# What is a “block” in an abelian category?

In the literature and in some posts here, there has been variation in the undefined use of the term "block" for a category of modules over a ring, or more abstractly an abelian category (all of which are categories of modules by Freyd-Mitchell). This raises the natural question:

What is meant by a "block" in an abelian category?

The concept originates to some extent in the modular representation theory of finite groups or their group algebras pioneered by Richard Brauer. Here a block is just an indecomposable two-sided ideal of the group algebra, corresponding to a primitive central idempotent. But in later developments the language of homological algebra plays a greater role than the group algebra or its center: the category of modules decomposes into a direct sum of subcategories, which are as small as possible relative to permitting no nontrivial extensions among their simple objects (irreducible representations). This approach generalizes well to other situations, where a center or central characters may be elusive and where the decomposition may be infinite, etc. By now "blocks" occur in many areas of representation theory influenced by classical Lie theory: algebraic groups, restricted enveloping algebras, quantum analogues, finite $W$-algebras, Cherednik algebras, Kac-Moody algebras and groups, Lie superalgebras.

There is some inconsistency in the literature about allowing "blocks" which might be further decomposed into direct sums. In classical or Kac-Moody Lie theory this usually reflects the special influence of infinitesimal/central characters. But full centers in Lie theory and related quantum groups may be unknown or unneeded, e.g., the approach Kac took to his analogue of the Weyl character formula for integrable modules of an affine Lie algebra relied just on a single Casimir-type operator (which works equally well in the classical finite-dimensional case). In Jantzen's book Representations of Algebraic Groups (AMS, 2003), the discussion of blocks for algebraic group schemes in II.7.1 is careful but not completely general.

In practice looser definitions of "block" than the homological one work well enough in many settings, but it creates some confusion when the word is used with no definition at all. Is there a single convention which reduces in familiar cases to older usage? In a recent comment to another post I paraphrased Humpty Dumpty ("my remote ancestor"), who actually said: "When I use a word it means just what I choose it to mean --- neither more nor less." But communication is better when the short and convenient word "block" starts out with a common meaning.

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By "module category" do you mean "the category of modules over a ring"? This isn't the usual definition of a "module category" so I'm going to edit your question accordingly. Please feel free to revert and clarify if I've messed up your question. –  Noah Snyder Apr 24 '10 at 0:05
I think this is a good question, but as written I think it is too discussion-y/argumentative? Rarely do I think a question can be improved by writing less: maybe it would be better to more clearly separate your (very good) question from your thoughts on it and other motivation? –  Theo Johnson-Freyd Apr 24 '10 at 6:08
Oh, Jim, did you intend to make this question CW? You've tagged it as such, but didn't click the button, which is something only you and moderators can do. –  Theo Johnson-Freyd Apr 24 '10 at 6:11
Jim, For Kac-Moody algebra, is it enough to show semisimplicity of integrable representation by only using Casimir operator? –  xiyu Apr 24 '10 at 9:32
@Noah: The examples I'm thinking about arise in various branches of representation theory, but "abelian category" is a good setting in view of Freyd-Mitchell embedding and recent work in other areas. –  Jim Humphreys Apr 24 '10 at 12:58
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Here's a definition of blocks taken from Comes-Ostrik (which just happened to be the first paper that came to mind that I knew talks about blocks, it's not a standard reference for this):

Let A denote an arbitrary F-linear category. Consider the weakest equivalence relation on the set of isomorphism classes of indecomposable objects in A where two indecomposable objects are equivalent whenever there exists a nonzero morphism between them. We call the equivalence classes in this relation blocks. We will also use the term block to refer to a full subcategory of A generated by the indecomposable objects in a single block.

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It's hard to say what is a "standard reference", which partly motivates my question. The approach of Comes-Ostrik seems reasonable, leaving aside their underlying field of characteristic 0. They emphasize indecomposable objects and morphisms rather than simple objects and extensions; both versions amount to the same thing in familiar examples, but their version may be more flexible. –  Jim Humphreys Apr 24 '10 at 13:17
Jim, we gave this definition for the same reason you are asking this question: different people mean different things when they are talking about blocks. The characteristic 0 assumption is irrelevant for this definition (but it is very important for other parts of our paper). Our category in question was not abelian, so we had no option to talk about simple objects. And I totally agree with Torsten: this definition is not very reasonable for categories where Krull-Schmidt theorem fails. –  Victor Ostrik Apr 24 '10 at 16:09
@Victor: Thanks for the clarification. It's good to have a notion of block in a general situation, but always with the understanding that (for instance) finiteness conditions on the category may lead to better results. For me it's been confusing to encounter inconsistent uses of the label "block", with some meanings looser than others. –  Jim Humphreys Apr 25 '10 at 13:21
@Noah: I prefer the generality of the Comes-Ostrik viewpoint, as explained in my comment to Torsten. My question was motivated by the unsolved problem of determining blocks for the parabolic subcategories of the BGG category, if "block" is defined in a general way. It's tempting to solve such a problem just by giving a definition to fit the situation. Given the finiteness properties in the BGG case one wants a parabolic subcategory to be a direct sum of indecomposable subcategories fitting a general notion of block. Describing those subcategories may be tricky. –  Jim Humphreys Apr 28 '10 at 22:18

It seems clear to me that blocks should have something to do with the decomposition of the category as a direct product of subcategories. A decomposition into a product of two factors corresponds exactly to an idempotent in the center of the category (recall that the center of an abelian category is the ring of natural transformations of the identity, it equals the center of a ring when the category is the category of all modules over the ring). Hence, one could define a block to be such an idempotent and a primitive block to be a primitive idempotent. Thus a block is a subcategory that is a direct factor and a primitive block is an indecomposable such direct factor.

