Post on 05-Oct-2020
transcript
On the abelianization of derivedcategories and a negative solution to
Rosicky’s problem
Silvana Bazzoni
Joint work with Jan Stovıcek
Maurice Auslander Distinguished Lectures and InternationalConference
April 14 - 19, 2011Woods Hole, Massachusetts, USA
Silvana Bazzoni Negative solution to Rosicky’s problem
Outline
I The problem and motivations
I Purity and generalized purity in Mod-R.
I Brown-Adams representability theorems.
I The Walker’s modules Pβ.
I The counterexamples.
I Application to abelianizations of derived categories.
Silvana Bazzoni Negative solution to Rosicky’s problem
Outline
I The problem and motivations
I Purity and generalized purity in Mod-R.
I Brown-Adams representability theorems.
I The Walker’s modules Pβ.
I The counterexamples.
I Application to abelianizations of derived categories.
Silvana Bazzoni Negative solution to Rosicky’s problem
Outline
I The problem and motivations
I Purity and generalized purity in Mod-R.
I Brown-Adams representability theorems.
I The Walker’s modules Pβ.
I The counterexamples.
I Application to abelianizations of derived categories.
Silvana Bazzoni Negative solution to Rosicky’s problem
Outline
I The problem and motivations
I Purity and generalized purity in Mod-R.
I Brown-Adams representability theorems.
I The Walker’s modules Pβ.
I The counterexamples.
I Application to abelianizations of derived categories.
Silvana Bazzoni Negative solution to Rosicky’s problem
Outline
I The problem and motivations
I Purity and generalized purity in Mod-R.
I Brown-Adams representability theorems.
I The Walker’s modules Pβ.
I The counterexamples.
I Application to abelianizations of derived categories.
Silvana Bazzoni Negative solution to Rosicky’s problem
Outline
I The problem and motivations
I Purity and generalized purity in Mod-R.
I Brown-Adams representability theorems.
I The Walker’s modules Pβ.
I The counterexamples.
I Application to abelianizations of derived categories.
Silvana Bazzoni Negative solution to Rosicky’s problem
Outline
I The problem and motivations
I Purity and generalized purity in Mod-R.
I Brown-Adams representability theorems.
I The Walker’s modules Pβ.
I The counterexamples.
I Application to abelianizations of derived categories.
Silvana Bazzoni Negative solution to Rosicky’s problem
The problem and motivations
Rosicky’s problemGiven a ring R, is there a regular cardinal λ such that the λ-pureglobal dimension of Mod-R is ≤ 1?
(a cardinal λ is regular if it has cofinality λ)
MotivationIf there is a regular cardinal λ satisfying Rosicky’s problem thenthe derived category D(R) satisfies:
[ARMλ] Adams λ representability theorem for morphisms.
Silvana Bazzoni Negative solution to Rosicky’s problem
The problem and motivations
Rosicky’s problemGiven a ring R, is there a regular cardinal λ such that the λ-pureglobal dimension of Mod-R is ≤ 1?
(a cardinal λ is regular if it has cofinality λ)
MotivationIf there is a regular cardinal λ satisfying Rosicky’s problem thenthe derived category D(R) satisfies:
[ARMλ] Adams λ representability theorem for morphisms.
Silvana Bazzoni Negative solution to Rosicky’s problem
The problem and motivations
Rosicky’s problemGiven a ring R, is there a regular cardinal λ such that the λ-pureglobal dimension of Mod-R is ≤ 1?
(a cardinal λ is regular if it has cofinality λ)
MotivationIf there is a regular cardinal λ satisfying Rosicky’s problem thenthe derived category D(R) satisfies:
[ARMλ] Adams λ representability theorem for morphisms.
Silvana Bazzoni Negative solution to Rosicky’s problem
The problem and motivations
Rosicky’s problemGiven a ring R, is there a regular cardinal λ such that the λ-pureglobal dimension of Mod-R is ≤ 1?
(a cardinal λ is regular if it has cofinality λ)
MotivationIf there is a regular cardinal λ satisfying Rosicky’s problem thenthe derived category D(R) satisfies:
[ARMλ] Adams λ representability theorem for morphisms.
Silvana Bazzoni Negative solution to Rosicky’s problem
Purity and λ-purity in Mod-R
Let λ be a regular cardinal. An exact sequence
0→ A→ B → C → 0
is (λ-)pure exact if for every finitely presented (< λ-presented)module X , every morphism f : X → C can be lifted to B, i.e.
B // C
X
f
OO
g
__@@
@@
A is (λ-)pure in B if 0→ A→ B → C → 0 is (λ-)pure exact.So, the usual notion of purity is the λ-purity for λ = ℵ0.
Silvana Bazzoni Negative solution to Rosicky’s problem
Purity and λ-purity in Mod-R
Let λ be a regular cardinal.
An exact sequence
0→ A→ B → C → 0
is (λ-)pure exact if for every finitely presented (< λ-presented)module X , every morphism f : X → C can be lifted to B, i.e.
B // C
X
f
OO
g
__@@
@@
A is (λ-)pure in B if 0→ A→ B → C → 0 is (λ-)pure exact.So, the usual notion of purity is the λ-purity for λ = ℵ0.
Silvana Bazzoni Negative solution to Rosicky’s problem
Purity and λ-purity in Mod-R
Let λ be a regular cardinal. An exact sequence
0→ A→ B → C → 0
is (λ-)pure exact if for every finitely presented (< λ-presented)module X , every morphism f : X → C can be lifted to B, i.e.
B // C
X
f
OO
g
__@@
@@
A is (λ-)pure in B if 0→ A→ B → C → 0 is (λ-)pure exact.So, the usual notion of purity is the λ-purity for λ = ℵ0.
Silvana Bazzoni Negative solution to Rosicky’s problem
Purity and λ-purity in Mod-R
Let λ be a regular cardinal. An exact sequence
0→ A→ B → C → 0
is (λ-)pure exact if for every finitely presented (< λ-presented)module X , every morphism f : X → C can be lifted to B, i.e.
B // C
X
f
OO
g
__@@
@@
A is (λ-)pure in B if 0→ A→ B → C → 0 is (λ-)pure exact.
So, the usual notion of purity is the λ-purity for λ = ℵ0.
Silvana Bazzoni Negative solution to Rosicky’s problem
Purity and λ-purity in Mod-R
Let λ be a regular cardinal. An exact sequence
0→ A→ B → C → 0
is (λ-)pure exact if for every finitely presented (< λ-presented)module X , every morphism f : X → C can be lifted to B, i.e.
B // C
X
f
OO
g
__@@
@@
A is (λ-)pure in B if 0→ A→ B → C → 0 is (λ-)pure exact.So, the usual notion of purity is the λ-purity for λ = ℵ0.
Silvana Bazzoni Negative solution to Rosicky’s problem
Other characterizations of purity.
I A is pure in B if every linear system∑1≤j≤n;1≤i≤m
rijxj = bi
of m equations in n unknowns with rij ∈ R and bi ∈ B, issolvable in A whenever it is solvable in B.
I An exact sequence 0→ A→ B → C → 0 of left R-modules ispure if
0→ A⊗R X → B ⊗R X → C ⊗R X → 0
is exact for every finitely presented right R-module X .
• EXAMPLE R a PID. An R-module A is pure in B if and only ifevery equation rx = b (r ∈ R, b ∈ B) solvable in B is also solvablein A, i.e. rB ∩ A = rA.
Silvana Bazzoni Negative solution to Rosicky’s problem
Other characterizations of purity.
I A is pure in B if every linear system∑1≤j≤n;1≤i≤m
rijxj = bi
of m equations in n unknowns with rij ∈ R and bi ∈ B,
issolvable in A whenever it is solvable in B.
I An exact sequence 0→ A→ B → C → 0 of left R-modules ispure if
0→ A⊗R X → B ⊗R X → C ⊗R X → 0
is exact for every finitely presented right R-module X .
• EXAMPLE R a PID. An R-module A is pure in B if and only ifevery equation rx = b (r ∈ R, b ∈ B) solvable in B is also solvablein A, i.e. rB ∩ A = rA.
Silvana Bazzoni Negative solution to Rosicky’s problem
Other characterizations of purity.
I A is pure in B if every linear system∑1≤j≤n;1≤i≤m
rijxj = bi
of m equations in n unknowns with rij ∈ R and bi ∈ B, issolvable in A whenever it is solvable in B.
I An exact sequence 0→ A→ B → C → 0 of left R-modules ispure if
0→ A⊗R X → B ⊗R X → C ⊗R X → 0
is exact for every finitely presented right R-module X .
• EXAMPLE R a PID. An R-module A is pure in B if and only ifevery equation rx = b (r ∈ R, b ∈ B) solvable in B is also solvablein A, i.e. rB ∩ A = rA.
Silvana Bazzoni Negative solution to Rosicky’s problem
Other characterizations of purity.
I A is pure in B if every linear system∑1≤j≤n;1≤i≤m
rijxj = bi
of m equations in n unknowns with rij ∈ R and bi ∈ B, issolvable in A whenever it is solvable in B.
I An exact sequence 0→ A→ B → C → 0 of left R-modules ispure if
0→ A⊗R X → B ⊗R X → C ⊗R X → 0
is exact for every finitely presented right R-module X .
• EXAMPLE R a PID. An R-module A is pure in B if and only ifevery equation rx = b (r ∈ R, b ∈ B) solvable in B is also solvablein A, i.e. rB ∩ A = rA.
Silvana Bazzoni Negative solution to Rosicky’s problem
Other characterizations of purity.
I A is pure in B if every linear system∑1≤j≤n;1≤i≤m
rijxj = bi
of m equations in n unknowns with rij ∈ R and bi ∈ B, issolvable in A whenever it is solvable in B.
I An exact sequence 0→ A→ B → C → 0 of left R-modules ispure if
0→ A⊗R X → B ⊗R X → C ⊗R X → 0
is exact for every finitely presented right R-module X .
• EXAMPLE R a PID. An R-module A is pure in B if and only if
every equation rx = b (r ∈ R, b ∈ B) solvable in B is also solvablein A, i.e. rB ∩ A = rA.
Silvana Bazzoni Negative solution to Rosicky’s problem
Other characterizations of purity.
I A is pure in B if every linear system∑1≤j≤n;1≤i≤m
rijxj = bi
of m equations in n unknowns with rij ∈ R and bi ∈ B, issolvable in A whenever it is solvable in B.
I An exact sequence 0→ A→ B → C → 0 of left R-modules ispure if
0→ A⊗R X → B ⊗R X → C ⊗R X → 0
is exact for every finitely presented right R-module X .
• EXAMPLE R a PID. An R-module A is pure in B if and only ifevery equation rx = b (r ∈ R, b ∈ B) solvable in B is also solvablein A,
i.e. rB ∩ A = rA.
Silvana Bazzoni Negative solution to Rosicky’s problem
Other characterizations of purity.
I A is pure in B if every linear system∑1≤j≤n;1≤i≤m
rijxj = bi
of m equations in n unknowns with rij ∈ R and bi ∈ B, issolvable in A whenever it is solvable in B.
I An exact sequence 0→ A→ B → C → 0 of left R-modules ispure if
0→ A⊗R X → B ⊗R X → C ⊗R X → 0
is exact for every finitely presented right R-module X .
• EXAMPLE R a PID. An R-module A is pure in B if and only ifevery equation rx = b (r ∈ R, b ∈ B) solvable in B is also solvablein A, i.e. rB ∩ A = rA.
Silvana Bazzoni Negative solution to Rosicky’s problem
• (λ-)pure projective: projective property with respect to the(λ-)pure exact sequences. They are summands of direct sums offinitely presented (< λ-presented) modules.• There are enough λ-pure projective, i.e. for every module Mthere is a λ-pure exact sequence:
0→ K → ⊕X (Hom(X ,M)) → M → 0
where X vary in the set of representatives of < λ-presentedmodules.
• We can define the λ-pure projective dimension of a module andthe λ-pure global dimension of Mod-R.
• Every module is a λ-directed limit of < λ-presented modules.(A partially ordered set is λ-directed if every subset of cardinalityless than λ has an upper bound.)
Silvana Bazzoni Negative solution to Rosicky’s problem
• (λ-)pure projective: projective property with respect to the(λ-)pure exact sequences. They are summands of direct sums offinitely presented (< λ-presented) modules.• There are enough λ-pure projective, i.e. for every module Mthere is a λ-pure exact sequence:
0→ K → ⊕X (Hom(X ,M)) → M → 0
where X vary in the set of representatives of < λ-presentedmodules.
• We can define the λ-pure projective dimension of a module andthe λ-pure global dimension of Mod-R.
• Every module is a λ-directed limit of < λ-presented modules.(A partially ordered set is λ-directed if every subset of cardinalityless than λ has an upper bound.)
Silvana Bazzoni Negative solution to Rosicky’s problem
• (λ-)pure projective: projective property with respect to the(λ-)pure exact sequences. They are summands of direct sums offinitely presented (< λ-presented) modules.• There are enough λ-pure projective, i.e. for every module Mthere is a λ-pure exact sequence:
0→ K → ⊕X (Hom(X ,M)) → M → 0
where X vary in the set of representatives of < λ-presentedmodules.
• We can define the λ-pure projective dimension of a module andthe λ-pure global dimension of Mod-R.
• Every module is a λ-directed limit of < λ-presented modules.(A partially ordered set is λ-directed if every subset of cardinalityless than λ has an upper bound.)
Silvana Bazzoni Negative solution to Rosicky’s problem
Brown-Adams Representability Theorems
T a triangulated category with arbitrary coproducts.
[BR] T is said to satisfy Brown representability, if everycontravariant cohomological functor F : T → Ab which sendscoproducts to products is isomorphic to HomR(−,X ) forsome X ∈ T .
