Idempotents and Krull–Schmidt categories

Let $\mathcal C$ be an additive category. An endomorphism $e\colon X\to X$ is idempotent if $e^2=e$. It splits if there are an object $Y$ and morphisms

\[Y\xrightarrow{i}X\xrightarrow{p}Y\]

such that

\[\label{eq:idempotent-splitting} p\circ i=\operatorname{id}_Y, \qquad i\circ p=e.\]

An additive category is Karoubian, or idempotent complete, if every idempotent splits. In an abelian category this is automatic: one may take $Y=\operatorname{im}(e)$.

Idempotents encode direct-sum decompositions. If

\[\label{eq:direct-sum-components} X\cong X_1\oplus\cdots\oplus X_n\]

has inclusions $i_r\colon X_r\to X$ and projections $p_r\colon X\to X_r$, then

\[\label{eq:orthogonal-idempotents} p_r\circ i_r=\operatorname{id}_{X_r}, \qquad p_r\circ i_s=0\quad(r\ne s), \qquad \operatorname{id}_X=\sum_{r=1}^n i_r\circ p_r.\]

Consequently, $e_r=i_r\circ p_r$ are pairwise orthogonal idempotents whose sum is $\operatorname{id}_X$. Conversely, splitting any such family recovers the direct sum in equation \eqref{eq:direct-sum-components}. Decomposing an object into indecomposable summands therefore amounts to decomposing its identity into primitive pairwise orthogonal idempotents.

An additive category is a Krull–Schmidt category if every object is a finite direct sum of objects with local endomorphism rings. These summands are indecomposable, and the decomposition is unique up to permutation and isomorphism. In this setting an object is indecomposable precisely when its endomorphism ring is local. An additive category with split idempotents and semiperfect endomorphism rings is Krull–Schmidt (Krause, 2015; Corollary 4.4). In particular, a Hom-finite $k$-linear additive category is Krull–Schmidt precisely when it is idempotent complete. Every locally finite abelian category is therefore Krull–Schmidt; compare (Etingof et al., 2015; Definition 1.8.1 and the paragraph following it). In particular, $\operatorname{Rep}_k(G)$ is Krull–Schmidt for every finite group $G$ and every field $k$.

The number of indecomposable isomorphism classes is a separate finiteness question. A finite-dimensional $k$-algebra has finite representation type if it has only finitely many isomorphism classes of finite-dimensional indecomposable modules; the same terminology is used for its representation category. A finite abelian category has only finitely many simple isomorphism classes, but it may have infinitely many indecomposable ones.

Every finite cyclic group has finite representation type over every field. In characteristic zero this follows from Maschke's theorem. In characteristic $p>0$ it follows from Higman's theorem, which more generally states that $\operatorname{Rep}_k(G)$ has finite representation type precisely when the Sylow $p$-subgroups of $G$ are cyclic (Higman, 1954). The Klein four group in characteristic $2$ therefore has infinitely many indecomposable representations.

For an arbitrary additive category $\mathcal C$, its Karoubi envelope formally adjoins objects $(X,e)$ for idempotents $e\in\operatorname{End}_{\mathcal C}(X)$; a morphism $(X,e)\to(Y,d)$ is a morphism $f\colon X\to Y$ satisfying $f=d\circ f\circ e$. The Karoubi envelope is Karoubian and contains $\mathcal C$ as a full subcategory.

Constructing it effectively can nevertheless be difficult. One must compute idempotents in endomorphism algebras, find their primitive decompositions, construct their images as objects, and recognize isomorphic summands. These are category-dependent algorithmic problems.

The interface

As with simple-object computations, determining indecomposability and finding a decomposition are generally difficult, category-specific problems. An implementation may provide:

The function karoubian_envelope(C) requires a category-specific construction; the abstract interface cannot manufacture the necessary idempotents and their images.

Over a finite field, the generic decomposition backend forms $A=\operatorname{End}_{\mathcal C}(X)$ and decomposes the regular right $A$-module. Projecting $1_A$ onto its indecomposable summands gives primitive idempotents in $A$, and their images give the indecomposable summands of $X$. Finite-group representations have a specialized implementation; see the representation-category inventory page for its scope and backend choices.

Scalar extension can create new idempotents in endomorphism algebras and hence new decompositions of formerly indecomposable objects. This is one reason that the scalar-extension construction generally includes a Karoubi envelope.

Example: Modular group representations

Let $G=C_5$ and $k=\mathbb F_5$. This category has one simple isomorphism class, while the unipotent Jordan blocks $J_r$, for $1\leq r\leq5$, are its five indecomposable isomorphism classes. The two-dimensional block $J_2$ is not simple, and $\operatorname{End}(J_2)\cong k[t]/(t^2)$ is local.

using TensorCategories, Oscar

function jordan_representation(C, n)
    J = identity_matrix(base_ring(C), n)
    for i in 1:n-1
        J[i, i+1] = 1
    end
    Representation(C, gens(base_group(C)), [J])
end

C = representation_category(GF(5), cyclic_group(5))
J1 = jordan_representation(C, 1)
J2 = jordan_representation(C, 2)
@assert is_indecomposable(J2) && !is_simple(J2)

X, inclusions, projections = direct_sum(J1, J2)
e1 = inclusions[1] ∘ projections[1]
e2 = inclusions[2] ∘ projections[2]
@assert e1 ∘ e1 == e1 && e2 ∘ e2 == e2
@assert e1 ∘ e2 == zero_morphism(X, X)
@assert e2 ∘ e1 == zero_morphism(X, X)
@assert e1 + e2 == id(X)

sort([(int_dim(Y), m) for (Y, m) in decompose(X)])

2-element Vector{Tuple{Int64, Int64}}:
 (1, 1)
 (2, 1)

The two primitive idempotents split $X$ into summands of dimensions $1$ and $2$, and decompose(X) returns [(1,1),(2,1)] after recording each summand by its dimension and multiplicity. This is a direct-sum decomposition into indecomposables; it is different from the composition series of $J_2$ discussed in the preceding section.

Continue with finite and semisimple categories.