on a decent behaviour of gorenstein at and gorenstein ... · gorenstein modules over general rings...

On a decent behaviour of Gorenstein flat andGorenstein injective modules over a general ring

ICART 2018 (HAMRC), RabatJuly 2–5, 2018

Jan Saroch (Charles University, Prague)(joint work with Jan St’ovıcek)

Fixing notation

R an associative ring with enough idempotents

Mod-R the category of all (unitary right R-)modules

mod-R the category of all finitely presented modules, i.e. moduleswhose covariant hom-functors commute with lim−→ in Mod-R

P0, I0,F0 the classes of all projective, injective, flat modules,respectively.

Gorenstein modules over general rings

Gorenstein analogs of projective, injective and flat modules naturallyemerge over Iwanaga-Gorenstein rings. First extensive treatment overarbitrary (unital) rings was done by Holm in his JPAA paper Gorensteinhomological dimensions from 2004.

Definition

A module M is called Gorenstein injective (or shortly GI) if it is a syzygymodule in a long exact sequence

· · · → I−1 → I 0 → I 1 → I 2 → · · · (1)

consisting of injective modules which remains exact after applying thecovariant functor HomR(E ,−) for arbitrary injective module E .

Gorenstein projective modules (aka GP-modules) are defined dually.

Gorenstein analogs of projective, injective and flat modules naturallyemerge over Iwanaga-Gorenstein rings.

First extensive treatment overarbitrary (unital) rings was done by Holm in his JPAA paper Gorensteinhomological dimensions from 2004.

Definition

· · · → I−1 → I 0 → I 1 → I 2 → · · · (1)

Definition

· · · → I−1 → I 0 → I 1 → I 2 → · · · (1)

Definition

· · · → I−1 → I 0 → I 1 → I 2 → · · · (1)

Definition

· · · → I−1 → I 0 → I 1 → I 2 → · · · (1)

Gorenstein modules over general rings II

Definition

A module M is called Gorenstein flat (or just GF) if it is a syzygy modulein a long exact sequence

· · · → F−1 → F 0 → F 1 → F 2 → · · · (2)

consisting of flat modules which remains exact after applying the tensorfunctor −⊗R E for arbitrary injective left module E .

We denote the classes of GP-, GI- and GF-modules by GP,GI and GF ,respectively

Examples

If R is of finite global dimension, then GP = P0,GI = I0 andGF = F0.

If R is an IF-ring, then GF = Mod-R, GP-modules are exactly theFP-projective ones and GI-modules are just the (Enochs) cotorsionmodules.

Definition

· · · → F−1 → F 0 → F 1 → F 2 → · · · (2)

Examples

Definition

· · · → F−1 → F 0 → F 1 → F 2 → · · · (2)

Examples

Holm’s paper

Recall, that a class C of modules is resolving if it contains all projectivemodules and is closed under taking kernels of epimorphisms andextensions. (The dual notion is coresolving.)

Basic closure properties of Gorenstein classes

GP is resolving and closed under direct sums and direct summands.

GI is coresolving and closed under products and direct summands.

GF is closed under direct sums. Moreover, if R is left coherent, thenM ∈ GF if and only if the character dual left module M∗ isGorenstein injective. Consequently, GF is resolving and closed underpure-epimorphic images.

Moreover, some mild versions of approximation properties of theGorenstein classes were proven.

Holm’s paper

Recall, that a class C of modules is resolving if it contains all projectivemodules and is closed under taking kernels of epimorphisms andextensions.

(The dual notion is coresolving.)

Holm’s paper

GF is closed under direct sums.

Moreover, if R is left coherent, thenM ∈ GF if and only if the character dual left module M∗ isGorenstein injective. Consequently, GF is resolving and closed underpure-epimorphic images.

Holm’s paper

GF is closed under direct sums. Moreover, if R is left coherent, thenM ∈ GF if and only if the character dual left module M∗ isGorenstein injective.

Consequently, GF is resolving and closed underpure-epimorphic images.

Holm’s paper

Module approximations

Definition

A class of modules A is precovering if for each module M there isf ∈ HomR(A,M) with A ∈ A such that each f ′ ∈ HomR(A′,M) withA′ ∈ A factorizes through f :

Af // M

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The map f is an A–precover of M.

If f is moreover right minimal (that is, f factorizes through itself only byan automorphism of A), then f is an A–cover of M.

If A provides for covers of all modules, then A is called a covering class.

In general, A-(pre)covers need not be onto.

Definition

Af // M

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Definition

Af // M

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Definition

Af // M

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Definition

Af // M

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Cotorsion pairs

Let C ⊆ Mod-R. Put C⊥ = {M ∈ Mod-R | (∀C ∈ C) Ext1R (C ,M) = 0}.Similarly, let’s denote ⊥C ={M ∈ Mod-R | (∀C ∈ C) Ext1R (M,C ) = 0}.

Definition

A pair C = (A,B) of classes of modules is called a cotorsion pair ifA⊥ = B and ⊥B = A. If C⊥ = B for a class C ⊆ Mod-R, we say that thecotorsion pair C is generated by the class C.

Remark

A is closed under transfinite extensions (Eklof Lemma) and B is closedunder products. Both classes are closed under direct summands.

Examples

(P0,Mod-R), (Mod-R, I0), (F0, EC); (FP-proj , FP-inj)

If A is closed under kernels of epimorphisms, C is called hereditary.

Cotorsion pairs

Let C ⊆ Mod-R. Put C⊥ = {M ∈ Mod-R | (∀C ∈ C) Ext1R (C ,M) = 0}.