Theorem (Brown 1962)
The homotopy category of spectra satisfies [BR].
Silvana Bazzoni Negative solution to Rosicky’s problem
Brown-Adams Representability Theorems
T a triangulated category with arbitrary coproducts.
[BR] T is said to satisfy Brown representability, if everycontravariant cohomological functor F : T → Ab which sendscoproducts to products is isomorphic to HomR(−,X ) forsome X ∈ T .
Theorem (Brown 1962)
The homotopy category of spectra satisfies [BR].
Silvana Bazzoni Negative solution to Rosicky’s problem
Brown-Adams Representability Theorems
T a triangulated category with arbitrary coproducts.
[BR] T is said to satisfy Brown representability, if everycontravariant cohomological functor F : T → Ab which sendscoproducts to products is isomorphic to HomR(−,X ) forsome X ∈ T .
Theorem (Brown 1962)
The homotopy category of spectra satisfies [BR].
Silvana Bazzoni Negative solution to Rosicky’s problem
• X ∈ T is called compact if the functor HomT (X ,−) commuteswith coproducts.
• T c full subcategory of compact objects of T .
• If T = D(R), the compact objects are the perfect complexes.
Theorem (Neeman 1992, 1996, 2001)
I Compactly generated triangulated categories satisfy [BR].
I Localizations of compactly generated triangulated categoriessatisfy [BR].
I Well generated triangulated categories satisfy [BR], that isλ-compactly generated for a regular cardinal λ.
Silvana Bazzoni Negative solution to Rosicky’s problem
• X ∈ T is called compact if the functor HomT (X ,−) commuteswith coproducts.
• T c full subcategory of compact objects of T .
• If T = D(R), the compact objects are the perfect complexes.
Theorem (Neeman 1992, 1996, 2001)
I Compactly generated triangulated categories satisfy [BR].
I Localizations of compactly generated triangulated categoriessatisfy [BR].
I Well generated triangulated categories satisfy [BR], that isλ-compactly generated for a regular cardinal λ.
Silvana Bazzoni Negative solution to Rosicky’s problem
• X ∈ T is called compact if the functor HomT (X ,−) commuteswith coproducts.
• T c full subcategory of compact objects of T .
• If T = D(R), the compact objects are the perfect complexes.
Theorem (Neeman 1992, 1996, 2001)
I Compactly generated triangulated categories satisfy [BR].
I Localizations of compactly generated triangulated categoriessatisfy [BR].
I Well generated triangulated categories satisfy [BR], that isλ-compactly generated for a regular cardinal λ.
Silvana Bazzoni Negative solution to Rosicky’s problem
• X ∈ T is called compact if the functor HomT (X ,−) commuteswith coproducts.
• T c full subcategory of compact objects of T .
• If T = D(R), the compact objects are the perfect complexes.
Theorem (Neeman 1992, 1996, 2001)
I Compactly generated triangulated categories satisfy [BR].
I Localizations of compactly generated triangulated categoriessatisfy [BR].
I Well generated triangulated categories satisfy [BR], that isλ-compactly generated for a regular cardinal λ.
Silvana Bazzoni Negative solution to Rosicky’s problem
• X ∈ T is called compact if the functor HomT (X ,−) commuteswith coproducts.
• T c full subcategory of compact objects of T .
• If T = D(R), the compact objects are the perfect complexes.
Theorem (Neeman 1992, 1996, 2001)
I Compactly generated triangulated categories satisfy [BR].
I Localizations of compactly generated triangulated categoriessatisfy [BR].
I Well generated triangulated categories satisfy [BR], that isλ-compactly generated for a regular cardinal λ.
Silvana Bazzoni Negative solution to Rosicky’s problem
• X ∈ T is called compact if the functor HomT (X ,−) commuteswith coproducts.
• T c full subcategory of compact objects of T .
• If T = D(R), the compact objects are the perfect complexes.
Theorem (Neeman 1992, 1996, 2001)
I Compactly generated triangulated categories satisfy [BR].
I Localizations of compactly generated triangulated categoriessatisfy [BR].
I Well generated triangulated categories satisfy [BR], that isλ-compactly generated for a regular cardinal λ.
Silvana Bazzoni Negative solution to Rosicky’s problem
• X ∈ T is called compact if the functor HomT (X ,−) commuteswith coproducts.
• T c full subcategory of compact objects of T .
• If T = D(R), the compact objects are the perfect complexes.
Theorem (Neeman 1992, 1996, 2001)
I Compactly generated triangulated categories satisfy [BR].
I Localizations of compactly generated triangulated categoriessatisfy [BR].
I Well generated triangulated categories satisfy [BR], that isλ-compactly generated for a regular cardinal λ.
Silvana Bazzoni Negative solution to Rosicky’s problem
Adams Representability Theorems
T a triangulated category. T c the subcategory of compact objects.
[ARO] T is said to satisfy Adams representability for objects if everycontravariant cohomological functor F : T c → Ab isisomorphic to HomT (−,X )|T c for some X ∈ T .
[ARM] T satisfies Adams representability for morphisms if everynatural transformationη : HomT (−,X )|T c −→ HomT (−,Y )|T c
is induced by a morphism X → Y in T .
Theorem (Adams 1971)
The homotopy category of spectra satisfies [ARO] and [ARM].
Silvana Bazzoni Negative solution to Rosicky’s problem
Adams Representability Theorems
T a triangulated category. T c the subcategory of compact objects.
[ARO] T is said to satisfy Adams representability for objects if everycontravariant cohomological functor F : T c → Ab isisomorphic to HomT (−,X )|T c for some X ∈ T .
[ARM] T satisfies Adams representability for morphisms if everynatural transformationη : HomT (−,X )|T c −→ HomT (−,Y )|T c
is induced by a morphism X → Y in T .
Theorem (Adams 1971)
The homotopy category of spectra satisfies [ARO] and [ARM].
Silvana Bazzoni Negative solution to Rosicky’s problem
Adams Representability Theorems
T a triangulated category. T c the subcategory of compact objects.
[ARO] T is said to satisfy Adams representability for objects if everycontravariant cohomological functor F : T c → Ab isisomorphic to HomT (−,X )|T c for some X ∈ T .
[ARM] T satisfies Adams representability for morphisms if everynatural transformationη : HomT (−,X )|T c −→ HomT (−,Y )|T c
is induced by a morphism X → Y in T .
Theorem (Adams 1971)
The homotopy category of spectra satisfies [ARO] and [ARM].
Silvana Bazzoni Negative solution to Rosicky’s problem
Adams Representability Theorems
T a triangulated category. T c the subcategory of compact objects.
[ARO] T is said to satisfy Adams representability for objects if everycontravariant cohomological functor F : T c → Ab isisomorphic to HomT (−,X )|T c for some X ∈ T .
[ARM] T satisfies Adams representability for morphisms if everynatural transformationη : HomT (−,X )|T c −→ HomT (−,Y )|T c
is induced by a morphism X → Y in T .
Theorem (Adams 1971)
The homotopy category of spectra satisfies [ARO] and [ARM].
Silvana Bazzoni Negative solution to Rosicky’s problem
Adams Representability Theorems
T a triangulated category. T c the subcategory of compact objects.
[ARO] T is said to satisfy Adams representability for objects if everycontravariant cohomological functor F : T c → Ab isisomorphic to HomT (−,X )|T c for some X ∈ T .
[ARM] T satisfies Adams representability for morphisms if everynatural transformationη : HomT (−,X )|T c −→ HomT (−,Y )|T c
is induced by a morphism X → Y in T .
Theorem (Adams 1971)
The homotopy category of spectra satisfies [ARO] and [ARM].
Silvana Bazzoni Negative solution to Rosicky’s problem
Theorem (Neeman 1997)
I [ARO] and [ARM] hold in T if and only if
proj. dimA(T ) HomT (−,X )|T c ≤ 1, for all X ∈ T
A(T ): abelian category of all contravariant cohomological functor
F : T c → Ab.
I D(C[x , y ]) does not satisfy [ARM].
• Beligiannis 2000: [ARM] ⇒ [ARO]
Theorem (Christensen, Keller, Neeman 2001)
Let T be a compactly generated triangulated category (eg: D(R)).Then:
I If global proj. dimA(T ) ≤ 2, then T satisfies [ARO],
I [ARM] is satisfied for T , if and only ifproj. dimA(T ) HomT (−,X )|T c ≤ 1 for all X ∈ T .
Silvana Bazzoni Negative solution to Rosicky’s problem
Theorem (Neeman 1997)
I [ARO] and [ARM] hold in T if and only if
proj. dimA(T ) HomT (−,X )|T c ≤ 1, for all X ∈ T
A(T ): abelian category of all contravariant cohomological functor
F : T c → Ab.
I D(C[x , y ]) does not satisfy [ARM].
• Beligiannis 2000: [ARM] ⇒ [ARO]
Theorem (Christensen, Keller, Neeman 2001)
Let T be a compactly generated triangulated category (eg: D(R)).Then:
I If global proj. dimA(T ) ≤ 2, then T satisfies [ARO],
I [ARM] is satisfied for T , if and only ifproj. dimA(T ) HomT (−,X )|T c ≤ 1 for all X ∈ T .
Silvana Bazzoni Negative solution to Rosicky’s problem
Theorem (Neeman 1997)
I [ARO] and [ARM] hold in T if and only if
proj. dimA(T ) HomT (−,X )|T c ≤ 1, for all X ∈ T
A(T ): abelian category of all contravariant cohomological functor
F : T c → Ab.
I D(C[x , y ]) does not satisfy [ARM].
• Beligiannis 2000: [ARM] ⇒ [ARO]
Theorem (Christensen, Keller, Neeman 2001)
Let T be a compactly generated triangulated category (eg: D(R)).Then:
I If global proj. dimA(T ) ≤ 2, then T satisfies [ARO],
I [ARM] is satisfied for T , if and only ifproj. dimA(T ) HomT (−,X )|T c ≤ 1 for all X ∈ T .
Silvana Bazzoni Negative solution to Rosicky’s problem
Theorem (Neeman 1997)
I [ARO] and [ARM] hold in T if and only if
proj. dimA(T ) HomT (−,X )|T c ≤ 1, for all X ∈ T
A(T ): abelian category of all contravariant cohomological functor
F : T c → Ab.
I D(C[x , y ]) does not satisfy [ARM].
• Beligiannis 2000: [ARM] ⇒ [ARO]
Theorem (Christensen, Keller, Neeman 2001)
Let T be a compactly generated triangulated category (eg: D(R)).Then:
I If global proj. dimA(T ) ≤ 2, then T satisfies [ARO],
I [ARM] is satisfied for T , if and only ifproj. dimA(T ) HomT (−,X )|T c ≤ 1 for all X ∈ T .
Silvana Bazzoni Negative solution to Rosicky’s problem
Theorem (Neeman 1997)
I [ARO] and [ARM] hold in T if and only if
proj. dimA(T ) HomT (−,X )|T c ≤ 1, for all X ∈ T
A(T ): abelian category of all contravariant cohomological functor
F : T c → Ab.
I D(C[x , y ]) does not satisfy [ARM].
• Beligiannis 2000: [ARM] ⇒ [ARO]
Theorem (Christensen, Keller, Neeman 2001)
Let T be a compactly generated triangulated category (eg: D(R)).Then:
I If global proj. dimA(T ) ≤ 2, then T satisfies [ARO],
I [ARM] is satisfied for T , if and only ifproj. dimA(T ) HomT (−,X )|T c ≤ 1 for all X ∈ T .
Silvana Bazzoni Negative solution to Rosicky’s problem
Theorem (Neeman 1997)
I [ARO] and [ARM] hold in T if and only if
proj. dimA(T ) HomT (−,X )|T c ≤ 1, for all X ∈ T
A(T ): abelian category of all contravariant cohomological functor
F : T c → Ab.
I D(C[x , y ]) does not satisfy [ARM].
• Beligiannis 2000: [ARM] ⇒ [ARO]
Theorem (Christensen, Keller, Neeman 2001)
Let T be a compactly generated triangulated category (eg: D(R)).Then:
I If global proj. dimA(T ) ≤ 2, then T satisfies [ARO],
I [ARM] is satisfied for T , if and only ifproj. dimA(T ) HomT (−,X )|T c ≤ 1 for all X ∈ T .
Silvana Bazzoni Negative solution to Rosicky’s problem
Theorem (Neeman 1997)
I [ARO] and [ARM] hold in T if and only if
proj. dimA(T ) HomT (−,X )|T c ≤ 1, for all X ∈ T
A(T ): abelian category of all contravariant cohomological functor
F : T c → Ab.
I D(C[x , y ]) does not satisfy [ARM].
• Beligiannis 2000: [ARM] ⇒ [ARO]
Theorem (Christensen, Keller, Neeman 2001)
Let T be a compactly generated triangulated category (eg: D(R)).Then:
I If global proj. dimA(T ) ≤ 2, then T satisfies [ARO],
I [ARM] is satisfied for T , if and only ifproj. dimA(T ) HomT (−,X )|T c ≤ 1 for all X ∈ T .
Silvana Bazzoni Negative solution to Rosicky’s problem
Theorem (Neeman 1997)
I [ARO] and [ARM] hold in T if and only if
proj. dimA(T ) HomT (−,X )|T c ≤ 1, for all X ∈ T
A(T ): abelian category of all contravariant cohomological functor
F : T c → Ab.
I D(C[x , y ]) does not satisfy [ARM].
• Beligiannis 2000: [ARM] ⇒ [ARO]
Theorem (Christensen, Keller, Neeman 2001)
Let T be a compactly generated triangulated category (eg: D(R)).Then:
I If global proj. dimA(T ) ≤ 2, then T satisfies [ARO],
I [ARM] is satisfied for T , if and only ifproj. dimA(T ) HomT (−,X )|T c ≤ 1 for all X ∈ T .
