Structural decompositions in module theory and their constraints

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Structural decompositions in module theory and their constraints

9th International Algebraic Conference in Ukraine Lviv, July 8, 2013

Jan Trlifaj (Univerzita Karlova, Praha)

(IACU’2013) Constraints for structural decompositions 1 / 27

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Overview

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Overview

Part I: Decomposable classes

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Overview

Part I: Decomposable classes (the rare jewels)

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Overview

1 Classic decomposition theorems.

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Overview

Part II: Deconstructible classes

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Overview

Part II: Deconstructible classes (the ubiquitous mainstream)

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Overview

1 Filtrations and transfinite extensions.

2 Deconstructibility and approximations.

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Overview

Part III: Non-deconstructibility

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Overview

Part III: Non-deconstructibility (reaching the limits)

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Overview

Part III: Non-deconstructibility (reaching the limits)

1 The basic example: Mittag-Leffler modules.

2 Trees and locally free modules.

3 Non-deconstructibility and infinite dimensional tilting theory.

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Part I: Decomposable classes

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Part I: Decomposable classes (blocks put in a row)

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Definition

A class of modules C is decomposable, provided there is a cardinalκ such that each module in C is a direct sum of < κ-generated modules fromC.

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Definition

Some classic examples

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Definition

[Gruson-Jensen’73], [Huisgen-Zimmermann’79]

Mod-R is decomposable, iff R is right pure-semisimple.

Uniformly: κ=ℵ₀ sufficient for all such R;

uniqueness by Krull-Schmidt-Azumaya.

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Definition

[Kaplansky’58] The class P₀ is decomposable.

Uniformly: κ=ℵ1 sufficient for all R, but no uniqueness in general.

E.g.,I⊕I⁻¹ ∼=R⁽²⁾ for each non-principal idealI of a Dedekind domainR.

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Definition

[Kaplansky’58] The class P₀ is decomposable.

Uniformly: κ=ℵ1 sufficient for all R, but no uniqueness in general.

E.g.,I⊕I⁻¹ ∼=R⁽²⁾ for each non-principal idealI of a Dedekind domainR.

[Faith-Walker’67] The class I0 of all injective modules is decomposable, iff R is right noetherian.

Here, κ depends R; uniqueness by Krull-Schmidt-Azumaya.

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Part II: Deconstructible classes

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Part II: Deconstructible classes (blocks put on top of other blocks)

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Definition

Let C ⊆Mod-R. A moduleM is C-filtered(or atransfinite extension of the modules inC), provided that there exists an increasing sequence (Mα|α≤σ) consisting of submodules ofM such thatM₀ = 0,Mσ =M,

Mα =S

β<αMβ for each limit ordinal α≤σ, and

for eachα < σ,Mα+1/Mα is isomorphic to an element ofC.

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Definition

Mα =S

Notation: M ∈Filt(C).

A class Ais closed under transfinite extensions, if Filt(A)⊆ A.

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Definition

Mα =S

Notation: M ∈Filt(C).

A class Ais closed under transfinite extensions, if Filt(A)⊆ A.

Eklof Lemma

The class ^⊥C:= KerExt¹_R(−,C) is closed under transfinite extensions for each class of modules C.

In particular, so are the classes Pn and Fn of all modules of projective and flat dimension ≤n, for eachn< ω.

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The ubiquity of deconstructible classes

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The ubiquity of deconstructible classes

Definition (Eklof’06)

A class of modules Ais deconstructible, provided there is a cardinalκ such that A ⊆Filt(A^<κ), where A^<κ denotes the class of all

< κ-presented modules fromA.

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The ubiquity of deconstructible classes

All decomposable classes are deconstructible (but not vice versa).

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The ubiquity of deconstructible classes

[Enochs et al.’01]

For each n < ω, the classesPn andFn are deconstructible.

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The ubiquity of deconstructible classes

[Enochs et al.’01]

For each n < ω, the classesPn andFn are deconstructible.