I think that if one wants a completely general definition this may be the only way to go as there are situations where there are lots of idempotents but no primitive idempotents. (Consider for instance modules over the Boolean algebra of subsets of an infinite set modulo the finite subsets.) In particular I don't think that the definition quoted by Noah would be suitable in the case when there are no indecomposables. (If every object is a sum of indecomposables I think the definition gives what I propose to call primitive blocks.)

A comment on the case of Lie algebra representations. The enveloping algebra of a Lie algebra of course contains no non-trivial idempotents and thus neither does its center. What one does however is to look at various subcategories where (some?) elements of the center have generalised eigenspace decompositions. This introduces idempotents in the centre of the category (which actually comes from idempotents in some suitable completion of the center of the enveloping algebra).

[Added] When every object in the category has finite length blocks are in bijection with subclasses $S$ of the class of simple objects closed under the relation of having non-trivial extensions (in either direction). Indeed, the only non-trivial part is to show that any object is the direct sum of one object all of whose Jordan-Hölder factors are in $S$ and another one none of whose Jordan-Hölder factors are in $S$. If $M$ is an object all of whose Jordan-Hölder factors are in $S$ and $P$ is a simple module not in $S$, then all extensions of $M$ by $S$ are trivial (as is shown by induction over the length of $M$) and the same for $M$ having no Jordan-Hölder factor in $S$ and $P$ being in $S$. The splitting of an object as such a direct sum is now done by induction over the length of the object.

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In your definition, two indecomposable objects lie in the same block if there is no nonsplit extension; in the definition quoted by Noah, two indecomposable objects are in the same block if there is a nonzero morphism from the one to the other. It seems the two definitions are very different, so when and why they coincide? –  xiyu Apr 24 '10 at 9:27
I do not see that what you say is true. Note that in the case of finite length categories two modules in different blocks have no common simple for their Jordan-Hölder factors so clearly there are no non-zero morphisms between them. –  Torsten Ekedahl Apr 24 '10 at 13:21
@Torsten: Your helpful added comment is in the spirit of Jantzen's discussion, using finiteness conditions. I'd like to avoid idempotents or centers, which can make it easier (and more interesting) in some special cases to find block decompositions and describe the blocks. Without indecomposables the whole notion of "block" does lose interest. Still I'm inclined to start with the most general language and then verify in special cases that the category is a direct sum of blocks even if they are hard to classify. I'd prefer not to solve that problem by ad hoc definition. –  Jim Humphreys Apr 28 '10 at 22:06

The classic situation in which blocks are defined is for an artinian ring $A$. In this case, writing $1=e_1+\cdots+e_n$ as a sum of central primitive idempotents, the blocks are the (two-sided) ideals $A_i=Ae_i$. For each non-zero indecomposable module $M$, there is a unique $1 \leq i \leq n$ for which $e_iM \neq 0$, and in this case $M$ belongs to the block $A_i=eA_i$. In particular this partitions the set of irreducible $A$-modules into blocks (more categorically, decomposes $A$-mod as a direct sum of $A_i$-mod's).

There are (at least) two ways in which one might try to generalize this setup:

(1) Weaken the Artinian hypothesis.

(2) Weaken the hypothesis that we are looking at the category of modules for a ring.

It looks to me like (2) is treated in Torsten's answer, but it looks like no one has mentioned the (by now standard, at least for Noetherian ring theorists) answer for (1): here the replacement for blocks is "cliques".

Suppose $R$ is a Noetherian ring. "Ideal" will mean two-sided unless otherwise specified. Prime ideals $P$ and $Q$ of $R$ are linked if there is a an ideal $I$ with $QP \subseteq I \subset P \cap Q$, and $P \cap Q/I$ non-zero, torsion free as a right $R/P$-module and as a left $R/Q$-module. The cliques of $R$ are the connected components of the graph with vertex set $\mathrm{Spec}(R)$ and with edges given by links. (The graph is really directed b/c of the assymetry in the definition: really it should be an arrow from $Q$ to $P$ but for defining cliques this makes no difference).

To make the definition a little more intuitive, assume $R$ is an algebra over a field $F$, that $M$ and $N$ are irreducible $R$-modules which are $F$-finite dimensional, and that $P$ and $Q$ are the annihilators of $M$ and $N$. Then it should be a not-too-hard exercise to check that $P$ and $Q$ are linked iff there is a non-trivial extension between $M$ and $N$.

In an Artinian ring, the cliques are the same thing as the blocks (see e.g. pages 142 to 144 of Jategaonkar's book "Localization in Noetherian Rings"). For Noetherian ring theory, the original purpose of introducing cliques was to figure out which multiplicative subsets were "good" for localizing (K. Brown's survey "Ore Sets in Noetherian Rings" is readable and discusses this motivation).

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