Silvana Bazzoni Negative solution to Rosicky’s problem
Generalization to larger cardinals
λ regular infinite cardinal.Replace T c by the category T λ of λ-compact objects (T ℵ0 = T c .)λ-compact objects X are in particular λ-small
(i.e. every f : X →∐i∈I
Yi factors through∐i∈J
Yi with | J |< λ).
• If R is a ring, D(R)λ is the full triangulated subcategory of D(R)containing R and closed under coproducts of < λ summands.
[AROλ] T satisfies Adams λ representability for objects if everycontravariant cohomological functor F : T λ → Ab whichsends coproducts with fewer than λ summands to products isisomorphic to HomT (−,X ) � T λ for some X ∈ T .
[ARMλ] T satisfies Adams λ representability for morphisms if everynatural transformation
η : HomT (−,X ) � T λ −→ HomT (−,Y ) � T λ
is induced by a morphism X → Y in T .
Silvana Bazzoni Negative solution to Rosicky’s problem
Generalization to larger cardinals
λ regular infinite cardinal.
Replace T c by the category T λ of λ-compact objects (T ℵ0 = T c .)λ-compact objects X are in particular λ-small
(i.e. every f : X →∐i∈I
Yi factors through∐i∈J
Yi with | J |< λ).
• If R is a ring, D(R)λ is the full triangulated subcategory of D(R)containing R and closed under coproducts of < λ summands.
[AROλ] T satisfies Adams λ representability for objects if everycontravariant cohomological functor F : T λ → Ab whichsends coproducts with fewer than λ summands to products isisomorphic to HomT (−,X ) � T λ for some X ∈ T .
[ARMλ] T satisfies Adams λ representability for morphisms if everynatural transformation
η : HomT (−,X ) � T λ −→ HomT (−,Y ) � T λ
is induced by a morphism X → Y in T .
Silvana Bazzoni Negative solution to Rosicky’s problem
Generalization to larger cardinals
λ regular infinite cardinal.Replace T c by the category T λ of λ-compact objects
(T ℵ0 = T c .)λ-compact objects X are in particular λ-small
(i.e. every f : X →∐i∈I
Yi factors through∐i∈J
Yi with | J |< λ).
• If R is a ring, D(R)λ is the full triangulated subcategory of D(R)containing R and closed under coproducts of < λ summands.
[AROλ] T satisfies Adams λ representability for objects if everycontravariant cohomological functor F : T λ → Ab whichsends coproducts with fewer than λ summands to products isisomorphic to HomT (−,X ) � T λ for some X ∈ T .
[ARMλ] T satisfies Adams λ representability for morphisms if everynatural transformation
η : HomT (−,X ) � T λ −→ HomT (−,Y ) � T λ
is induced by a morphism X → Y in T .
Silvana Bazzoni Negative solution to Rosicky’s problem
Generalization to larger cardinals
λ regular infinite cardinal.Replace T c by the category T λ of λ-compact objects (T ℵ0 = T c .)
λ-compact objects X are in particular λ-small
(i.e. every f : X →∐i∈I
Yi factors through∐i∈J
Yi with | J |< λ).
• If R is a ring, D(R)λ is the full triangulated subcategory of D(R)containing R and closed under coproducts of < λ summands.
[AROλ] T satisfies Adams λ representability for objects if everycontravariant cohomological functor F : T λ → Ab whichsends coproducts with fewer than λ summands to products isisomorphic to HomT (−,X ) � T λ for some X ∈ T .
[ARMλ] T satisfies Adams λ representability for morphisms if everynatural transformation
η : HomT (−,X ) � T λ −→ HomT (−,Y ) � T λ
is induced by a morphism X → Y in T .
Silvana Bazzoni Negative solution to Rosicky’s problem
Generalization to larger cardinals
λ regular infinite cardinal.Replace T c by the category T λ of λ-compact objects (T ℵ0 = T c .)λ-compact objects X are in particular λ-small
(i.e. every f : X →∐i∈I
Yi factors through∐i∈J
Yi with | J |< λ).
• If R is a ring, D(R)λ is the full triangulated subcategory of D(R)containing R and closed under coproducts of < λ summands.
[AROλ] T satisfies Adams λ representability for objects if everycontravariant cohomological functor F : T λ → Ab whichsends coproducts with fewer than λ summands to products isisomorphic to HomT (−,X ) � T λ for some X ∈ T .
[ARMλ] T satisfies Adams λ representability for morphisms if everynatural transformation
η : HomT (−,X ) � T λ −→ HomT (−,Y ) � T λ
is induced by a morphism X → Y in T .
Silvana Bazzoni Negative solution to Rosicky’s problem
Generalization to larger cardinals
λ regular infinite cardinal.Replace T c by the category T λ of λ-compact objects (T ℵ0 = T c .)λ-compact objects X are in particular λ-small
(i.e. every f : X →∐i∈I
Yi factors through∐i∈J
Yi with | J |< λ).
• If R is a ring, D(R)λ is the full triangulated subcategory of D(R)containing R and closed under coproducts of < λ summands.
[AROλ] T satisfies Adams λ representability for objects if everycontravariant cohomological functor F : T λ → Ab whichsends coproducts with fewer than λ summands to products isisomorphic to HomT (−,X ) � T λ for some X ∈ T .
[ARMλ] T satisfies Adams λ representability for morphisms if everynatural transformation
η : HomT (−,X ) � T λ −→ HomT (−,Y ) � T λ
is induced by a morphism X → Y in T .
Silvana Bazzoni Negative solution to Rosicky’s problem
Generalization to larger cardinals
λ regular infinite cardinal.Replace T c by the category T λ of λ-compact objects (T ℵ0 = T c .)λ-compact objects X are in particular λ-small
(i.e. every f : X →∐i∈I
Yi factors through∐i∈J
Yi with | J |< λ).
• If R is a ring, D(R)λ is the full triangulated subcategory of D(R)containing R and closed under coproducts of < λ summands.
[AROλ] T satisfies Adams λ representability for objects if everycontravariant cohomological functor F : T λ → Ab whichsends coproducts with fewer than λ summands to products isisomorphic to HomT (−,X ) � T λ for some X ∈ T .
[ARMλ] T satisfies Adams λ representability for morphisms if everynatural transformation
η : HomT (−,X ) � T λ −→ HomT (−,Y ) � T λ
is induced by a morphism X → Y in T .
Silvana Bazzoni Negative solution to Rosicky’s problem
Generalization to larger cardinals
λ regular infinite cardinal.Replace T c by the category T λ of λ-compact objects (T ℵ0 = T c .)λ-compact objects X are in particular λ-small
(i.e. every f : X →∐i∈I
Yi factors through∐i∈J
Yi with | J |< λ).
• If R is a ring, D(R)λ is the full triangulated subcategory of D(R)containing R and closed under coproducts of < λ summands.
[AROλ] T satisfies Adams λ representability for objects if everycontravariant cohomological functor F : T λ → Ab whichsends coproducts with fewer than λ summands to products isisomorphic to HomT (−,X ) � T λ for some X ∈ T .
[ARMλ] T satisfies Adams λ representability for morphisms if everynatural transformation
η : HomT (−,X ) � T λ −→ HomT (−,Y ) � T λ
is induced by a morphism X → Y in T .
Silvana Bazzoni Negative solution to Rosicky’s problem
Generalization to larger cardinals
λ regular infinite cardinal.Replace T c by the category T λ of λ-compact objects (T ℵ0 = T c .)λ-compact objects X are in particular λ-small
(i.e. every f : X →∐i∈I
Yi factors through∐i∈J
Yi with | J |< λ).
• If R is a ring, D(R)λ is the full triangulated subcategory of D(R)containing R and closed under coproducts of < λ summands.
[AROλ] T satisfies Adams λ representability for objects if everycontravariant cohomological functor F : T λ → Ab whichsends coproducts with fewer than λ summands to products isisomorphic to HomT (−,X ) � T λ for some X ∈ T .
[ARMλ] T satisfies Adams λ representability for morphisms if everynatural transformation
η : HomT (−,X ) � T λ −→ HomT (−,Y ) � T λ
is induced by a morphism X → Y in T .
Silvana Bazzoni Negative solution to Rosicky’s problem
Negative answer to Rosicky’s problem
Neeman 1997: D(C[x , y ]) does not satisfy [ARM]=([ARMℵ0 ]).
• Rosicky in 2005 claimed that for every ring R there is an infiniteregular cardinal λ such that D(R) satisfies [ARMλ].
• Rosicky’s claim is equivalent to:
• For every ring R there is an infinite regular cardinal λ such thatλ-pure global dimension Mod-R ≤ 1.
Theorem (B. Stovıcek 2010)
Let k be a field and λ an infinite regular cardinal. If R is one ofthe following:
1 k(·⇒ ·) with k uncountable.
2 k[x , y ] with k uncountable.
3 k[[x , y ]]
Then λ-pure global dimension Mod-R ≥ 2.
Silvana Bazzoni Negative solution to Rosicky’s problem
Negative answer to Rosicky’s problem
Neeman 1997: D(C[x , y ]) does not satisfy [ARM]=([ARMℵ0 ]).
• Rosicky in 2005 claimed that for every ring R there is an infiniteregular cardinal λ such that D(R) satisfies [ARMλ].
• Rosicky’s claim is equivalent to:
• For every ring R there is an infinite regular cardinal λ such thatλ-pure global dimension Mod-R ≤ 1.
Theorem (B. Stovıcek 2010)
Let k be a field and λ an infinite regular cardinal. If R is one ofthe following:
1 k(·⇒ ·) with k uncountable.
2 k[x , y ] with k uncountable.
3 k[[x , y ]]
Then λ-pure global dimension Mod-R ≥ 2.
Silvana Bazzoni Negative solution to Rosicky’s problem
Negative answer to Rosicky’s problem
Neeman 1997: D(C[x , y ]) does not satisfy [ARM]=([ARMℵ0 ]).
• Rosicky in 2005 claimed that for every ring R there is an infiniteregular cardinal λ such that D(R) satisfies [ARMλ].
• Rosicky’s claim is equivalent to:
• For every ring R there is an infinite regular cardinal λ such thatλ-pure global dimension Mod-R ≤ 1.
Theorem (B. Stovıcek 2010)
Let k be a field and λ an infinite regular cardinal. If R is one ofthe following:
1 k(·⇒ ·) with k uncountable.
2 k[x , y ] with k uncountable.
3 k[[x , y ]]
Then λ-pure global dimension Mod-R ≥ 2.
Silvana Bazzoni Negative solution to Rosicky’s problem
Negative answer to Rosicky’s problem
Neeman 1997: D(C[x , y ]) does not satisfy [ARM]=([ARMℵ0 ]).
• Rosicky in 2005 claimed that for every ring R there is an infiniteregular cardinal λ such that D(R) satisfies [ARMλ].
• Rosicky’s claim is equivalent to:
• For every ring R there is an infinite regular cardinal λ such thatλ-pure global dimension Mod-R ≤ 1.
Theorem (B. Stovıcek 2010)
Let k be a field and λ an infinite regular cardinal. If R is one ofthe following:
1 k(·⇒ ·) with k uncountable.
2 k[x , y ] with k uncountable.
3 k[[x , y ]]
Then λ-pure global dimension Mod-R ≥ 2.
Silvana Bazzoni Negative solution to Rosicky’s problem
Negative answer to Rosicky’s problem
Neeman 1997: D(C[x , y ]) does not satisfy [ARM]=([ARMℵ0 ]).
• Rosicky in 2005 claimed that for every ring R there is an infiniteregular cardinal λ such that D(R) satisfies [ARMλ].
• Rosicky’s claim is equivalent to:
• For every ring R there is an infinite regular cardinal λ such thatλ-pure global dimension Mod-R ≤ 1.
Theorem (B. Stovıcek 2010)
Let k be a field and λ an infinite regular cardinal. If R is one ofthe following:
1 k(·⇒ ·) with k uncountable.
2 k[x , y ] with k uncountable.
3 k[[x , y ]]
Then λ-pure global dimension Mod-R ≥ 2.
Silvana Bazzoni Negative solution to Rosicky’s problem
Negative answer to Rosicky’s problem
Neeman 1997: D(C[x , y ]) does not satisfy [ARM]=([ARMℵ0 ]).
• Rosicky in 2005 claimed that for every ring R there is an infiniteregular cardinal λ such that D(R) satisfies [ARMλ].
• Rosicky’s claim is equivalent to:
• For every ring R there is an infinite regular cardinal λ such thatλ-pure global dimension Mod-R ≤ 1.
Theorem (B. Stovıcek 2010)
Let k be a field and λ an infinite regular cardinal. If R is one ofthe following:
1 k(·⇒ ·) with k uncountable.
2 k[x , y ] with k uncountable.
3 k[[x , y ]]
Then λ-pure global dimension Mod-R ≥ 2.
Silvana Bazzoni Negative solution to Rosicky’s problem
Negative answer to Rosicky’s problem
Neeman 1997: D(C[x , y ]) does not satisfy [ARM]=([ARMℵ0 ]).
• Rosicky in 2005 claimed that for every ring R there is an infiniteregular cardinal λ such that D(R) satisfies [ARMλ].
• Rosicky’s claim is equivalent to:
• For every ring R there is an infinite regular cardinal λ such thatλ-pure global dimension Mod-R ≤ 1.
Theorem (B. Stovıcek 2010)
Let k be a field and λ an infinite regular cardinal. If R is one ofthe following:
1 k(·⇒ ·) with k uncountable.
2 k[x , y ] with k uncountable.
3 k[[x , y ]]
Then λ-pure global dimension Mod-R ≥ 2.Silvana Bazzoni Negative solution to Rosicky’s problem
Discrete valuation domains
• R discrete valuation domain (ex. Zp; k[[x ]], k a field), p a prime.
