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Orbit sums of primitive split central idempotents descend

Statement

Let $E / F$ be finite Galois with group $Γ$ , let $A$ be a finite-dimensional unital $F$ -algebra, and assume $B = E \otimes_{F} A$ is split semisimple. Its primitive central idempotents are the coordinate identities in a split matrix decomposition. They correspond bijectively to simple left $B$ -module classes by support. The action $σ_{B}$ permutes these idempotents, and $c$ supports $W$ if and only if $σ_{B} (c)$ supports $^{σ} W$ .

For every orbit $O$ of primitive central idempotents there is a unique nonzero central idempotent $e_{O} \in A$ with $1 \otimes e_{O} = c_{O} : = \sum_{c \in O} c .$ Distinct $e_{O}$ are orthogonal and their sum is $1_{A}$ . For $A = 0$ , both families are empty and this last equality means $0 = 1_{A}$ .

Facts & Assumptions

[F1]

Tensor algebra actions, twists, support and split semisimplicity are defined in Semilinear Galois actions, twists, and split central idempotents.

[F2]

Canonical fixed tensors are exactly $1 \otimes A$ , and $A \to E \otimes_{F} A$ is injective: Galois fixed points recover finite-dimensional scalar extensions.

[F3]

$Z (M_{n} (E)) = E I_{n}$ for $n \geq 1$ : The center of $M_{n} (k)$ consists of the scalar matrices.

[F4]

A finite nonempty product of full matrix rings over division rings has exactly one simple left-module class per factor, its column module: Simple modules over a product of matrix rings over division rings.

Proof

Given: $B ≅ \prod_{j = 1}^{r} M_{d_{j}} (E)$ with every $d_{j} \geq 1$ , allowing $r = 0$ .

1.1

For $r > 0$ , a central element has a scalar matrix in each coordinate, since commuting can be tested one coordinate at a time. The equation $λ^{2} = λ$ in the field $E$ forces $λ = 0$ or $1$ . Hence all central idempotents are unique subset sums of the coordinate identities $c_{1}, \dots, c_{r}$ . Such a nonzero sum is primitive exactly when its subset is a singleton: a larger subset splits into two nonempty subsets, whereas a singleton cannot.

F3algebra

2.1

The simple module for factor $j$ is $E^{d_{j}}$ , on which $c_{j}$ acts as identity and every other $c_{k}$ as zero. F4 says these give every class exactly once. An automorphism $σ_{B}$ preserves centrality, idempotence, nonzeroness and orthogonal splittings in both directions (use its inverse), so it permutes primitive central idempotents. The twist equation gives $σ_{B} (c) ⋆ w = c w$ . Thus identity action of $c$ on $W$ is equivalent to identity action of $σ_{B} (c)$ on $^{σ} W$ , proving both directions of the claimed compatibility.

F1F4step 1.1algebra

3.1

Each orbit $O$ is a nonempty subset of this finite family. The sum $c_{O}$ is nonzero, central and idempotent, since its terms are nonzero orthogonal coordinate identities. Every $σ_{B}$ permutes its terms, so it is fixed. F2 gives a unique $e_{O} \in A$ such that $1 \otimes e_{O} = c_{O}$ .

F2step 1.1step 2.1algebra

4.1

The identities $1 \otimes (e_{O}^{2} - e_{O}) = c_{O}^{2} - c_{O} = 0$ and $1 \otimes (e_{O} a - a e_{O}) = c_{O} (1 \otimes a) - (1 \otimes a) c_{O} = 0$ for every $a \in A$ imply idempotence and centrality by injectivity. Nonzeroness follows from $c_{O} \neq 0$ . Disjoint orbits have disjoint coordinate supports, so $c_{O} c_{O^{'}} = 0$ for $O \neq O^{'}$ , and all orbit sums add to $1_{B}$ . Injectivity reflects these two equalities to $A$ .

F1F2step 3.1algebra

5.1

If $r = 0$ , $B = 0$ and injectivity gives $A = 0$ . A unital module over the zero ring is zero because $w = 1 w = 0 w = 0$ , so there are no simple modules and no nonzero primitive central idempotents. The families are empty, with sum $0 = 1_{A}$ . If an orbit has one element, step 3.1 descends that element itself, with the same nonzeroness and uniqueness proof. [F2, step 3.1, step 4.1, algebra] QED

Remarks

The subset-of-factors argument combines the local matrix-center and simple-module results with Zheng, Theorem 3.8.1, pp.132–133. It does not assert that an individual simple module descends with multiplicity one.

Depends on

Used by

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Sources

Weizhe Zheng, Lectures on Algebra (10 January 2025) (standard reference, not scraped)
Gábor Wiese, Galois Representations (standard reference, not scraped)