[Eklof-T.’01], [ˇSˇtov´ıˇcek-T.’09]

For each set of modules S, the class ^⊥(S^⊥) is deconstructible.

Here, S^⊥:= KerExt¹_R(S,−).

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Approximations of modules

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Approximations of modules

A class of modules Ais precovering if for each moduleM there is f ∈HomR(A,M) with A∈ Asuch that each f^′ ∈HomR(A^′,M) with A^′ ∈ Ahas a factorization throughf:

A ^f ^//M

A^′

OO f^′

>>

}} }} }} }

The mapf is called an A–precoverof M.

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Approximations of modules

A class of modules Ais precovering if for each moduleM there is f ∈HomR(A,M) with A∈ Asuch that each f^′ ∈HomR(A^′,M) with A^′ ∈ Ahas a factorization throughf:

A ^f ^//M

A^′

OO f^′

>>

}} }} }} }

The mapf is called an A–precoverof M.

[Saor´ın-ˇSˇtov´ıˇcek’11], [Enochs’12]

All deconstructible classes closed under transfinite extensions are precovering.

In particular, so are the classes ^⊥(S^⊥) for all sets of modulesS.

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Some questions

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Some questions

Is each class of modules closed under transfinite extensions deconstructible/precovering?

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Some questions

Is each class of modules closed under transfinite extensions deconstructible/precovering?

What about the classes of the form^⊥C?

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Part III: Non-deconstructible classes

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Part III: Non-deconstructible classes (no block pattern at all)

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First examples

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First examples

[Eklof-Shelah’03]

Let W :=^⊥{Z} denote the class of all Whitehead groups.

It is independent of ZFC whether W is precovering (or deconstructible).

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First examples

[Eklof-Shelah’03]

A result in ZFC

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First examples

[Eklof-Shelah’03]

A result in ZFC

A module M is flat Mittag-Lefflerprovided the functor M⊗_R− is exact, and for each system of left R-modules (N_i |i ∈I), the canonical map M ⊗R Q

i∈INi →Q

i∈IM ⊗R Ni is monic.

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First examples

[Eklof-Shelah’03]

A result in ZFC

A module M is flat Mittag-Lefflerprovided the functor M⊗_R− is exact, and for each system of left R-modules (N_i |i ∈I), the canonical map M ⊗R Q

i∈INi →Q

i∈IM ⊗R Ni is monic.

Assume that R is not right perfect.

[Herbera-T.’12] The class FMof all flat Mittag-Leffler modules is closed under transfinite extensions, but it is not deconstructible.

[ˇSaroch-T.’12], [Bazzoni-ˇSˇtov´ıˇcek’12] IfR is countable, thenFMis not precovering.

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Further questions

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Further questions

Is non-deconstructibility a more general phenomenon?

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Further questions

Is non-deconstructibility a more general phenomenon?

Still open

Can the class ^⊥C be non-deconstructible/non-precovering in ZFC?

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Locally F -free modules

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Locally F -free modules

Let R be a ring, and F a class of countably presented modules.

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Locally F -free modules

Definition

A module M is locallyF-free, ifM possesses a subsetS consisting of countably F-filtered modules, such that

each countable subset ofM is contained in an element of S, 0∈ S, and S is closed under unions of countable chains.

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Locally F -free modules

Definition

Let L denote the class of all locallyF-free modules.

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Locally F -free modules

Definition

Let L denote the class of all locallyF-free modules.

Note: IfM is countably generated, thenM is locallyF-free, iffM is countably F-filtered.

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Flat Mittag-Leffler modules are locally F -free

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Flat Mittag-Leffler modules are locally F -free

Theorem (Herbera-T.’12)

Let F =be the class of all countably presented projective modules. Then the notions of a locally F-free module and a flat Mittag-Leffler module coincide for any ring R.

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Flat Mittag-Leffler modules are locally F -free

Theorem (Herbera-T.’12)

Let F =be the class of all countably presented projective modules. Then the notions of a locally F-free module and a flat Mittag-Leffler module coincide for any ring R.