• G ∈ Mod-R, σ an ordinal
I p0G = G .
I pσ+1G = p(pσG ),
I pσG =⋂ρ<σ
pρG , for σ limit.
• length of G = l(G ) =min{λ | pλG = pλ+1G}. For such λ, pλGis divisible and pλG = 0 if and only if G is reduced.
Silvana Bazzoni Negative solution to Rosicky’s problem
Discrete valuation domains
• R discrete valuation domain (ex. Zp; k[[x ]], k a field), p a prime.
• G ∈ Mod-R, σ an ordinal
I p0G = G .
I pσ+1G = p(pσG ),
I pσG =⋂ρ<σ
pρG , for σ limit.
• length of G = l(G ) =min{λ | pλG = pλ+1G}. For such λ, pλGis divisible and pλG = 0 if and only if G is reduced.
Silvana Bazzoni Negative solution to Rosicky’s problem
Discrete valuation domains
• R discrete valuation domain (ex. Zp; k[[x ]], k a field), p a prime.
• G ∈ Mod-R, σ an ordinal
I p0G = G .
I pσ+1G = p(pσG ),
I pσG =⋂ρ<σ
pρG , for σ limit.
• length of G = l(G ) =min{λ | pλG = pλ+1G}. For such λ, pλGis divisible and pλG = 0 if and only if G is reduced.
Silvana Bazzoni Negative solution to Rosicky’s problem
Discrete valuation domains
• R discrete valuation domain (ex. Zp; k[[x ]], k a field), p a prime.
• G ∈ Mod-R, σ an ordinal
I p0G = G .
I pσ+1G = p(pσG ),
I pσG =⋂ρ<σ
pρG , for σ limit.
• length of G = l(G ) =min{λ | pλG = pλ+1G}. For such λ, pλGis divisible and pλG = 0 if and only if G is reduced.
Silvana Bazzoni Negative solution to Rosicky’s problem
The Walker’s modules Pβ 1974
R a DVR, p a prime element and β an ordinal.• Pβ is generated by the finite sequences:
ββ1β2 . . . βn such that β > β1 > β2 > · · · > βn
• Relations:
p · ββ1β2 . . . βnβn+1 = ββ1β2 . . . βn and p · β = 0.
Silvana Bazzoni Negative solution to Rosicky’s problem
The Walker’s modules Pβ 1974
R a DVR, p a prime element and β an ordinal.• Pβ is generated by the finite sequences:
ββ1β2 . . . βn such that β > β1 > β2 > · · · > βn
• Relations:
p · ββ1β2 . . . βnβn+1 = ββ1β2 . . . βn and p · β = 0.
Silvana Bazzoni Negative solution to Rosicky’s problem
The Walker’s modules Pβ 1974
R a DVR, p a prime element and β an ordinal.• Pβ is generated by the finite sequences:
ββ1β2 . . . βn such that β > β1 > β2 > · · · > βn
• Relations:
p · ββ1β2 . . . βnβn+1 = ββ1β2 . . . βn and p · β = 0.
Silvana Bazzoni Negative solution to Rosicky’s problem
The Walker’s modules Pβ 1974
R a DVR, p a prime element and β an ordinal.• Pβ is generated by the finite sequences:
ββ1β2 . . . βn such that β > β1 > β2 > · · · > βn
• Relations:
p · ββ1β2 . . . βnβn+1 = ββ1β2 . . . βn and p · β = 0.
Silvana Bazzoni Negative solution to Rosicky’s problem
The Walker’s modules Pβ 1974
R a DVR, p a prime element and β an ordinal.• Pβ is generated by the finite sequences:
ββ1β2 . . . βn such that β > β1 > β2 > · · · > βn
• Relations:
p · ββ1β2 . . . βnβn+1 = ββ1β2 . . . βn and p · β = 0.
Silvana Bazzoni Negative solution to Rosicky’s problem
• Pβ is torsion and reduced. If β is infinite, Pβ is | β |-presented.
•Pβ〈β〉
=⊕α<β
Pα; l(Pβ) = β + 1.
• β < λ, λ a regular cardinal, then the maps:
ββ1β2 . . . βn → λβ1β2 . . . βn
induce embeddings Pβ → Pλ.
• Pλ is a λ-directed union of the images of Pβ.
Silvana Bazzoni Negative solution to Rosicky’s problem
• Pβ is torsion and reduced. If β is infinite, Pβ is | β |-presented.
•Pβ〈β〉
=⊕α<β
Pα; l(Pβ) = β + 1.
• β < λ, λ a regular cardinal, then the maps:
ββ1β2 . . . βn → λβ1β2 . . . βn
induce embeddings Pβ → Pλ.
• Pλ is a λ-directed union of the images of Pβ.
Silvana Bazzoni Negative solution to Rosicky’s problem
• Pβ is torsion and reduced. If β is infinite, Pβ is | β |-presented.
•Pβ〈β〉
=⊕α<β
Pα; l(Pβ) = β + 1.
• β < λ, λ a regular cardinal, then the maps:
ββ1β2 . . . βn → λβ1β2 . . . βn
induce embeddings Pβ → Pλ.
• Pλ is a λ-directed union of the images of Pβ.
Silvana Bazzoni Negative solution to Rosicky’s problem
• Pβ is torsion and reduced. If β is infinite, Pβ is | β |-presented.
•Pβ〈β〉
=⊕α<β
Pα; l(Pβ) = β + 1.
• β < λ, λ a regular cardinal, then the maps:
ββ1β2 . . . βn → λβ1β2 . . . βn
induce embeddings Pβ → Pλ.
• Pλ is a λ-directed union of the images of Pβ.
Silvana Bazzoni Negative solution to Rosicky’s problem
λ-pure projective dimension of Pλ
PropositionLet R be a DVR, λ a regular cardinal. There is a λ-pure exactsequence
0→ K →⊕β<λ
P(Hom(Pβ ,Pλ))β → Pλ → 0
The sequence is λ-pure exact since for every < λ-presented moduleX every morphism from X to Pλ factors through some Pβ. This isbecause Pλ is a λ-directed limit of the Pβ’s, β < λ.
• To conclude that for every regular uncountable cardinal λ, theλ-pure proj. dim of Pλ is greater than 1, we need to show that Kcannot be (a summand of) a direct sum of < λ-presented modules.For this we use a topology on torsion modules.
Silvana Bazzoni Negative solution to Rosicky’s problem
λ-pure projective dimension of Pλ
PropositionLet R be a DVR, λ a regular cardinal. There is a λ-pure exactsequence
0→ K →⊕β<λ
P(Hom(Pβ ,Pλ))β → Pλ → 0
The sequence is λ-pure exact since for every < λ-presented moduleX every morphism from X to Pλ factors through some Pβ. This isbecause Pλ is a λ-directed limit of the Pβ’s, β < λ.
• To conclude that for every regular uncountable cardinal λ, theλ-pure proj. dim of Pλ is greater than 1, we need to show that Kcannot be (a summand of) a direct sum of < λ-presented modules.For this we use a topology on torsion modules.
Silvana Bazzoni Negative solution to Rosicky’s problem
λ-pure projective dimension of Pλ
PropositionLet R be a DVR, λ a regular cardinal. There is a λ-pure exactsequence
0→ K →⊕β<λ
P(Hom(Pβ ,Pλ))β → Pλ → 0
The sequence is λ-pure exact since for every < λ-presented moduleX every morphism from X to Pλ factors through some Pβ.
This isbecause Pλ is a λ-directed limit of the Pβ’s, β < λ.
• To conclude that for every regular uncountable cardinal λ, theλ-pure proj. dim of Pλ is greater than 1, we need to show that Kcannot be (a summand of) a direct sum of < λ-presented modules.For this we use a topology on torsion modules.
Silvana Bazzoni Negative solution to Rosicky’s problem
λ-pure projective dimension of Pλ
PropositionLet R be a DVR, λ a regular cardinal. There is a λ-pure exactsequence
0→ K →⊕β<λ
P(Hom(Pβ ,Pλ))β → Pλ → 0
The sequence is λ-pure exact since for every < λ-presented moduleX every morphism from X to Pλ factors through some Pβ. This isbecause Pλ is a λ-directed limit of the Pβ’s, β < λ.
• To conclude that for every regular uncountable cardinal λ, theλ-pure proj. dim of Pλ is greater than 1, we need to show that Kcannot be (a summand of) a direct sum of < λ-presented modules.For this we use a topology on torsion modules.
Silvana Bazzoni Negative solution to Rosicky’s problem
λ-pure projective dimension of Pλ
PropositionLet R be a DVR, λ a regular cardinal. There is a λ-pure exactsequence
0→ K →⊕β<λ
P(Hom(Pβ ,Pλ))β → Pλ → 0
The sequence is λ-pure exact since for every < λ-presented moduleX every morphism from X to Pλ factors through some Pβ. This isbecause Pλ is a λ-directed limit of the Pβ’s, β < λ.
• To conclude that for every regular uncountable cardinal λ, theλ-pure proj. dim of Pλ is greater than 1, we need to show that Kcannot be (a summand of) a direct sum of < λ-presented modules.
For this we use a topology on torsion modules.
Silvana Bazzoni Negative solution to Rosicky’s problem
λ-pure projective dimension of Pλ
PropositionLet R be a DVR, λ a regular cardinal. There is a λ-pure exactsequence
0→ K →⊕β<λ
P(Hom(Pβ ,Pλ))β → Pλ → 0
The sequence is λ-pure exact since for every < λ-presented moduleX every morphism from X to Pλ factors through some Pβ. This isbecause Pλ is a λ-directed limit of the Pβ’s, β < λ.
• To conclude that for every regular uncountable cardinal λ, theλ-pure proj. dim of Pλ is greater than 1, we need to show that Kcannot be (a summand of) a direct sum of < λ-presented modules.For this we use a topology on torsion modules.
Silvana Bazzoni Negative solution to Rosicky’s problem
The pλ-adic topology
Let R be a DVR, λ and ordinal.The pλ-adic topology on a torsion module G is a linear topologywith U = {pσG | σ < λ} as basis of neighborhoods of 0(studied by Mines in 1968 for abelian p-groups.)Assume that G is reduced, then:• pλ-adic topology is discrete ⇐⇒ l(G ) < λ• pλ-adic topology is Hausdorff ⇐⇒ l(G ) ≤ λ
Proposition (Salce 1980)
• Let R be a DVR, λ a regular uncountable cardinal.• G (a summand of) a direct sum of < λ-presented modules (i.e.G λ-pure projective)Then G is complete in the pλ-adic topology.
Silvana Bazzoni Negative solution to Rosicky’s problem
The pλ-adic topology
Let R be a DVR, λ and ordinal.
The pλ-adic topology on a torsion module G is a linear topologywith U = {pσG | σ < λ} as basis of neighborhoods of 0(studied by Mines in 1968 for abelian p-groups.)Assume that G is reduced, then:• pλ-adic topology is discrete ⇐⇒ l(G ) < λ• pλ-adic topology is Hausdorff ⇐⇒ l(G ) ≤ λ
Proposition (Salce 1980)
• Let R be a DVR, λ a regular uncountable cardinal.• G (a summand of) a direct sum of < λ-presented modules (i.e.G λ-pure projective)Then G is complete in the pλ-adic topology.
Silvana Bazzoni Negative solution to Rosicky’s problem
The pλ-adic topology
Let R be a DVR, λ and ordinal.The pλ-adic topology on a torsion module G is a linear topologywith U = {pσG | σ < λ} as basis of neighborhoods of 0
(studied by Mines in 1968 for abelian p-groups.)Assume that G is reduced, then:• pλ-adic topology is discrete ⇐⇒ l(G ) < λ• pλ-adic topology is Hausdorff ⇐⇒ l(G ) ≤ λ
Proposition (Salce 1980)
• Let R be a DVR, λ a regular uncountable cardinal.• G (a summand of) a direct sum of < λ-presented modules (i.e.G λ-pure projective)Then G is complete in the pλ-adic topology.
Silvana Bazzoni Negative solution to Rosicky’s problem
The pλ-adic topology
Let R be a DVR, λ and ordinal.The pλ-adic topology on a torsion module G is a linear topologywith U = {pσG | σ < λ} as basis of neighborhoods of 0(studied by Mines in 1968 for abelian p-groups.)
Assume that G is reduced, then:• pλ-adic topology is discrete ⇐⇒ l(G ) < λ• pλ-adic topology is Hausdorff ⇐⇒ l(G ) ≤ λ
Proposition (Salce 1980)
• Let R be a DVR, λ a regular uncountable cardinal.• G (a summand of) a direct sum of < λ-presented modules (i.e.G λ-pure projective)Then G is complete in the pλ-adic topology.
Silvana Bazzoni Negative solution to Rosicky’s problem
The pλ-adic topology
Let R be a DVR, λ and ordinal.The pλ-adic topology on a torsion module G is a linear topologywith U = {pσG | σ < λ} as basis of neighborhoods of 0(studied by Mines in 1968 for abelian p-groups.)Assume that G is reduced, then:
• pλ-adic topology is discrete ⇐⇒ l(G ) < λ• pλ-adic topology is Hausdorff ⇐⇒ l(G ) ≤ λ
Proposition (Salce 1980)
• Let R be a DVR, λ a regular uncountable cardinal.• G (a summand of) a direct sum of < λ-presented modules (i.e.G λ-pure projective)Then G is complete in the pλ-adic topology.
