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Algebraic consequences of the Farrell–Jones conjecture: bass_conjecture_integral_domain

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Statement

Bass conjecture for commutative integral domains, in idempotent-matrix form.

Let A be an idempotent matrix over R[G]. If the order of g is not invertible in R, then the Hattori–Stallings trace of the projective module represented by A vanishes at the conjugacy class of g. This includes every infinite-order g, whose orderOf is zero. This is Conjecture 0.6 of [BLR08]. By Theorems 0.5 (ii) and 0.7 of [BLR08] it follows from the Farrell–Jones conjecture with coefficients in every field of prime characteristic.

Lean statement

theorem bass_conjecture_integral_domain {R : Type*} [CommRing R] [IsDomain R] {G : Type*}
    [Group G] {n : Type*} [Fintype n] (A : Matrix n n (MonoidAlgebra R G))
    (hA : A * A = A) (g : G) (hg : ¬IsUnit (orderOf g : R)) :
    matrixHattoriStallingsTraceAt A g = 0 := by
  sorry

Sources(1)

Openness evidence

Tagged category research open in google-deepmind/formal-conjectures at commit a64e959c32c2. Definitions used by the statement are in FormalConjectures/Arxiv/math.0703548/FarrellJonesConsequences.lean.

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