What is a "block" in an abelian category?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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).

Answer 4

In a talk I went to today on Lie superalgebras, the word "block" was defined as follows. Consider the graph whose vertices are (isomorphism classes of) simple objects in your category, and draw an arrow between any two objects if there is a non-split extension between them. Then the blocks are the connected components of this graph.

I don't know how standard this definition is, nor have I thought about whether it agrees with other definitions proposed above.