Silvana Bazzoni Negative solution to Rosicky’s problem
The pλ-adic topology
Let R be a DVR, λ and ordinal.The pλ-adic topology on a torsion module G is a linear topologywith U = {pσG | σ < λ} as basis of neighborhoods of 0(studied by Mines in 1968 for abelian p-groups.)Assume that G is reduced, then:• pλ-adic topology is discrete ⇐⇒ l(G ) < λ• pλ-adic topology is Hausdorff ⇐⇒ l(G ) ≤ λ
Proposition (Salce 1980)
• Let R be a DVR, λ a regular uncountable cardinal.• G (a summand of) a direct sum of < λ-presented modules (i.e.G λ-pure projective)Then G is complete in the pλ-adic topology.
Silvana Bazzoni Negative solution to Rosicky’s problem
The pλ-adic topology
Let R be a DVR, λ and ordinal.The pλ-adic topology on a torsion module G is a linear topologywith U = {pσG | σ < λ} as basis of neighborhoods of 0(studied by Mines in 1968 for abelian p-groups.)Assume that G is reduced, then:• pλ-adic topology is discrete ⇐⇒ l(G ) < λ• pλ-adic topology is Hausdorff ⇐⇒ l(G ) ≤ λ
Proposition (Salce 1980)
• Let R be a DVR, λ a regular uncountable cardinal.
• G (a summand of) a direct sum of < λ-presented modules (i.e.G λ-pure projective)Then G is complete in the pλ-adic topology.
Silvana Bazzoni Negative solution to Rosicky’s problem
The pλ-adic topology
Let R be a DVR, λ and ordinal.The pλ-adic topology on a torsion module G is a linear topologywith U = {pσG | σ < λ} as basis of neighborhoods of 0(studied by Mines in 1968 for abelian p-groups.)Assume that G is reduced, then:• pλ-adic topology is discrete ⇐⇒ l(G ) < λ• pλ-adic topology is Hausdorff ⇐⇒ l(G ) ≤ λ
Proposition (Salce 1980)
• Let R be a DVR, λ a regular uncountable cardinal.• G (a summand of) a direct sum of < λ-presented modules (i.e.G λ-pure projective)
Then G is complete in the pλ-adic topology.
Silvana Bazzoni Negative solution to Rosicky’s problem
The pλ-adic topology
Let R be a DVR, λ and ordinal.The pλ-adic topology on a torsion module G is a linear topologywith U = {pσG | σ < λ} as basis of neighborhoods of 0(studied by Mines in 1968 for abelian p-groups.)Assume that G is reduced, then:• pλ-adic topology is discrete ⇐⇒ l(G ) < λ• pλ-adic topology is Hausdorff ⇐⇒ l(G ) ≤ λ
Proposition (Salce 1980)
• Let R be a DVR, λ a regular uncountable cardinal.• G (a summand of) a direct sum of < λ-presented modules (i.e.G λ-pure projective)Then G is complete in the pλ-adic topology.
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-pure projective dimension of Pλ
TheoremLet R be a DVR, λ a regular uncountable cardinal.The λ-pure projective dimension of Pλ is > 1.
• In fact: in the sequence
0→ K → T =⊕β<λ
P(Hom(Pβ ,Pλ))β → Pλ → 0
T is complete and K is not closed in T , hence not complete. ThusK cannot be λ-pure projective.
Consequence:• If R is DVR the λ-pure global dimension of Mod-R is greaterthan 1 for every uncountable regular cardinal λ.• But the (ℵ0) pure global dimension of Mod-R is 1.• We need to find a ring S with pure global dimension > 1 suchthat the category T of torsion modules over DVR can beembedded in Mod-S .
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-pure projective dimension of Pλ
TheoremLet R be a DVR, λ a regular uncountable cardinal.The λ-pure projective dimension of Pλ is > 1.
• In fact: in the sequence
0→ K → T =⊕β<λ
P(Hom(Pβ ,Pλ))β → Pλ → 0
T is complete and K is not closed in T , hence not complete. ThusK cannot be λ-pure projective.
Consequence:• If R is DVR the λ-pure global dimension of Mod-R is greaterthan 1 for every uncountable regular cardinal λ.• But the (ℵ0) pure global dimension of Mod-R is 1.• We need to find a ring S with pure global dimension > 1 suchthat the category T of torsion modules over DVR can beembedded in Mod-S .
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-pure projective dimension of Pλ
TheoremLet R be a DVR, λ a regular uncountable cardinal.The λ-pure projective dimension of Pλ is > 1.
• In fact: in the sequence
0→ K → T =⊕β<λ
P(Hom(Pβ ,Pλ))β → Pλ → 0
T is complete and K is not closed in T , hence not complete. ThusK cannot be λ-pure projective.
Consequence:• If R is DVR the λ-pure global dimension of Mod-R is greaterthan 1 for every uncountable regular cardinal λ.• But the (ℵ0) pure global dimension of Mod-R is 1.• We need to find a ring S with pure global dimension > 1 suchthat the category T of torsion modules over DVR can beembedded in Mod-S .
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-pure projective dimension of Pλ
TheoremLet R be a DVR, λ a regular uncountable cardinal.The λ-pure projective dimension of Pλ is > 1.
• In fact: in the sequence
0→ K → T =⊕β<λ
P(Hom(Pβ ,Pλ))β → Pλ → 0
T is complete and K is not closed in T , hence not complete. ThusK cannot be λ-pure projective.
Consequence:• If R is DVR the λ-pure global dimension of Mod-R is greaterthan 1 for every uncountable regular cardinal λ.
• But the (ℵ0) pure global dimension of Mod-R is 1.• We need to find a ring S with pure global dimension > 1 suchthat the category T of torsion modules over DVR can beembedded in Mod-S .
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-pure projective dimension of Pλ
TheoremLet R be a DVR, λ a regular uncountable cardinal.The λ-pure projective dimension of Pλ is > 1.
• In fact: in the sequence
0→ K → T =⊕β<λ
P(Hom(Pβ ,Pλ))β → Pλ → 0
T is complete and K is not closed in T , hence not complete. ThusK cannot be λ-pure projective.
Consequence:• If R is DVR the λ-pure global dimension of Mod-R is greaterthan 1 for every uncountable regular cardinal λ.• But the (ℵ0) pure global dimension of Mod-R is 1.
• We need to find a ring S with pure global dimension > 1 suchthat the category T of torsion modules over DVR can beembedded in Mod-S .
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-pure projective dimension of Pλ
TheoremLet R be a DVR, λ a regular uncountable cardinal.The λ-pure projective dimension of Pλ is > 1.
• In fact: in the sequence
0→ K → T =⊕β<λ
P(Hom(Pβ ,Pλ))β → Pλ → 0
T is complete and K is not closed in T , hence not complete. ThusK cannot be λ-pure projective.
Consequence:• If R is DVR the λ-pure global dimension of Mod-R is greaterthan 1 for every uncountable regular cardinal λ.• But the (ℵ0) pure global dimension of Mod-R is 1.• We need to find a ring S with pure global dimension > 1 suchthat the category T of torsion modules over DVR can beembedded in Mod-S .
Silvana Bazzoni Negative solution to Rosicky’s problem
Accessible categories
Definitionλ infinite regular cardinal. A an additive category with λ-directedcolimits. An object X ∈ A is called λ-presentable if, for anyλ-directed system (Yi | i ∈ I ), we have an isomorphism
lim−→HomA(X ,Yi ) ∼= HomA(X , lim−→Yi ).
The category A is λ-accessible if it admits a set S of λ-presentableobjects such that each Y ∈ A is a λ-directed colimit of objectsfrom S.If λ = ℵ0, A is called a finitely accessible category.
• Mod-R is finitely accessible.• Any additive λ-accessible category is a subcategory of a“module category”Mod-S: additive contravariant functors S → Ab
Silvana Bazzoni Negative solution to Rosicky’s problem
Accessible categories
Definitionλ infinite regular cardinal. A an additive category with λ-directedcolimits. An object X ∈ A is called λ-presentable if, for anyλ-directed system (Yi | i ∈ I ), we have an isomorphism
lim−→HomA(X ,Yi ) ∼= HomA(X , lim−→Yi ).
The category A is λ-accessible if it admits a set S of λ-presentableobjects such that each Y ∈ A is a λ-directed colimit of objectsfrom S.If λ = ℵ0, A is called a finitely accessible category.
• Mod-R is finitely accessible.• Any additive λ-accessible category is a subcategory of a“module category”Mod-S: additive contravariant functors S → Ab
Silvana Bazzoni Negative solution to Rosicky’s problem
Accessible categories
Definitionλ infinite regular cardinal. A an additive category with λ-directedcolimits. An object X ∈ A is called λ-presentable if, for anyλ-directed system (Yi | i ∈ I ), we have an isomorphism
lim−→HomA(X ,Yi ) ∼= HomA(X , lim−→Yi ).
The category A is λ-accessible if it admits a set S of λ-presentableobjects such that each Y ∈ A is a λ-directed colimit of objectsfrom S.
If λ = ℵ0, A is called a finitely accessible category.
• Mod-R is finitely accessible.• Any additive λ-accessible category is a subcategory of a“module category”Mod-S: additive contravariant functors S → Ab
Silvana Bazzoni Negative solution to Rosicky’s problem
Accessible categories
Definitionλ infinite regular cardinal. A an additive category with λ-directedcolimits. An object X ∈ A is called λ-presentable if, for anyλ-directed system (Yi | i ∈ I ), we have an isomorphism
lim−→HomA(X ,Yi ) ∼= HomA(X , lim−→Yi ).
The category A is λ-accessible if it admits a set S of λ-presentableobjects such that each Y ∈ A is a λ-directed colimit of objectsfrom S.If λ = ℵ0, A is called a finitely accessible category.
• Mod-R is finitely accessible.• Any additive λ-accessible category is a subcategory of a“module category”Mod-S: additive contravariant functors S → Ab
Silvana Bazzoni Negative solution to Rosicky’s problem
Accessible categories
Definitionλ infinite regular cardinal. A an additive category with λ-directedcolimits. An object X ∈ A is called λ-presentable if, for anyλ-directed system (Yi | i ∈ I ), we have an isomorphism
lim−→HomA(X ,Yi ) ∼= HomA(X , lim−→Yi ).
The category A is λ-accessible if it admits a set S of λ-presentableobjects such that each Y ∈ A is a λ-directed colimit of objectsfrom S.If λ = ℵ0, A is called a finitely accessible category.
• Mod-R is finitely accessible.
• Any additive λ-accessible category is a subcategory of a“module category”Mod-S: additive contravariant functors S → Ab
Silvana Bazzoni Negative solution to Rosicky’s problem
Accessible categories
Definitionλ infinite regular cardinal. A an additive category with λ-directedcolimits. An object X ∈ A is called λ-presentable if, for anyλ-directed system (Yi | i ∈ I ), we have an isomorphism
lim−→HomA(X ,Yi ) ∼= HomA(X , lim−→Yi ).
The category A is λ-accessible if it admits a set S of λ-presentableobjects such that each Y ∈ A is a λ-directed colimit of objectsfrom S.If λ = ℵ0, A is called a finitely accessible category.
• Mod-R is finitely accessible.• Any additive λ-accessible category is a subcategory of a“module category”
Mod-S: additive contravariant functors S → Ab
Silvana Bazzoni Negative solution to Rosicky’s problem
Accessible categories
Definitionλ infinite regular cardinal. A an additive category with λ-directedcolimits. An object X ∈ A is called λ-presentable if, for anyλ-directed system (Yi | i ∈ I ), we have an isomorphism
lim−→HomA(X ,Yi ) ∼= HomA(X , lim−→Yi ).
The category A is λ-accessible if it admits a set S of λ-presentableobjects such that each Y ∈ A is a λ-directed colimit of objectsfrom S.If λ = ℵ0, A is called a finitely accessible category.
• Mod-R is finitely accessible.• Any additive λ-accessible category is a subcategory of a“module category”Mod-S: additive contravariant functors S → Ab
Silvana Bazzoni Negative solution to Rosicky’s problem
• In a λ accessible category we can define λ-purity.
A sequence 0→ A→ B → C → 0 in A is λ-pure exact if
0→ HomA(X ,A)→ HomA(X ,B)→ HomA(X ,C )→ 0
is exact for all λ-presentable objects X ∈ A.• The class of λ-pure exact sequences gives rise to an exactstructure on A.
PropositionA additive λ-accessible category with coproducts. For everyY ∈ A there is a λ-pure exact sequence
0 −→ K −→ P ∼=⊕
X (HomA(X ,Y )) −→ Y −→ 0
X running over all isoclasses of λ-presentable objects of A.
• If B is a λ-accessible subcategory of A with coproducts, then
λ-pure gl. dimB ≤ λ-pure gl. dimA.
• A finitely accessible ⇒ A λ-accessible category for each infiniteregular cardinal λ.
Silvana Bazzoni Negative solution to Rosicky’s problem
• In a λ accessible category we can define λ-purity.A sequence 0→ A→ B → C → 0 in A is λ-pure exact if
0→ HomA(X ,A)→ HomA(X ,B)→ HomA(X ,C )→ 0
is exact for all λ-presentable objects X ∈ A.
• The class of λ-pure exact sequences gives rise to an exactstructure on A.
PropositionA additive λ-accessible category with coproducts. For everyY ∈ A there is a λ-pure exact sequence
0 −→ K −→ P ∼=⊕
X (HomA(X ,Y )) −→ Y −→ 0
X running over all isoclasses of λ-presentable objects of A.
• If B is a λ-accessible subcategory of A with coproducts, then
λ-pure gl. dimB ≤ λ-pure gl. dimA.
• A finitely accessible ⇒ A λ-accessible category for each infiniteregular cardinal λ.
Silvana Bazzoni Negative solution to Rosicky’s problem
• In a λ accessible category we can define λ-purity.A sequence 0→ A→ B → C → 0 in A is λ-pure exact if
0→ HomA(X ,A)→ HomA(X ,B)→ HomA(X ,C )→ 0
is exact for all λ-presentable objects X ∈ A.• The class of λ-pure exact sequences gives rise to an exactstructure on A.
