Scalar extension and splitting

Computations take place over a fixed coefficient field, and for exact computations it is usually preferable to choose this field as small as possible. An object which is indecomposable or simple over the chosen field may decompose after extending the field. This non-split phenomenon is not merely a technical complication: it records arithmetic structure which is invisible over an algebraically closed field. Scalar extension and splitting make this structure accessible.

Scalar extension

Let $\mathcal C$ be a $k$-linear category and let $\iota\colon k\hookrightarrow L$ be a field embedding. The Hom-space extension $\mathcal C\otimes_k^{\mathrm{Hom}}L$ has the same objects as $\mathcal C$ and morphism spaces

\[\label{eq:hom-space-scalar-extension} \operatorname{Hom}_{\mathcal C\otimes_k^{\mathrm{Hom}}L}(X,Y) = \operatorname{Hom}_{\mathcal C}(X,Y)\otimes_{k,\iota}L.\]

Composition is extended $L$-bilinearly. There is a canonical $k$-linear functor

\[\label{eq:hom-space-extension-functor} T^{\mathrm{Hom}}_{L/k}\colon\mathcal C \longrightarrow\mathcal C\otimes_k^{\mathrm{Hom}}L, \qquad X\longmapsto X, \qquad f\longmapsto f\otimes1.\]

If $\mathcal C$ is additive, then its Hom-space extension is additive and $L$-linear. However, when $\mathcal C$ is Karoubian, its Hom-space extension need not be Karoubian: an algebra $\operatorname{End}_{\mathcal C}(X)\otimes_kL$ may contain idempotents whose images are not represented by the old objects. We therefore define the scalar extension of an idempotent-complete additive category to be the Karoubian closure

\[\label{eq:completed-scalar-extension} \mathcal C_L = \operatorname{Kar}\!\left(\mathcal C\otimes_k^{\mathrm{Hom}}L\right).\]

The Karoubian closure was described in the previous section. Some sources write $\mathcal C\otimes_kL$ for this completed category; others use that notation first for the Hom-space extension and mention the Karoubi envelope separately (Morrison and Snyder, 2012; §2.1).

If $\mathcal C$ is a Hom-finite Krull–Schmidt category, then $\mathcal C_L$ is again Hom-finite and idempotent complete, hence Krull–Schmidt (Krause, 2015; Corollary 4.4). If $\mathcal C$ is finite semisimple and $k$ is perfect, then $\mathcal C_L$ is again finite semisimple. Indeed, over a perfect field every finite-dimensional division algebra is separable over $k$, and a separable algebra remains semisimple after any field extension. More generally, it is enough that the endomorphism algebras of the simple objects are separable over $k$ (Etingof and Gelaki, 2012; §3.1, footnote 1).

Absolute indecomposability and splitting

Let $\mathcal C$ now be Hom-finite and Krull–Schmidt. If $X$ is indecomposable, then $E_X=\operatorname{End}_{\mathcal C}(X)$ is local. Its residue algebra

\[\label{eq:residue-endomorphism-division-algebra} D_X=E_X/\operatorname{rad}(E_X)\]

is a division algebra. After extending scalars, $E_X\otimes_kL$ can acquire nontrivial idempotents, and $X_L$ can therefore decompose. Over a separable extension, this decomposition is governed by the primitive idempotents of $D_X\otimes_kL$, which lift to $E_X\otimes_kL$.

An indecomposable object is absolutely indecomposable if it remains indecomposable after extension to an algebraic closure. To split an object or a finite family means to choose an extension over which all resulting indecomposable summands are absolutely indecomposable.

For a finite semisimple category, the indecomposable objects are precisely the simple objects. A simple object is absolutely simple if it remains simple after every field extension, equivalently after extension to an algebraic closure. By Schur's lemma,

\[\label{eq:simple-endomorphism-division-algebra} D_S=\operatorname{End}_{\mathcal C}(S)\]

is a finite-dimensional division algebra for every simple $S$. The category is split over $k$ if the canonical map

\[\label{eq:split-simple-condition} k\longrightarrow D_S\]

is an isomorphism for every simple $S$. This is equivalent to every simple object being absolutely simple: after scalar extension, $\operatorname{End}(S_L)=D_S\otimes_kL$. A field $L$ is a splitting field for $\mathcal C$ if $\mathcal C_L$ is split; compare (Etingof et al., 2015; §4.16).

If $\mathcal C$ is finite split semisimple, then its Hom-space extension is already idempotent complete, so the Karoubi envelope introduces no new objects. When scalar extension remains semisimple, extending a simple object $S$ amounts to finding primitive idempotents in $D_S\otimes_kL$ and taking their images. Every simple object of $\mathcal C_L$ arises in this way from a simple object of $\mathcal C$.

The interface

Computing a splitting field is a category-dependent problem. It can require endomorphism algebras, their radicals and primitive idempotents, images of idempotents, and decompositions of the resulting objects. A user implementing a new category must provide the necessary operations, or a specialized split method, when such algorithms are available.

Over finite fields, the generic methods split(X) and split(objects) compute the required extension degree from the residue endomorphism algebras, choose one extension for the specified object or family, and return the extended objects together with their decompositions. They work for $\operatorname{Rep}_k(G)$ and for any other model which provides the required endomorphism, decomposition, and scalar-extension operations. The resulting summands are absolutely indecomposable; in a nonsemisimple category they need not be simple. A category-level split(C) remains model-dependent.

The predicate is_split_semisimple(C) first tests semisimplicity and then checks that every enumerated simple object has one-dimensional endomorphism algebra over the base field. It therefore depends on the implementation's ability to enumerate simple objects.

The two-dimensional simple representation of $C_3$ over $\mathbb F_2$ provides a small example. Its endomorphism field is $\mathbb F_4$. After extension to $\mathbb F_4$,

\[\mathbb F_4\otimes_{\mathbb F_2}\mathbb F_4 \cong \mathbb F_4\times\mathbb F_4.\]

The two primitive idempotents give the two nontrivial one-dimensional characters.

using TensorCategories, Oscar
G = cyclic_group(3)
C = representation_category(GF(2), G)
V = only(filter(X -> int_dim(X) == 2, simples(C)))

result = split(V)
summands = only(result.decompositions)

@assert result.extension_degree == 2
@assert order(result.field) == 4
@assert length(summands) == 2
@assert all(m == 1 && int_dim(X) == 1 for (X, m) in summands)
summands

2-element Vector{Tuple{GroupRepresentation, Int64}}:
 (1-dimensional group representation over Finite field of degree 2 and characteristic 2 of Pc group of order 3), 1)
 (1-dimensional group representation over Finite field of degree 2 and characteristic 2 of Pc group of order 3), 1)

Continue with functors and natural transformations.