A genus question for orders

Matthias Kuenzer

General  Example

R : principal ideal domain

R := Z

p ∈ R : prime  p := 5

Λ ⊆ R × ⋯ × R =: Γ : orders over R such that • Λ_{(p)} is local

• p^{m}Γ ⊆ Λ ⊆ Γ for some m ⩾ 0

Λ := \underbrace{\{ (z, z′) ∈ Z × Z : z ≡_{5} z′ \}}_{“_{Z _{5} Z} ”} ⊆ Z × Z =: Γ

Construct

α = (α_{i})_{i}

[figure]

Γ_{(p)}^{∗}

[figure]

normalised idèles \overbrace{R_{(p)}^{∗} × ⋯ × R_{(p)}^{∗}}

ᾱ Γ_{(p)}^{∗}/Λ_{(p)}^{∗}Γ^{∗} : \underbrace{Cl(Λ)}_{idèle class group} Calculate

Cl(Λ)

= \frac{Z_{(5)}^{∗} × Z_{(5)}^{∗}}{\{ (z, z′) ∈ Z_{(5)}^{∗}×Z_{(5)}^{∗} : z ≡_{5} z′ \} · (\{±1\}×\{±1\})}
= ⟨ \overline{(2, 1)} ⟩ ≃ C_{2} ≄ 1 ,

whence Krull-Schmidt fails in Λ -proj. Therefore, Λ being a certain endomorphism ring, it also fails in ZS_{5} -mod – as is well-known.

To α ∈ Γ_{(p)}^{∗}, we attach the projective indecomposable Λ-module

Λ(α) := { (x_{i})_{i} ∈ Γ : (x_{i}α_{i}^{−1})_{i} ∈ Λ_{(p)} } .

• All projective indecomposable Λ-modules are of this form.

• Λ(α) ≃ Λ(β) ⇔ ᾱ = β̄

• Hom_{Λ}(Λ(α), Λ(β)) ≃ Λ(α^{−1}β)

• As Λ_{(p)}-modules, we have Λ(α)_{(p)} ≃ Λ_{(p)} .

To α := (2, 1), 1 = (1, 1) ∈ Z_{(5)}^{∗} × Z_{(5)}^{∗} = Γ_{(p)}^{∗}, we attach

\begin{aligned}Λ(α) & = & \{ (z, z′) ∈ Z × Z : −2z ≡_{5} z′ \} \\ Λ(1) & = & \{ (z, z′) ∈ Z × Z : z ≡_{5} z′ \} = Λ . \end{aligned}

A ring Morita-equivalent to Λ is of the form

Ξ_{α_{1} ,..., α_{t}} := End_{Λ}(Λ(α_{1}) ⊕ ⋯ ⊕ Λ(α_{t})) for some t ⩾ 1 and some α_{i} ∈ Γ_{(p)}^{∗} for i ∈ [1, t]. E.g. Ξ_{1 ,..., 1} ≃ Λ^{t×t} . Note that

(Ξ_{α_{1} ,..., α_{t}})_{(p)} ≃ (Ξ_{1 ,..., 1})_{(p)} We obtain

Ξ_{α,1}

= End_{Λ}(Λ(α) ⊕ Λ(1))
≃ \{ ((\begin{matrix}a & b \\ c & d \end{matrix}) , (\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix})) ∈ Z^{2×2}×Z^{2×2} : \begin{matrix}a ≡_{5} a′, & 2b ≡_{5} b′, \\ −2c ≡_{5} c′, & d ≡_{5} d′ \end{matrix} \}
Ξ_{1,1}
= End_{Λ}(Λ(1) ⊕ Λ(1))
≃ \{ ((\begin{matrix}a & b \\ c & d \end{matrix}) , (\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix})) ∈ Z^{2×2}×Z^{2×2} : \begin{matrix}a ≡_{5} a′, & b ≡_{5} b′, \\ c ≡_{5} c′, & d ≡_{5} d′ \end{matrix} \} .

General (cont.)

Example (cont.)

Write Cl(Λ)^{t} := { ᾱ^{t} : ᾱ ∈ Cl(Λ) } ⩽ Cl(Λ). Suppose given t ⩾ 1 and α_{i} , β_{i} ∈ Γ_{(p)}^{∗} for i ∈ [1, t].

Proposition. We have

Ξ_{α_{1} ,..., α_{t}} ≃ Ξ_{β_{1} ,..., β_{t}}

as Λ-algebras if and only if

(ᾱ_{1} ⋯ ᾱ_{t})(β̄_{1} ⋯ β̄_{t})^{−1} ∈ Cl(Λ)^{t} .