PropositionA additive λ-accessible category with coproducts. For everyY ∈ A there is a λ-pure exact sequence
0 −→ K −→ P ∼=⊕
X (HomA(X ,Y )) −→ Y −→ 0
X running over all isoclasses of λ-presentable objects of A.
• If B is a λ-accessible subcategory of A with coproducts, then
λ-pure gl. dimB ≤ λ-pure gl. dimA.
• A finitely accessible ⇒ A λ-accessible category for each infiniteregular cardinal λ.
Silvana Bazzoni Negative solution to Rosicky’s problem
• In a λ accessible category we can define λ-purity.A sequence 0→ A→ B → C → 0 in A is λ-pure exact if
0→ HomA(X ,A)→ HomA(X ,B)→ HomA(X ,C )→ 0
is exact for all λ-presentable objects X ∈ A.• The class of λ-pure exact sequences gives rise to an exactstructure on A.
PropositionA additive λ-accessible category with coproducts.
For everyY ∈ A there is a λ-pure exact sequence
0 −→ K −→ P ∼=⊕
X (HomA(X ,Y )) −→ Y −→ 0
X running over all isoclasses of λ-presentable objects of A.
• If B is a λ-accessible subcategory of A with coproducts, then
λ-pure gl. dimB ≤ λ-pure gl. dimA.
• A finitely accessible ⇒ A λ-accessible category for each infiniteregular cardinal λ.
Silvana Bazzoni Negative solution to Rosicky’s problem
• In a λ accessible category we can define λ-purity.A sequence 0→ A→ B → C → 0 in A is λ-pure exact if
0→ HomA(X ,A)→ HomA(X ,B)→ HomA(X ,C )→ 0
is exact for all λ-presentable objects X ∈ A.• The class of λ-pure exact sequences gives rise to an exactstructure on A.
PropositionA additive λ-accessible category with coproducts. For everyY ∈ A there is a λ-pure exact sequence
0 −→ K −→ P ∼=⊕
X (HomA(X ,Y )) −→ Y −→ 0
X running over all isoclasses of λ-presentable objects of A.
• If B is a λ-accessible subcategory of A with coproducts, then
λ-pure gl. dimB ≤ λ-pure gl. dimA.
• A finitely accessible ⇒ A λ-accessible category for each infiniteregular cardinal λ.
Silvana Bazzoni Negative solution to Rosicky’s problem
• In a λ accessible category we can define λ-purity.A sequence 0→ A→ B → C → 0 in A is λ-pure exact if
0→ HomA(X ,A)→ HomA(X ,B)→ HomA(X ,C )→ 0
is exact for all λ-presentable objects X ∈ A.• The class of λ-pure exact sequences gives rise to an exactstructure on A.
PropositionA additive λ-accessible category with coproducts. For everyY ∈ A there is a λ-pure exact sequence
0 −→ K −→ P ∼=⊕
X (HomA(X ,Y )) −→ Y −→ 0
X running over all isoclasses of λ-presentable objects of A.
• If B is a λ-accessible subcategory of A with coproducts, then
λ-pure gl. dimB ≤ λ-pure gl. dimA.
• A finitely accessible ⇒ A λ-accessible category for each infiniteregular cardinal λ.
Silvana Bazzoni Negative solution to Rosicky’s problem
• In a λ accessible category we can define λ-purity.A sequence 0→ A→ B → C → 0 in A is λ-pure exact if
0→ HomA(X ,A)→ HomA(X ,B)→ HomA(X ,C )→ 0
is exact for all λ-presentable objects X ∈ A.• The class of λ-pure exact sequences gives rise to an exactstructure on A.
PropositionA additive λ-accessible category with coproducts. For everyY ∈ A there is a λ-pure exact sequence
0 −→ K −→ P ∼=⊕
X (HomA(X ,Y )) −→ Y −→ 0
X running over all isoclasses of λ-presentable objects of A.
• If B is a λ-accessible subcategory of A with coproducts, then
λ-pure gl. dimB ≤ λ-pure gl. dimA.
• A finitely accessible ⇒ A λ-accessible category for each infiniteregular cardinal λ.
Silvana Bazzoni Negative solution to Rosicky’s problem
• In a λ accessible category we can define λ-purity.A sequence 0→ A→ B → C → 0 in A is λ-pure exact if
0→ HomA(X ,A)→ HomA(X ,B)→ HomA(X ,C )→ 0
is exact for all λ-presentable objects X ∈ A.• The class of λ-pure exact sequences gives rise to an exactstructure on A.
PropositionA additive λ-accessible category with coproducts. For everyY ∈ A there is a λ-pure exact sequence
0 −→ K −→ P ∼=⊕
X (HomA(X ,Y )) −→ Y −→ 0
X running over all isoclasses of λ-presentable objects of A.
• If B is a λ-accessible subcategory of A with coproducts, then
λ-pure gl. dimB ≤ λ-pure gl. dimA.
• A finitely accessible ⇒ A λ-accessible category for each infiniteregular cardinal λ.
Silvana Bazzoni Negative solution to Rosicky’s problem
The counterexamples
Theorem (B. Stovıcek 2010)
(1) Let R = k[x , y ] with k uncountable or k[[x , y ]] or
(2) R = k(·⇒ ·) with k uncountable.
Then, λ-pure gl. dim(Mod-R) ≥ 2 ∀ infinite regular cardinal λ.
Sketch: (1) Q the quotient field of R.
• By Kaplansky 1966, Osofsky 1973, respectively, proj.dim. Q = 2.Hence ℵ0-pure global dimension of Mod-R ≥ 2.
• Let B be the category of torsion k[[x ]]-modules.For λ regular uncountable, λ-pure proj. dimB Pλ ≥ 2.
• B is a finitely accessible subcategory of Mod-R (trivial action ofy on modules in B). Hence:
2 ≤ λ-pure gl. dimB ≤ λ-pure gl. dim Mod-R.
Silvana Bazzoni Negative solution to Rosicky’s problem
The counterexamples
Theorem (B. Stovıcek 2010)
(1) Let R = k[x , y ] with k uncountable or k[[x , y ]] or
(2) R = k(·⇒ ·) with k uncountable.
Then, λ-pure gl. dim(Mod-R) ≥ 2 ∀ infinite regular cardinal λ.
Sketch: (1) Q the quotient field of R.
• By Kaplansky 1966, Osofsky 1973, respectively, proj.dim. Q = 2.Hence ℵ0-pure global dimension of Mod-R ≥ 2.
• Let B be the category of torsion k[[x ]]-modules.For λ regular uncountable, λ-pure proj. dimB Pλ ≥ 2.
• B is a finitely accessible subcategory of Mod-R (trivial action ofy on modules in B). Hence:
2 ≤ λ-pure gl. dimB ≤ λ-pure gl. dim Mod-R.
Silvana Bazzoni Negative solution to Rosicky’s problem
The counterexamples
Theorem (B. Stovıcek 2010)
(1) Let R = k[x , y ] with k uncountable or k[[x , y ]] or
(2) R = k(·⇒ ·) with k uncountable.
Then, λ-pure gl. dim(Mod-R) ≥ 2 ∀ infinite regular cardinal λ.
Sketch: (1) Q the quotient field of R.
• By Kaplansky 1966, Osofsky 1973, respectively, proj.dim. Q = 2.Hence ℵ0-pure global dimension of Mod-R ≥ 2.
• Let B be the category of torsion k[[x ]]-modules.For λ regular uncountable, λ-pure proj. dimB Pλ ≥ 2.
• B is a finitely accessible subcategory of Mod-R (trivial action ofy on modules in B). Hence:
2 ≤ λ-pure gl. dimB ≤ λ-pure gl. dim Mod-R.
Silvana Bazzoni Negative solution to Rosicky’s problem
The counterexamples
Theorem (B. Stovıcek 2010)
(1) Let R = k[x , y ] with k uncountable or k[[x , y ]] or
(2) R = k(·⇒ ·) with k uncountable.
Then, λ-pure gl. dim(Mod-R) ≥ 2 ∀ infinite regular cardinal λ.
Sketch: (1) Q the quotient field of R.
• By Kaplansky 1966, Osofsky 1973, respectively, proj.dim. Q = 2.Hence ℵ0-pure global dimension of Mod-R ≥ 2.
• Let B be the category of torsion k[[x ]]-modules.For λ regular uncountable, λ-pure proj. dimB Pλ ≥ 2.
• B is a finitely accessible subcategory of Mod-R (trivial action ofy on modules in B). Hence:
2 ≤ λ-pure gl. dimB ≤ λ-pure gl. dim Mod-R.
Silvana Bazzoni Negative solution to Rosicky’s problem
The counterexamples
Theorem (B. Stovıcek 2010)
(1) Let R = k[x , y ] with k uncountable or k[[x , y ]] or
(2) R = k(·⇒ ·) with k uncountable.
Then, λ-pure gl. dim(Mod-R) ≥ 2 ∀ infinite regular cardinal λ.
Sketch: (1) Q the quotient field of R.
• By Kaplansky 1966, Osofsky 1973, respectively, proj.dim. Q = 2.Hence ℵ0-pure global dimension of Mod-R ≥ 2.
• Let B be the category of torsion k[[x ]]-modules.For λ regular uncountable, λ-pure proj. dimB Pλ ≥ 2.
• B is a finitely accessible subcategory of Mod-R (trivial action ofy on modules in B). Hence:
2 ≤ λ-pure gl. dimB ≤ λ-pure gl. dim Mod-R.
Silvana Bazzoni Negative solution to Rosicky’s problem
The counterexamples
Theorem (B. Stovıcek 2010)
(1) Let R = k[x , y ] with k uncountable or k[[x , y ]] or
(2) R = k(·⇒ ·) with k uncountable.
Then, λ-pure gl. dim(Mod-R) ≥ 2 ∀ infinite regular cardinal λ.
Sketch: (1) Q the quotient field of R.
• By Kaplansky 1966, Osofsky 1973, respectively, proj.dim. Q = 2.
Hence ℵ0-pure global dimension of Mod-R ≥ 2.
• Let B be the category of torsion k[[x ]]-modules.For λ regular uncountable, λ-pure proj. dimB Pλ ≥ 2.
• B is a finitely accessible subcategory of Mod-R (trivial action ofy on modules in B). Hence:
2 ≤ λ-pure gl. dimB ≤ λ-pure gl. dim Mod-R.
Silvana Bazzoni Negative solution to Rosicky’s problem
The counterexamples
Theorem (B. Stovıcek 2010)
(1) Let R = k[x , y ] with k uncountable or k[[x , y ]] or
(2) R = k(·⇒ ·) with k uncountable.
Then, λ-pure gl. dim(Mod-R) ≥ 2 ∀ infinite regular cardinal λ.
Sketch: (1) Q the quotient field of R.
• By Kaplansky 1966, Osofsky 1973, respectively, proj.dim. Q = 2.Hence ℵ0-pure global dimension of Mod-R ≥ 2.
• Let B be the category of torsion k[[x ]]-modules.For λ regular uncountable, λ-pure proj. dimB Pλ ≥ 2.
• B is a finitely accessible subcategory of Mod-R (trivial action ofy on modules in B). Hence:
2 ≤ λ-pure gl. dimB ≤ λ-pure gl. dim Mod-R.
Silvana Bazzoni Negative solution to Rosicky’s problem
The counterexamples
Theorem (B. Stovıcek 2010)
(1) Let R = k[x , y ] with k uncountable or k[[x , y ]] or
(2) R = k(·⇒ ·) with k uncountable.
Then, λ-pure gl. dim(Mod-R) ≥ 2 ∀ infinite regular cardinal λ.
Sketch: (1) Q the quotient field of R.
• By Kaplansky 1966, Osofsky 1973, respectively, proj.dim. Q = 2.Hence ℵ0-pure global dimension of Mod-R ≥ 2.
• Let B be the category of torsion k[[x ]]-modules.
For λ regular uncountable, λ-pure proj. dimB Pλ ≥ 2.
• B is a finitely accessible subcategory of Mod-R (trivial action ofy on modules in B). Hence:
2 ≤ λ-pure gl. dimB ≤ λ-pure gl. dim Mod-R.
Silvana Bazzoni Negative solution to Rosicky’s problem
The counterexamples
Theorem (B. Stovıcek 2010)
(1) Let R = k[x , y ] with k uncountable or k[[x , y ]] or
(2) R = k(·⇒ ·) with k uncountable.
Then, λ-pure gl. dim(Mod-R) ≥ 2 ∀ infinite regular cardinal λ.
Sketch: (1) Q the quotient field of R.
• By Kaplansky 1966, Osofsky 1973, respectively, proj.dim. Q = 2.Hence ℵ0-pure global dimension of Mod-R ≥ 2.
• Let B be the category of torsion k[[x ]]-modules.For λ regular uncountable, λ-pure proj. dimB Pλ ≥ 2.
• B is a finitely accessible subcategory of Mod-R (trivial action ofy on modules in B). Hence:
2 ≤ λ-pure gl. dimB ≤ λ-pure gl. dim Mod-R.
Silvana Bazzoni Negative solution to Rosicky’s problem
The counterexamples
Theorem (B. Stovıcek 2010)
(1) Let R = k[x , y ] with k uncountable or k[[x , y ]] or
(2) R = k(·⇒ ·) with k uncountable.
Then, λ-pure gl. dim(Mod-R) ≥ 2 ∀ infinite regular cardinal λ.
Sketch: (1) Q the quotient field of R.
• By Kaplansky 1966, Osofsky 1973, respectively, proj.dim. Q = 2.Hence ℵ0-pure global dimension of Mod-R ≥ 2.
• Let B be the category of torsion k[[x ]]-modules.For λ regular uncountable, λ-pure proj. dimB Pλ ≥ 2.
