In modular representation theory, what is a block?

Question

In modular representation theory, the blocks are defined to be indecomposable two-sided ideals of a ring $R$ with unity element and we can write \begin{equation} \displaystyle R=\bigoplus_{i=1}^r B_i \end{equation}

for the unique decomposition of $R$ into a direct sum of indecomposable nonzero ideals (the blocks) of $R$.

Another definition by Walter Feit is the following:

If $\{e_i\}$ is the set of all centrally primitive idempotents of a ring $R$, then a block $B=B(e)$ associated with $e$ is the set (Feit says the category) of all finitely generated $R$-modules $V$ satisfying $Ve=V$.

I can't see why the two definitions are equivalent. Are $R$-modules with the above condition and two-sided indecomposable ideals of a ring the same thing ? I thank you for any suggestions.

Answer 1

There is a duality between primitive central orthogonal idempotents and indecomposable nonzero ideals. Let $1$ be the unity element, then you can pick a decomposition $$1 = e_1 + \dots + e_n$$

where each $e_i$ is a:

• primitive ($e = f + g$ with $fg=0$ implies $f=0$ or $g=0$)
• orthogonal ($e_ie_j=0$ if $i \neq j$)
• central ($e \in Z(R)$)
• idempotent ($e_i^2 = e_i$)

Note that this is always possible, because $1=1$ is a decomposition into orthogonal central idempotents.

This induces a decomposition of $R$ as follows: $$R = Re_1 \oplus Re_2 \oplus \dots \oplus Re_n$$ and each $Re_i$ is a two-sided nonzero ideal - a block (note how $Re_i = e_iRe_i$). Now given an indecomposable $R$-module $V$, there is a decomposition $$V = V_1 \oplus V_2 \oplus \dots \oplus V_n$$

induced by the one of $R$. Since we took an indecomposable $V$, then only one $V_i$ is nonzero, and we say that $V$ belongs to the block $Re_i$. Note that then $Ve_i = V$, which is the definition Feit gives for modules to belong to a block.

In short, studying the category of $R$-modules and studying the category of $Re_i$-modules for each $i$ is the same thing, because you can decompose each $R$-module and focus on the individual summands, and each summand is in just one block.

A very good introduction to modular representation theory is Peter Webb's book!