For instance, if R=Z, then an abelian group Ais flat Mittag-Leffler, iff all countable subgroups of Aare free.

In particular, the Baer-Specker group Z^κ is flat Mittag-Leffler for eachκ, but not free.

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Trees for locally F -free modules

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Trees for locally F -free modules

Let κ be an infinite cardinal, andT_κ be the set of all finite sequences of ordinals < κ, so

Tκ={τ :n →κ|n< ω}.

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Trees for locally F -free modules

Tκ={τ :n →κ|n< ω}.

Partially ordered by inclusion, Tκ is a tree, called thetree on κ.

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Trees for locally F -free modules

Tκ={τ :n →κ|n< ω}.

Let Br(Tκ) denote the set of all branches of Tκ. Each ν∈Br(Tκ) can be identified with an ω-sequence of ordinals < κ:

Br(T_κ) ={ν:ω→κ}.

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Trees for locally F -free modules

Tκ={τ :n →κ|n< ω}.

Let Br(Tκ) denote the set of all branches of Tκ. Each ν∈Br(Tκ) can be identified with an ω-sequence of ordinals < κ:

Br(T_κ) ={ν:ω→κ}.

cardTκ =κ and card Br(Tκ) =κ^ω.

Notation: ℓ(τ) denotes the length of τ for eachτ ∈Tκ.

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The Bass modules

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The Bass modules

Let R be a ring, and F be a class of countably presented modules.

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The Bass modules

lim−→^ωF denotes the class of allBass modules, i.e., the modules N that are countable direct limits of the modules from F. W.l.o.g., suchN is the direct limit of a chain

F₀ →^g⁰ F₁→^g¹ . . .^g→ⁱ⁻¹Fi gi

→F_i+1 ^g→ⁱ⁺¹ . . . with Fi ∈ F and gi ∈HomR(Fi,Fi+1) for all i < ω.

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The Bass modules

lim−→^ωF denotes the class of allBass modules, i.e., the modules N that are countable direct limits of the modules from F. W.l.o.g., suchN is the direct limit of a chain

F₀ →^g⁰ F₁→^g¹ . . .^g→ⁱ⁻¹Fi gi

→F_i+1 ^g→ⁱ⁺¹ . . . with Fi ∈ F and gi ∈HomR(Fi,Fi+1) for all i < ω.

Example

Let F be the class of all countably generated projective modules. Then the Bass modules coincide with the countably presented flat modules.

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Decorating trees by Bass modules

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Decorating trees by Bass modules

Let D:=L

τ∈TκF_ℓ(τ), andP :=Q

τ∈TκF_ℓ(τ).

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Decorating trees by Bass modules

Let D:=L

τ∈TκF_ℓ(τ).

For ν ∈Br(Tκ), i < ω, andx∈Fi, we definexνi ∈P by πν↾i(xνi) =x,

π_ν_↾j(x_νi) =g_j−1. . .g_i(x) for each i <j < ω, π_τ(x_νi) = 0 otherwise,

where πτ ∈HomR(P,F_ℓ(τ)) denotes theτth projection for each τ ∈Tκ.

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Decorating trees by Bass modules

Let D:=L

τ∈TκF_ℓ(τ).

For ν ∈Br(Tκ), i < ω, andx∈Fi, we definexνi ∈P by πν↾i(xνi) =x,

π_ν_↾j(x_νi) =g_j−1. . .g_i(x) for each i <j < ω, π_τ(x_νi) = 0 otherwise,

where πτ ∈HomR(P,F_ℓ(τ)) denotes theτth projection for each τ ∈Tκ. Let Xνi :={xνi |x ∈Fi}. Then Xνi is a submodule ofP isomorphic to Fi.

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The locally F -free module L

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The locally F -free module L

Let X_ν :=P

i<ωX_νi, andL:=P

ν∈Br(Tκ)X_ν.