• B is a finitely accessible subcategory of Mod-R (trivial action ofy on modules in B). Hence:
2 ≤ λ-pure gl. dimB ≤ λ-pure gl. dim Mod-R.
Silvana Bazzoni Negative solution to Rosicky’s problem
(2) By Lenzing 1984, the generic Kronecker module
G : k(x)x ·− //1·−
// k(x)
has ℵ0-pure projective dimension 2.
• B the class of all torsion k[[x ]]-modules.•The assignment
B 7→(
Bx ·− //1·−
// B)
induces an equivalence between B and a finitely accessiblesubcategory B′ of Mod-k(·⇒ ·).( B′ = lim−→(add t) where t is the tube corresponding to the regularmodule
k0 //1
// k )
2 ≤ λ-pure gl. dimB′ ≤ λ-pure gl. dim Mod-R.
Silvana Bazzoni Negative solution to Rosicky’s problem
(2) By Lenzing 1984, the generic Kronecker module
G : k(x)x ·− //1·−
// k(x)
has ℵ0-pure projective dimension 2.
• B the class of all torsion k[[x ]]-modules.
•The assignment
B 7→(
Bx ·− //1·−
// B)
induces an equivalence between B and a finitely accessiblesubcategory B′ of Mod-k(·⇒ ·).( B′ = lim−→(add t) where t is the tube corresponding to the regularmodule
k0 //1
// k )
2 ≤ λ-pure gl. dimB′ ≤ λ-pure gl. dim Mod-R.
Silvana Bazzoni Negative solution to Rosicky’s problem
(2) By Lenzing 1984, the generic Kronecker module
G : k(x)x ·− //1·−
// k(x)
has ℵ0-pure projective dimension 2.
• B the class of all torsion k[[x ]]-modules.•The assignment
B 7→(
Bx ·− //1·−
// B)
induces an equivalence between B and a finitely accessiblesubcategory B′ of Mod-k(·⇒ ·).
( B′ = lim−→(add t) where t is the tube corresponding to the regularmodule
k0 //1
// k )
2 ≤ λ-pure gl. dimB′ ≤ λ-pure gl. dim Mod-R.
Silvana Bazzoni Negative solution to Rosicky’s problem
(2) By Lenzing 1984, the generic Kronecker module
G : k(x)x ·− //1·−
// k(x)
has ℵ0-pure projective dimension 2.
• B the class of all torsion k[[x ]]-modules.•The assignment
B 7→(
Bx ·− //1·−
// B)
induces an equivalence between B and a finitely accessiblesubcategory B′ of Mod-k(·⇒ ·).( B′ = lim−→(add t) where t is the tube corresponding to the regularmodule
k0 //1
// k )
2 ≤ λ-pure gl. dimB′ ≤ λ-pure gl. dim Mod-R.
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-abelianization of a triangulatedcategory
Representability is related to abelianizations.
DefinitionT triangulated category with coproducts, λ a regular cardinal.Aλ(T ): category of contravariant functors F : T λ −→ Abwhich send coproducts with < λ summands to products.Aλ(T ) is a locally λ-presentable abelian category with enoughprojectives, with exact products, coproducts and λ-filtered colimits.The λ-abelianization of T is defined as the Yoneda functor
Hλ : T −→ Aλ(T ), X ∈ T 7→ HomT (−,X ) � T λ.
[ARMλ] holds if and only if the functor Hλ is full.
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-abelianization of a triangulatedcategory
Representability is related to abelianizations.
DefinitionT triangulated category with coproducts, λ a regular cardinal.Aλ(T ): category of contravariant functors F : T λ −→ Abwhich send coproducts with < λ summands to products.Aλ(T ) is a locally λ-presentable abelian category with enoughprojectives, with exact products, coproducts and λ-filtered colimits.The λ-abelianization of T is defined as the Yoneda functor
Hλ : T −→ Aλ(T ), X ∈ T 7→ HomT (−,X ) � T λ.
[ARMλ] holds if and only if the functor Hλ is full.
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-abelianization of a triangulatedcategory
Representability is related to abelianizations.
DefinitionT triangulated category with coproducts, λ a regular cardinal.
Aλ(T ): category of contravariant functors F : T λ −→ Abwhich send coproducts with < λ summands to products.Aλ(T ) is a locally λ-presentable abelian category with enoughprojectives, with exact products, coproducts and λ-filtered colimits.The λ-abelianization of T is defined as the Yoneda functor
Hλ : T −→ Aλ(T ), X ∈ T 7→ HomT (−,X ) � T λ.
[ARMλ] holds if and only if the functor Hλ is full.
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-abelianization of a triangulatedcategory
Representability is related to abelianizations.
DefinitionT triangulated category with coproducts, λ a regular cardinal.Aλ(T ): category of contravariant functors F : T λ −→ Abwhich send coproducts with < λ summands to products.
Aλ(T ) is a locally λ-presentable abelian category with enoughprojectives, with exact products, coproducts and λ-filtered colimits.The λ-abelianization of T is defined as the Yoneda functor
Hλ : T −→ Aλ(T ), X ∈ T 7→ HomT (−,X ) � T λ.
[ARMλ] holds if and only if the functor Hλ is full.
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-abelianization of a triangulatedcategory
Representability is related to abelianizations.
DefinitionT triangulated category with coproducts, λ a regular cardinal.Aλ(T ): category of contravariant functors F : T λ −→ Abwhich send coproducts with < λ summands to products.Aλ(T ) is a locally λ-presentable abelian category with enoughprojectives, with exact products, coproducts and λ-filtered colimits.The λ-abelianization of T is defined as the Yoneda functor
Hλ : T −→ Aλ(T ), X ∈ T 7→ HomT (−,X ) � T λ.
[ARMλ] holds if and only if the functor Hλ is full.
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-abelianization of a triangulatedcategory
Representability is related to abelianizations.
DefinitionT triangulated category with coproducts, λ a regular cardinal.Aλ(T ): category of contravariant functors F : T λ −→ Abwhich send coproducts with < λ summands to products.Aλ(T ) is a locally λ-presentable abelian category with enoughprojectives, with exact products, coproducts and λ-filtered colimits.The λ-abelianization of T is defined as the Yoneda functor
Hλ : T −→ Aλ(T ), X ∈ T 7→ HomT (−,X ) � T λ.
[ARMλ] holds if and only if the functor Hλ is full.
Silvana Bazzoni Negative solution to Rosicky’s problem
Projective dimension in Aλ(T )
Generalization of the result by Christensen, Keller, Neeman in2001, from ℵ0 to λ is:
Proposition (Muro, Raventos 2010)
Let T be a well generated triangulated category and λ a regularcardinal such that T is generated by T λ. Then:
I If global proj. dimAλ(T ) ≤ 2, then T satisfies [AROλ],
I [ARMλ] is satisfied for T , if and only ifproj. dimAλ(T ) Hλ(X ) ≤ 1 for all X ∈ T .
• If [ARMλ] holds the abelianization functor Hλ is a Rosicky’sfunctor.
Silvana Bazzoni Negative solution to Rosicky’s problem
Projective dimension in Aλ(T )
Generalization of the result by Christensen, Keller, Neeman in2001, from ℵ0 to λ is:
Proposition (Muro, Raventos 2010)
Let T be a well generated triangulated category and λ a regularcardinal such that T is generated by T λ. Then:
I If global proj. dimAλ(T ) ≤ 2, then T satisfies [AROλ],
I [ARMλ] is satisfied for T , if and only ifproj. dimAλ(T ) Hλ(X ) ≤ 1 for all X ∈ T .
• If [ARMλ] holds the abelianization functor Hλ is a Rosicky’sfunctor.
Silvana Bazzoni Negative solution to Rosicky’s problem
Projective dimension in Aλ(T )
Generalization of the result by Christensen, Keller, Neeman in2001, from ℵ0 to λ is:
Proposition (Muro, Raventos 2010)
Let T be a well generated triangulated category and λ a regularcardinal such that T is generated by T λ. Then:
I If global proj. dimAλ(T ) ≤ 2, then T satisfies [AROλ],
I [ARMλ] is satisfied for T , if and only ifproj. dimAλ(T ) Hλ(X ) ≤ 1 for all X ∈ T .
• If [ARMλ] holds the abelianization functor Hλ is a Rosicky’sfunctor.
Silvana Bazzoni Negative solution to Rosicky’s problem
Projective dimension in Aλ(T )
Generalization of the result by Christensen, Keller, Neeman in2001, from ℵ0 to λ is:
Proposition (Muro, Raventos 2010)
Let T be a well generated triangulated category and λ a regularcardinal such that T is generated by T λ. Then:
I If global proj. dimAλ(T ) ≤ 2, then T satisfies [AROλ],
I [ARMλ] is satisfied for T , if and only ifproj. dimAλ(T ) Hλ(X ) ≤ 1 for all X ∈ T .
• If [ARMλ] holds the abelianization functor Hλ is a Rosicky’sfunctor.
Silvana Bazzoni Negative solution to Rosicky’s problem
Projective dimension in Aλ(T )
Generalization of the result by Christensen, Keller, Neeman in2001, from ℵ0 to λ is:
Proposition (Muro, Raventos 2010)
Let T be a well generated triangulated category and λ a regularcardinal such that T is generated by T λ. Then:
I If global proj. dimAλ(T ) ≤ 2, then T satisfies [AROλ],
I [ARMλ] is satisfied for T , if and only ifproj. dimAλ(T ) Hλ(X ) ≤ 1 for all X ∈ T .
• If [ARMλ] holds the abelianization functor Hλ is a Rosicky’sfunctor.
Silvana Bazzoni Negative solution to Rosicky’s problem
Projective dimension in Aλ(T )
Generalization of the result by Christensen, Keller, Neeman in2001, from ℵ0 to λ is:
Proposition (Muro, Raventos 2010)
Let T be a well generated triangulated category and λ a regularcardinal such that T is generated by T λ. Then:
I If global proj. dimAλ(T ) ≤ 2, then T satisfies [AROλ],
I [ARMλ] is satisfied for T , if and only ifproj. dimAλ(T ) Hλ(X ) ≤ 1 for all X ∈ T .
• If [ARMλ] holds the abelianization functor Hλ is a Rosicky’sfunctor.
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-pure global dimension
PropositionR a ring, T = D(R) and λ a regular cardinal.
I If R is right coherent ring and every finitely presented rightmodule has proj.dim. <∞, then
λ-pure gl. dim Mod-R ≤ sup{proj. dimAλ(T ) Hλ(X ) | X ∈ D(R)}.
I If R is right hereditary, then equality in the above formulaholds.
• Our examples of rings R are such that D(R) does not satisfy[ARMλ] for any infinite regular cardinal λ.
• Hence the abelianization functor Hλ is NOT a Rosicky’sfunctor.
• One has to find other ways for proving the existence of aRosicky’s functor.
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-pure global dimension
PropositionR a ring, T = D(R) and λ a regular cardinal.
I If R is right coherent ring and every finitely presented rightmodule has proj.dim. <∞, then
λ-pure gl. dim Mod-R ≤ sup{proj. dimAλ(T ) Hλ(X ) | X ∈ D(R)}.
I If R is right hereditary, then equality in the above formulaholds.
• Our examples of rings R are such that D(R) does not satisfy[ARMλ] for any infinite regular cardinal λ.
• Hence the abelianization functor Hλ is NOT a Rosicky’sfunctor.
• One has to find other ways for proving the existence of aRosicky’s functor.
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-pure global dimension
PropositionR a ring, T = D(R) and λ a regular cardinal.
I If R is right coherent ring and every finitely presented rightmodule has proj.dim. <∞, then
λ-pure gl. dim Mod-R ≤ sup{proj. dimAλ(T ) Hλ(X ) | X ∈ D(R)}.
I If R is right hereditary, then equality in the above formulaholds.
• Our examples of rings R are such that D(R) does not satisfy[ARMλ] for any infinite regular cardinal λ.
• Hence the abelianization functor Hλ is NOT a Rosicky’sfunctor.
• One has to find other ways for proving the existence of aRosicky’s functor.
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-pure global dimension
PropositionR a ring, T = D(R) and λ a regular cardinal.
I If R is right coherent ring and every finitely presented rightmodule has proj.dim. <∞, then
λ-pure gl. dim Mod-R ≤ sup{proj. dimAλ(T ) Hλ(X ) | X ∈ D(R)}.
I If R is right hereditary, then equality in the above formulaholds.
• Our examples of rings R are such that D(R) does not satisfy[ARMλ] for any infinite regular cardinal λ.
• Hence the abelianization functor Hλ is NOT a Rosicky’sfunctor.
• One has to find other ways for proving the existence of aRosicky’s functor.
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-pure global dimension
PropositionR a ring, T = D(R) and λ a regular cardinal.
I If R is right coherent ring and every finitely presented rightmodule has proj.dim. <∞, then
λ-pure gl. dim Mod-R ≤ sup{proj. dimAλ(T ) Hλ(X ) | X ∈ D(R)}.
I If R is right hereditary, then equality in the above formulaholds.
• Our examples of rings R are such that D(R) does not satisfy[ARMλ] for any infinite regular cardinal λ.
• Hence the abelianization functor Hλ is NOT a Rosicky’sfunctor.
• One has to find other ways for proving the existence of aRosicky’s functor.
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-pure global dimension
PropositionR a ring, T = D(R) and λ a regular cardinal.
I If R is right coherent ring and every finitely presented rightmodule has proj.dim. <∞, then
λ-pure gl. dim Mod-R ≤ sup{proj. dimAλ(T ) Hλ(X ) | X ∈ D(R)}.
I If R is right hereditary, then equality in the above formulaholds.
• Our examples of rings R are such that D(R) does not satisfy[ARMλ] for any infinite regular cardinal λ.
