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216fb31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 | import OAI.Algebra.Finitistic.Dimension.PDResidueSemisimple
import OAI.Algebra.Finitistic.Dimension.PDSimpleTensor
import OAI.Algebra.Finitistic.Dimension.PDLowerBound
import OAI.Algebra.Finitistic.Tensor.BalancedTensorMaps
namespace OAI
noncomputable section
namespace LittleFinitistic.PDTensor
open CategoryTheory CategoryTheory.Limits Submodule
universe u
variable {k A : Type u} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A]
/-- A nonzero finite module is detected by tensor with an actual simple right
submodule of A/rad A. This is an existence proof, not an assumed simple detector. -/
theorem exists_simple_tensor_nonzero
(N : Type u) [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
[Module.Finite A N] [Nontrivial N] :
∃ S : Submodule Aᵐᵒᵖ (Residue (A := A)), IsSimpleModule Aᵐᵒᵖ S ∧
¬ Subsingleton (BalancedTensor.Tensor k A S N) := by
have := residue_semisimple (k := k) (A := A)
by_contra h
have hS (S : Submodule Aᵐᵒᵖ (Residue (A := A))) (hs : IsSimpleModule Aᵐᵒᵖ S) :
Subsingleton (BalancedTensor.Tensor k A S N) := by
by_contra hn
exact h ⟨S, hs, hn⟩
let K : Submodule Aᵐᵒᵖ (Residue (A := A)) :=
{ carrier := {u | ∀ n : N, BalancedTensor.mk k A (Residue (A := A)) N u n=0}
zero_mem' := fun n => LinearMap.congr_fun (map_zero (BalancedTensor.mk k A (Residue (A := A)) N)) n
add_mem' := fun hu hv n => by
change (BalancedTensor.mk k A (Residue (A := A)) N ( _ + _)) n=0
rw [map_add, LinearMap.add_apply, hu, hv, add_zero]
smul_mem' := fun a u hu n => by
rw [show a = MulOpposite.op a.unop from rfl, BalancedTensor.balance]
exact hu _ }
have hK : (⊤ : Submodule Aᵐᵒᵖ (Residue (A := A))) ≤ K := by
rw [← IsSemisimpleModule.sSup_simples_eq_top Aᵐᵒᵖ (Residue (A := A))]
apply sSup_le
intro S hs u hu n
let f := BalancedTensor.mapLeft (k := k) N S.subtype
have hz : BalancedTensor.mk k A S N ⟨u,hu⟩ n = 0 := (hS S hs).elim _ _
have hz' := congrArg f hz
simpa [f] using hz'
have hz (z : BalancedTensor.Tensor k A (Residue (A := A)) N) : z=0 := by
induction z using BalancedTensor.induction_on with
| hz => rfl
| ht u n => exact hK mem_top n
| ha x y hx hy => rw [hx,hy,add_zero]
have := residueTensor_nontrivial (k := k) (A := A) N
obtain ⟨x,y,hxy⟩ := exists_pair_ne (BalancedTensor.Tensor k A (Residue (A := A)) N)
exact hxy ((hz x).trans (hz y).symm)
variable (M : PDMinimal.FiniteModule k A)
/-- The exact simple-module form of projective-dimension detection. The test
modules here are actual simple right submodules of the fixed radical quotient. -/
theorem pdLE_iff_simpleTor_zero (n : ℕ) :
HasProjectiveDimensionLE M.cat n ↔
∀ S : Submodule Aᵐᵒᵖ (Residue (A := A)), IsSimpleModule Aᵐᵒᵖ S →
IsZero (Tor (k := k) S M.cat (n+1)) := by
constructor
· intro h S _
exact tor_zero_of_pdLT S M (n+1) h
· intro h
apply (PDMinimal.pdLE_iff_term_zero M n).mpr
by_contra ht
have : Nontrivial (PDMinimal.term M (n+1)) := not_subsingleton_iff_nontrivial.mp ht
have : Nontrivial (ScalarModule k A (PDMinimal.term M (n+1)).cat) :=
show Nontrivial (PDMinimal.term M (n+1)) from inferInstance
have : Module.Finite A (ScalarModule k A (PDMinimal.term M (n+1)).cat) :=
show Module.Finite A (PDMinimal.term M (n+1)) from inferInstance
obtain ⟨S, hS, hnonzero⟩ := exists_simple_tensor_nonzero (k := k) (A := A)
(ScalarModule k A (PDMinimal.term M (n+1)).cat)
have := hS
have hz := (h S hS).of_iso (simpleTorIsoTerm S M (n+1)).symm
exact hnonzero (ModuleCat.isZero_iff_subsingleton.mp hz)
end LittleFinitistic.PDTensor
end
end OAI
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