# In modular representation theory, what is a block?

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 $$$$\displaystyle R=\bigoplus_{i=1}^r B_i$$$$

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.

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!