• Hence the abelianization functor Hλ is NOT a Rosicky’sfunctor.
• One has to find other ways for proving the existence of aRosicky’s functor.
Silvana Bazzoni Negative solution to Rosicky’s problem
The λ-pure global dimension
PropositionR a ring, T = D(R) and λ a regular cardinal.
I If R is right coherent ring and every finitely presented rightmodule has proj.dim. <∞, then
λ-pure gl. dim Mod-R ≤ sup{proj. dimAλ(T ) Hλ(X ) | X ∈ D(R)}.
I If R is right hereditary, then equality in the above formulaholds.
• Our examples of rings R are such that D(R) does not satisfy[ARMλ] for any infinite regular cardinal λ.
• Hence the abelianization functor Hλ is NOT a Rosicky’sfunctor.
• One has to find other ways for proving the existence of aRosicky’s functor.
Silvana Bazzoni Negative solution to Rosicky’s problem
Abundance of examples
propositionA a locally finitely presentable Grothendieck category, λ anuncountable regular cardinal.
If A contains a tube or is strictlywild. Then λ-pure gl. dimA ≥ 2.
DefinitionA tube in A is a finitely presented object S such that
1. k = EndA(S) is a skew-field;
2. Ext1A(S ,S) ∼= k and Ext2A(S ,S) = 0.
Examples• R Dedekind domain A = Mod-R, S any simple module.• R tame hereditary artin algebra, A = Mod-R, S a quasi-simpleregular module.• A = Qco(X ), Xprojective or affine curve, S the simple coherentsheaf corresponding to a non-singular closed point p ∈ X
Silvana Bazzoni Negative solution to Rosicky’s problem
Abundance of examples
propositionA a locally finitely presentable Grothendieck category, λ anuncountable regular cardinal. If A contains a tube or is strictlywild. Then λ-pure gl. dimA ≥ 2.
DefinitionA tube in A is a finitely presented object S such that
1. k = EndA(S) is a skew-field;
2. Ext1A(S ,S) ∼= k and Ext2A(S ,S) = 0.
Examples• R Dedekind domain A = Mod-R, S any simple module.• R tame hereditary artin algebra, A = Mod-R, S a quasi-simpleregular module.• A = Qco(X ), Xprojective or affine curve, S the simple coherentsheaf corresponding to a non-singular closed point p ∈ X
Silvana Bazzoni Negative solution to Rosicky’s problem
Abundance of examples
propositionA a locally finitely presentable Grothendieck category, λ anuncountable regular cardinal. If A contains a tube or is strictlywild. Then λ-pure gl. dimA ≥ 2.
DefinitionA tube in A is a finitely presented object S such that
1. k = EndA(S) is a skew-field;
2. Ext1A(S , S) ∼= k and Ext2A(S , S) = 0.
Examples• R Dedekind domain A = Mod-R, S any simple module.• R tame hereditary artin algebra, A = Mod-R, S a quasi-simpleregular module.• A = Qco(X ), Xprojective or affine curve, S the simple coherentsheaf corresponding to a non-singular closed point p ∈ X
Silvana Bazzoni Negative solution to Rosicky’s problem
Abundance of examples
propositionA a locally finitely presentable Grothendieck category, λ anuncountable regular cardinal. If A contains a tube or is strictlywild. Then λ-pure gl. dimA ≥ 2.
DefinitionA tube in A is a finitely presented object S such that
1. k = EndA(S) is a skew-field;
2. Ext1A(S , S) ∼= k and Ext2A(S , S) = 0.
Examples• R Dedekind domain A = Mod-R, S any simple module.
• R tame hereditary artin algebra, A = Mod-R, S a quasi-simpleregular module.• A = Qco(X ), Xprojective or affine curve, S the simple coherentsheaf corresponding to a non-singular closed point p ∈ X
Silvana Bazzoni Negative solution to Rosicky’s problem
Abundance of examples
propositionA a locally finitely presentable Grothendieck category, λ anuncountable regular cardinal. If A contains a tube or is strictlywild. Then λ-pure gl. dimA ≥ 2.
DefinitionA tube in A is a finitely presented object S such that
1. k = EndA(S) is a skew-field;
2. Ext1A(S , S) ∼= k and Ext2A(S , S) = 0.
Examples• R Dedekind domain A = Mod-R, S any simple module.• R tame hereditary artin algebra, A = Mod-R, S a quasi-simpleregular module.
• A = Qco(X ), Xprojective or affine curve, S the simple coherentsheaf corresponding to a non-singular closed point p ∈ X
Silvana Bazzoni Negative solution to Rosicky’s problem
Abundance of examples
propositionA a locally finitely presentable Grothendieck category, λ anuncountable regular cardinal. If A contains a tube or is strictlywild. Then λ-pure gl. dimA ≥ 2.
DefinitionA tube in A is a finitely presented object S such that
1. k = EndA(S) is a skew-field;
2. Ext1A(S , S) ∼= k and Ext2A(S , S) = 0.
Examples• R Dedekind domain A = Mod-R, S any simple module.• R tame hereditary artin algebra, A = Mod-R, S a quasi-simpleregular module.• A = Qco(X ), Xprojective or affine curve, S the simple coherentsheaf corresponding to a non-singular closed point p ∈ X
Silvana Bazzoni Negative solution to Rosicky’s problem
DefinitionA loc. fin. pres. Grothendieck is called strictly wild if there is afield k such that given any finite dimensional k-algebra R, there isa fully faithful functor Φ: mod-R → fpA.
Examples• A = Mod-kQ, Q a wild finite quiver without oriented cycles.• Klinger, Levy, 2006: Mod-R for some commutative noetherianrings are strictly wild.
Silvana Bazzoni Negative solution to Rosicky’s problem
DefinitionA loc. fin. pres. Grothendieck is called strictly wild if there is afield k such that given any finite dimensional k-algebra R, there isa fully faithful functor Φ: mod-R → fpA.
Examples• A = Mod-kQ, Q a wild finite quiver without oriented cycles.
• Klinger, Levy, 2006: Mod-R for some commutative noetherianrings are strictly wild.
Silvana Bazzoni Negative solution to Rosicky’s problem
DefinitionA loc. fin. pres. Grothendieck is called strictly wild if there is afield k such that given any finite dimensional k-algebra R, there isa fully faithful functor Φ: mod-R → fpA.
Examples• A = Mod-kQ, Q a wild finite quiver without oriented cycles.• Klinger, Levy, 2006: Mod-R for some commutative noetherianrings are strictly wild.
Silvana Bazzoni Negative solution to Rosicky’s problem
Tools:
PropositionR hereditary noetherian uniserial ring, λ uncountable regularcardinal. There is a semiartinian right R-module Pλ such thatλ-pure proj. dim Pλ ≥ 2.
S a tube in A Grothendieck loc. fin. pres.S ⊆ A full subcategory of objects X such that:
0 = X0 ⊆ X1 ⊆ · · · ⊆ X` = X
with Xi+1/Xi∼= S , for each 0 ≤ i < `.
S is the unique simple in S and S is uniserialB = lim−→S is loc. finite Grothendieck.E injective envelope of S in B. R = EndB(E ) is an hereditarynoetherian uniserial complete ring,B = lim−→S is finitely accessible and equivalent to the category ofsemiartinian right R-modules.
Silvana Bazzoni Negative solution to Rosicky’s problem
Tools:
PropositionR hereditary noetherian uniserial ring, λ uncountable regularcardinal. There is a semiartinian right R-module Pλ such thatλ-pure proj. dim Pλ ≥ 2.
S a tube in A Grothendieck loc. fin. pres.S ⊆ A full subcategory of objects X such that:
0 = X0 ⊆ X1 ⊆ · · · ⊆ X` = X
with Xi+1/Xi∼= S , for each 0 ≤ i < `.
S is the unique simple in S and S is uniserialB = lim−→S is loc. finite Grothendieck.E injective envelope of S in B. R = EndB(E ) is an hereditarynoetherian uniserial complete ring,B = lim−→S is finitely accessible and equivalent to the category ofsemiartinian right R-modules.
Silvana Bazzoni Negative solution to Rosicky’s problem
Tools:
PropositionR hereditary noetherian uniserial ring, λ uncountable regularcardinal. There is a semiartinian right R-module Pλ such thatλ-pure proj. dim Pλ ≥ 2.
S a tube in A Grothendieck loc. fin. pres.S ⊆ A full subcategory of objects X such that:
0 = X0 ⊆ X1 ⊆ · · · ⊆ X` = X
with Xi+1/Xi∼= S , for each 0 ≤ i < `.
S is the unique simple in S and S is uniserialB = lim−→S is loc. finite Grothendieck.E injective envelope of S in B. R = EndB(E ) is an hereditarynoetherian uniserial complete ring,B = lim−→S is finitely accessible and equivalent to the category ofsemiartinian right R-modules.
Silvana Bazzoni Negative solution to Rosicky’s problem
Tools:
PropositionR hereditary noetherian uniserial ring, λ uncountable regularcardinal. There is a semiartinian right R-module Pλ such thatλ-pure proj. dim Pλ ≥ 2.
S a tube in A Grothendieck loc. fin. pres.S ⊆ A full subcategory of objects X such that:
0 = X0 ⊆ X1 ⊆ · · · ⊆ X` = X
with Xi+1/Xi∼= S , for each 0 ≤ i < `.
S is the unique simple in S and S is uniserialB = lim−→S is loc. finite Grothendieck.
E injective envelope of S in B. R = EndB(E ) is an hereditarynoetherian uniserial complete ring,B = lim−→S is finitely accessible and equivalent to the category ofsemiartinian right R-modules.
Silvana Bazzoni Negative solution to Rosicky’s problem
Tools:
PropositionR hereditary noetherian uniserial ring, λ uncountable regularcardinal. There is a semiartinian right R-module Pλ such thatλ-pure proj. dim Pλ ≥ 2.
S a tube in A Grothendieck loc. fin. pres.S ⊆ A full subcategory of objects X such that:
0 = X0 ⊆ X1 ⊆ · · · ⊆ X` = X
with Xi+1/Xi∼= S , for each 0 ≤ i < `.
S is the unique simple in S and S is uniserialB = lim−→S is loc. finite Grothendieck.E injective envelope of S in B. R = EndB(E ) is an hereditarynoetherian uniserial complete ring,
B = lim−→S is finitely accessible and equivalent to the category ofsemiartinian right R-modules.
Silvana Bazzoni Negative solution to Rosicky’s problem
Tools:
PropositionR hereditary noetherian uniserial ring, λ uncountable regularcardinal. There is a semiartinian right R-module Pλ such thatλ-pure proj. dim Pλ ≥ 2.
S a tube in A Grothendieck loc. fin. pres.S ⊆ A full subcategory of objects X such that:
0 = X0 ⊆ X1 ⊆ · · · ⊆ X` = X
with Xi+1/Xi∼= S , for each 0 ≤ i < `.
S is the unique simple in S and S is uniserialB = lim−→S is loc. finite Grothendieck.E injective envelope of S in B. R = EndB(E ) is an hereditarynoetherian uniserial complete ring,B = lim−→S is finitely accessible and equivalent to the category ofsemiartinian right R-modules.
Silvana Bazzoni Negative solution to Rosicky’s problem
Let A be a loc. fin. pres. Grothendieck category strictly wild.
Then there is a field k and a fully faithful functor
mod-k(·⇒ ·) −→ fpA,
which extends to a fully faithful and direct limit preserving functor
Mod-k(·⇒ ·) −→ A.
whose essential image is a finitely accessible subcategory of A.The category of all torsion k[[x ]]-modules embeds as a finitelyaccessible subcategory in Mod-k(·⇒ ·), hence also in A.
Silvana Bazzoni Negative solution to Rosicky’s problem
Let A be a loc. fin. pres. Grothendieck category strictly wild.Then there is a field k and a fully faithful functor
mod-k(·⇒ ·) −→ fpA,
which extends to a fully faithful and direct limit preserving functor
Mod-k(·⇒ ·) −→ A.
whose essential image is a finitely accessible subcategory of A.The category of all torsion k[[x ]]-modules embeds as a finitelyaccessible subcategory in Mod-k(·⇒ ·), hence also in A.
Silvana Bazzoni Negative solution to Rosicky’s problem
Let A be a loc. fin. pres. Grothendieck category strictly wild.Then there is a field k and a fully faithful functor
mod-k(·⇒ ·) −→ fpA,
which extends to a fully faithful and direct limit preserving functor
Mod-k(·⇒ ·) −→ A.
whose essential image is a finitely accessible subcategory of A.The category of all torsion k[[x ]]-modules embeds as a finitelyaccessible subcategory in Mod-k(·⇒ ·), hence also in A.
Silvana Bazzoni Negative solution to Rosicky’s problem
Let A be a loc. fin. pres. Grothendieck category strictly wild.Then there is a field k and a fully faithful functor
mod-k(·⇒ ·) −→ fpA,
which extends to a fully faithful and direct limit preserving functor
Mod-k(·⇒ ·) −→ A.
whose essential image is a finitely accessible subcategory of A.
The category of all torsion k[[x ]]-modules embeds as a finitelyaccessible subcategory in Mod-k(·⇒ ·), hence also in A.
Silvana Bazzoni Negative solution to Rosicky’s problem
Let A be a loc. fin. pres. Grothendieck category strictly wild.Then there is a field k and a fully faithful functor
mod-k(·⇒ ·) −→ fpA,
which extends to a fully faithful and direct limit preserving functor
Mod-k(·⇒ ·) −→ A.
whose essential image is a finitely accessible subcategory of A.The category of all torsion k[[x ]]-modules embeds as a finitelyaccessible subcategory in Mod-k(·⇒ ·), hence also in A.
Silvana Bazzoni Negative solution to Rosicky’s problem