Lemma 7.4 and Proposition 7.5 for a representation-finite algebra #
lem:integer-weight-lifting and prop:primitive-parameter-characters
(weights-and-parameters.tex), for the algebra A₀ of a basic representation-finite
algebra A, connected as in the source (IsConnectedRing: only the central idempotents 0
and 1). Connectedness passes to A₀ (a0Conn_ar), which gives the lifted walks from a base
projective used in the proofs (hK_of_A0Conn; the source cites [ARS, Theorem VII.2.1] for the
connectedness of Γ_A). Every mesh of Γ_A has a nonzero middle term (ar_hmid, from the
exactness of the mesh data).
Every mesh of Γ_A has a nonzero middle term.
A connected basic algebra has a connected A₀: the splitting of A over A₀
(Lemma 6.4 for the radical filtration, posFiltA) carries the vertex idempotents of A₀ to a
complete set of nonzero orthogonal idempotents of A and their corners to corners, so the
corner graph of A₀ is connected (eqvGen_corner).
Lemma 7.4 (lem:integer-weight-lifting) for A₀ = kP of a basic connected A: a
mesh weight restricts to a ray weight (MeshWt.res), and
0 → {ζ : ζ_P = 0} → H^mesh → H → 0 is exact (with values in any abelian group M):
every ray weight is a restriction; a mesh weight restricts to zero exactly when it is
w(a) = ζ_{target a} − ζ_{source a}, ν_v = ζ_v − ζ_{τv} for object values vanishing at the
projectives; and such object values are determined by their differences.
Proposition 7.5 (prop:primitive-parameter-characters) for A₀ = kP of A with
basic and connected A, in characteristic two: for rows p_i at v_i as in Proposition 6.1(2) (at
periodic vertices, one per component, spanning the cokernels), the residual characters
χ_i = w(p_i) − ν_{v_i} of the universal mesh weight w (restricting to the identity of
L = W(P), Lemma 7.4) are part of an integer basis of W(P).