Since (ᾱ · ¯1)(¯1 · ¯1)^{−1} = ᾱ ∉ \underbrace{Cl(Λ)^{2}}_{≃ 1} ⩽ \underbrace{Cl(Λ)}_{= ⟨ᾱ⟩ ≃ C_{2}},

we have

Ξ_{α,1} ≄ Ξ_{1,1} .

Since Cl(Λ)^{3} = Cl(Λ), we have

Ξ_{α,1,1} ≃ Ξ_{1,1,1} .

Since Cl(Λ)^{4} ≃ 1, we have

Ξ_{α,1,1,1} ≄ Ξ_{1,1,1,1} .

Thus we partition the Morita-equivalence class of Λ into isoclasses.

Representatives of the isoclasses within the Morita-equivalence class of Λ are given by

Ξ_{1} (= Λ), Ξ_{1,1} , Ξ_{α,1} , Ξ_{1,1,1} , Ξ_{1,1,1,1} , Ξ_{α,1,1,1} , …

Main ingredient :

Jacobinski’s Cancellation Theorem.

In this context it states that given P, Q finitely generated projective Λ-modules, we have the implication

P ⊕ Λ ≃ Q ⊕ Λ ⟹ P ≃ Q .

(Without Jacobinski, it is possible to give a complicated proof for the case t = 2.)

Motivation

1. Data R : principal ideal domain with char R = 0 G : finite group such that R is large enough for G π(G) : set of prime divisors of |G| p ∈ π(G) : prime divisor of |G|

2. Using simple RG-lattices for a Wedderburn isomorphism ω_{p}, we obtain

[figure: R_{(p)}G; KG; ∼; ω_{p}; ∏_{i} K^{n_{i}×n_{i}}; RG_{[p]}; Φ; ∼; ∏_{i} R^{n_{i}×n_{i}}; RG; injective]

A hook in a quadrangle indicates a pullback. Here RG_{[p]} := Φ ∩ R_{(p)}G is called the naive localisation of RG at p. It depends on the choice of the maximal order Φ, i.e. on the choice of RG-lattices used for the Wedderburn isomorphism ω_{p}.

3. We aim to find a “good” Wedderburn embedding ω_{q} : RG_{[q]} → ∏_{i} R^{n_{i}×n_{i}} for each q ∈ π(G) separately, using the same RG-lattices for all q ∈ π(G), but different R-linear bases (with base change matrices of determinant 1 between them). “Good” means that its image allows a simple description via congruences. Since SL_{n_{i}}(R) → ∏_{q∈π(G)} SL_{n_{i}}(R/q^{v_{q}(|G|)}R) is surjective, the various ω_{q} can be patched together to yield

RG ≃ ⋂_{q∈π(G)} ω_{q}(RG_{[q]}) ⊆ ∏_{i} R^{n_{i}×n_{i}}

4. Two RG_{[p]}-lattices are in the same genus if they are isomorphic after localisation at (p). Let 1_{RG_{[p]}} = e_{1}+⋯+e_{n} be an orthogonal decomposition into primitive idempotents in RG_{[p]} , ordered such that indecomposable projective modules RG_{[p]}e_{i} in the same genus occur in intervals. Say, RG_{[p]}e_{1} , …, RG_{[p]}e_{t} are in the same genus, for some t ∈ [1, n]. Assume that the decomposition numbers that occur for R_{(p)}Ge_{1} are all in {0, 1}. Let

Λ := e_{1}RG_{[p]}e_{1}

be the endomorphism ring of the indecomposable projective RG_{[p]}-module RG_{[p]}e_{1} . Now the proposition above allows to decide whether

Ξ := (e_{1} + ⋯ + e_{t})RG_{[p]}(e_{1} + ⋯ + e_{t}) ≃^{?} Λ^{t×t} ,

provided the left hand side is sufficiently explicitly known. An affirmative answer is “good” for our Wedderburn embedding ω_{p}.

5. Using Specht lattices and their duals for the Wedderburn isomorphisms, the answers were all affirmative for (ZS_{n})_{[q]} for n ∈ [1, 6], q ∈ π(S_{n}).

To illustrate, there exists an endomorphism ring of an indecomposable projective (ZS_{6})_{[2]}-module isomorphic to

Λ := { (a, b, c, d, e, f) ∈ Z^{×6} : a ≡_{2} e, a + d − 2f ≡_{16} b + c − 2e ≡_{8} 0, e − f ≡_{4} c − d ≡_{2} 0 } ,

having Cl(Λ) ≃ C_{4} .