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The locally F -free module L

Let X_ν :=P

i<ωX_νi, andL:=P

ν∈Br(Tκ)X_ν. Lemma

D ⊆L⊆P.

L/D ∼=N⁽Br(T^κ)). L is locally F-free.

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The locally F -free module L

Let X_ν :=P

i<ωX_νi, andL:=P

ν∈Br(Tκ)X_ν. Lemma

D ⊆L⊆P.

L/D ∼=N⁽Br(T^κ)). L is locally F-free.

Lemma (Sl´avik-T.)

L is closed under transfinite extensions.

L^⊥ ⊆(lim−→^ωF)^⊥.

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Non-deconstructibility of locally F -free modules

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Non-deconstructibility of locally F -free modules

• F a class of countably presented modules,

• L the class of all locallyF-free modules,

• D the class of all direct summands of the modulesM that fit into an exact sequence

0→F^′ →M →C^′ →0,

where F^′ is a free module, andC^′ is countablyF-filtered.

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Non-deconstructibility of locally F -free modules

• F a class of countably presented modules,

• L the class of all locallyF-free modules,

• D the class of all direct summands of the modulesM that fit into an exact sequence

0→F^′ →M →C^′ →0,

where F^′ is a free module, andC^′ is countablyF-filtered.

Theorem (Sl´avik-T.)

Assume there exists a Bass module N ∈ D. Then the class/ Lis not deconstructible.

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Flat Mittag-Leffler modules revisited

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Flat Mittag-Leffler modules revisited

Corollary

FM is not deconstructible for each non-right perfect ring R.

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Flat Mittag-Leffler modules revisited

Corollary

Proof: IfR a non-right perfect ring, then there is a strictly decreasing chain of principal left ideals

Ra₀ )· · ·)Ran. . .a₀ )Ra_n+1an. . .ao ). . .

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Flat Mittag-Leffler modules revisited

Corollary

Ra₀ )· · ·)Ran. . .a₀ )Ra_n+1an. . .ao ). . .

Let F be the class of all countably presented projective modules. Consider the Bass moduleN which is a direct limit of the chain

R â→⁰^.R→â¹^.. . .â→ⁱ⁻¹^.R →âⁱ^.R â→ⁱ⁺¹^.. . .

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Flat Mittag-Leffler modules revisited

Corollary

Ra₀ )· · ·)Ran. . .a₀ )Ra_n+1an. . .ao ). . .

Let F be the class of all countably presented projective modules. Consider the Bass moduleN which is a direct limit of the chain

R â→⁰^.R→â¹^.. . .â→ⁱ⁻¹^.R →âⁱ^.R â→ⁱ⁺¹^.. . . Then there is a non-split pure-exact sequence

0→R^(ω)→^f R^(ω)→N→0,

where f(1i) = 1i −ai.1_i+1 for all i < ω. Then N∈ D/ =P₀.

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Infinite dimensional tilting modules

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Infinite dimensional tilting modules

Definition

T is atilting module provided that T has finite projective dimension,

Extⁱ_R(T,T^(κ)) = 0 for each cardinalκ, and

there exists an exact sequence 0→R→T₀ → · · · →T_r →0 such that r < ω, and for eachi <r,Ti ∈Add(T), i.e., Ti is a direct summand of a (possibly infinite) direct sum of copies of T .

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Infinite dimensional tilting modules

Definition

there exists an exact sequence 0→R→T₀ → · · · →T_r →0 such that r < ω, and for eachi <r,Ti ∈Add(T), i.e., Ti is a direct summand of a (possibly infinite) direct sum of copies of T . BT :={T}^⊥^∞ =T

1<iKerExtⁱ_R(T,−) theright tilting classof T.

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Infinite dimensional tilting modules

Definition

there exists an exact sequence 0→R→T₀ → · · · →T_r →0 such that r < ω, and for eachi <r,Ti ∈Add(T), i.e., Ti is a direct summand of a (possibly infinite) direct sum of copies of T . BT :={T}^⊥^∞ =T

1<iKerExtⁱ_R(T,−) theright tilting classof T. AT :=^⊥BT theleft tilting classof T.

