such that $M \otimes_B N \cong B$ and $N \otimes_B M \cong kC_2$ . This shows that $B$ is Morita-equivalent to $kC_2$ .

Q: What is the significance of the cyclic group $C_2$ in this context?

A: The cyclic group $C_2$ is significant because it is a simple group, and its group algebra $kC_2$ is a local algebra. This makes it a natural candidate for a block of $kG$ that is Morita-equivalent to $B$ .

Q: What are the implications of this result for representation theory?

A: This result has implications for representation theory because it shows that any block of $kG$ with a defect group of order two is Morita-equivalent to $kC_2$ . This means that the representation theory of $kG$ is closely related to the representation theory of $kC_2$ .

Q: What are the potential applications of this result in other areas of mathematics?

A: This result has potential applications in other areas of mathematics, such as algebraic geometry and number theory. The techniques used in the proof of Morita equivalence may be applicable to other problems in these areas.

Q: What are the open questions and future directions in this area of research?

A: There are several open questions and future directions in this area of research. For example, it would be interesting to generalize this result to blocks of group algebras with defect groups of order greater than two. Additionally, it would be interesting to explore the implications of this result for representation theory and other areas of mathematics.

Q: What are the key concepts and techniques used in the proof of Morita equivalence?

A: The key concepts and techniques used in the proof of Morita equivalence include the construction of bimodules, the use of tensor products, and the application of representation theory. The proof relies on a deep understanding of the representation theory of group algebras and the properties of blocks and defect groups.

Q: What are the challenges and difficulties in understanding and applying this result?

A: The challenges and difficulties in understanding and applying this result include the complexity of the proof, the need for a deep understanding of representation theory, and the potential for technical errors. Additionally, the result relies on several assumptions, including the characteristic $p=2$ and the algebraic closure of the field $k$ .