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