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Some infinite dimensional tilting theory

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Some infinite dimensional tilting theory

Theorem (A model-theoretic characterization of right tilting classes) Tilting classes are exactly the classes of finite type, i.e., the classes of the form S^⊥, whereS is a set of strongly finitely presented modules of bounded projective dimension.

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Some infinite dimensional tilting theory

Let S_T :=A_T ∩mod-R and ¯A_T := lim−→S. ThenA_T is the class of all direct summands of S_T-filtered modules, andA_T ⊆A¯_T.

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Some infinite dimensional tilting theory

Definition

The tilting moduleT is P

-pure split provided that ¯A_T =A_T, that is, the left tilting class ofT is closed under direct limits. Equivalently:

Each pure embedding T0 ֒→T1 such thatT0,T1 ∈Add(T) splits.

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Some infinite dimensional tilting theory

Definition

The tilting moduleT is P

-pure split provided that ¯A_T =A_T, that is, the left tilting class ofT is closed under direct limits. Equivalently:

Each pure embedding T0 ֒→T1 such thatT0,T1 ∈Add(T) splits.

Example (Bass)

Let T =R. Then T is a tilting module of projective dimension 0, and T is P

-pure split, iffR is a right perfect ring.

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Locally free modules and tilting

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Locally free modules and tilting

The setting

Let R be a countable ring, and T be a non-P

-pure-split tilting module.

Let FT be the class of all countably presented modules from AT, and LT the class of all locally FT-free modules.

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Locally free modules and tilting

The setting

Assume that LT ⊆ P₁,LT is closed under direct summands, and M ∈ L_T whenever M⊆L∈ L_T and L/M ∈A¯_T.

Then the class L_T is not precovering.

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Locally free modules and tilting

The setting

Assume that LT ⊆ P₁,LT is closed under direct summands, and M ∈ L_T whenever M⊆L∈ L_T and L/M ∈A¯_T.

Then the class L_T is not precovering.

Corollary

If R is countable and non-right perfect, then FMis not precovering.

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Finite dimensional hereditary algebras

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Finite dimensional hereditary algebras

Let R be an indecomposable hereditary artin algebra of infinite representation type, with the Auslander-Reiten translation τ.

Then there is a partition of all indecomposable finitely generated modules into three sets:

q := indecomposable preinjective modules

(i.e., indecomposable injectives and theirτ-shifts), p := indecomposable preprojective modules

(i.e., indecomposable projectives and their τ⁻-shifts), t := regular modules (the rest).

. . .

p t q

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The Lukas tilting module and the Baer modules

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The Lukas tilting module and the Baer modules

p^⊥ is a right tilting class.

M ∈p^⊥ iff M has no non-zero direct summands from p.

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The Lukas tilting module and the Baer modules

The tilting moduleL inducingp^⊥ is called theLukas tilting module.

The left tilting class of Lis the class of allBaer modules.

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The Lukas tilting module and the Baer modules

The tilting moduleL inducingp^⊥ is called theLukas tilting module.

The left tilting class of Lis the class of allBaer modules.

[Angeleri-Kerner-T.’10]

The class of all Baer modules coincides with Filt(p).

The Lukas tilting module Lis countably generated, but has no finite dimensional direct summands, and it is not P

-pure split.

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Non-deconstructibility in the realm of artin algebras

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Non-deconstructibility in the realm of artin algebras

Let FL be the class of all countably presented Baer modules.

The elements of L_L are called thelocally Baer modules.

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Non-deconstructibility in the realm of artin algebras

Let FL be the class of all countably presented Baer modules.

The elements of L_L are called thelocally Baer modules.

Let R be a countable indecomposable hereditary artin algebra of infinite representation type. Then the class LL is not precovering (and hence not deconstructible).

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A conjecture

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A conjecture

A ringR is right pure-semisimple, iff each class of rightR-modules closed under transfinite extensions and direct summands is deconstructible.