Similarly, let’s denote ⊥C ={M ∈ Mod-R | (∀C ∈ C) Ext1R (M,C ) = 0}.

Definition

Remark

Examples

Cotorsion pairs

Definition

Remark

Examples

Cotorsion pairs

Definition

Remark

Examples

Cotorsion pairs

Definition

Remark

Examples

Cotorsion pairs

Definition

Remark

Examples

Cotorsion pairs

Definition

Remark

Examples

Cotorsion pairs and approximations

Definition

An A-precover f : A→ M is called special if it is onto and Ker(f ) ∈ A⊥.If A provides for special precovers of all modules, we say that A isspecial precovering.

Eklof, Trlifaj; Enochs

If a cotorsion pair (A,B) is generated by a set of modules, then A isspecial precovering and B is special preenveloping. Moreover, if A isclosed under lim−→, then A is covering and B is enveloping. In this case,C is called perfect.

Open problem (Enochs)

Is a covering class of modules necessarily closed under lim−→?

Definition

If a cotorsion pair (A,B) is generated by a set of modules, then A isspecial precovering and B is special preenveloping.

Moreover, if A isclosed under lim−→, then A is covering and B is enveloping. In this case,C is called perfect.

Definition

Short history of approximation-related results

1 Over a left coherent ring, there is a perfect hereditary cotorsion pairGF = (GF ,GF⊥) (Enochs, Jenda, Lopez-Ramos ’04).

2 Over a right noetherian ring, there is a perfect hereditary cotorsionpair GI = (⊥GI,GI) (Krause ’05).

3 If GF is closed under extensions (aka R is right GF-closed), then itis resolving and closed under direct summands (Bennis ’08).

4 If GF is closed under extensions, then it is closed under lim−→ as well,and GF is a perfect hereditary cotorsion pair (Yang, Liu ’12).

Task: to prove the results over arbitrary ring R.

Projectively coresolved Gorenstein flat modules

An idea: show that GF is generated by the class F0 ∪ GP, whence GF isclosed under extensions.Problem: it is not known whether GP ⊆ GF in general.Solution: instead of GP consider the class PGF of modules which aresyzygies in a long exact sequence

· · · → P−1 → P0 → P1 → P2 → · · · (3)

consisting of projective modules which remains exact after applyingthe tensor functor −⊗R E for arbitrary injective left module E .Then trivially PGF ⊆ GF . Furthermore, we have

Theorem

PGF⊥ contains all modules in the definable closure of {RR}.In particular, PGF ⊆ GP. Moreover:

1 R is right perfect ⇐⇒ PGF = GF .

2 If GP ⊆ GF (for instance if R is right perfect and left coherent),then GP = PGF .

An idea: show that GF is generated by the class F0 ∪ GP, whence GF isclosed under extensions.

Problem: it is not known whether GP ⊆ GF in general.Solution: instead of GP consider the class PGF of modules which aresyzygies in a long exact sequence

· · · → P−1 → P0 → P1 → P2 → · · · (3)

Theorem

An idea: show that GF is generated by the class F0 ∪ GP, whence GF isclosed under extensions.Problem: it is not known whether GP ⊆ GF in general.

Solution: instead of GP consider the class PGF of modules which aresyzygies in a long exact sequence

· · · → P−1 → P0 → P1 → P2 → · · · (3)

Theorem

· · · → P−1 → P0 → P1 → P2 → · · · (3)

consisting of projective modules which remains exact after applyingthe tensor functor −⊗R E for arbitrary injective left module E .

Then trivially PGF ⊆ GF . Furthermore, we have

Theorem

· · · → P−1 → P0 → P1 → P2 → · · · (3)

Theorem

PGF⊥ contains all modules in the definable closure of {RR}.In particular, PGF ⊆ GP.

Moreover:

· · · → P−1 → P0 → P1 → P2 → · · · (3)

Theorem

· · · → P−1 → P0 → P1 → P2 → · · · (3)

Theorem

Approximation properties of the class PGF

Theorem

The pair PGF = (PGF ,PGF⊥) is a hereditary cotorsion pair generatedby a representative set of ν-presented modules from PGF where ν is theminimal infinite cardinal such that all finitely generated right ideals of Rare ν-presented (i.e. R is right ν-coherent). Moreover, the class PGF⊥is resolving, equivalently PGF ∩ PGF⊥ = P0.

The class PGF is currently the largest known subclass of GP whichpossesses nice approximation properties over an arbitrary ring.

Corollary

Let R be an Artin algebra. Then the cotorsion pair GP = (GP,GP⊥) isgenerated by a set of countably presented modules. Moreover, GP isgenerated by finitely presented modules if and only if GP⊥ = ⊥GI, i.e.R is virtually Gorenstein (Beligiannis, Krause ’08).

Theorem

The pair PGF = (PGF ,PGF⊥) is a hereditary cotorsion pair generatedby a representative set of ν-presented modules from PGF where ν is theminimal infinite cardinal such that all finitely generated right ideals of Rare ν-presented (i.e. R is right ν-coherent).

Moreover, the class PGF⊥is resolving, equivalently PGF ∩ PGF⊥ = P0.

Corollary

Theorem

Corollary

Theorem

Corollary

Theorem

Corollary

Let R be an Artin algebra. Then the cotorsion pair GP = (GP,GP⊥) isgenerated by a set of countably presented modules.

Moreover, GP isgenerated by finitely presented modules if and only if GP⊥ = ⊥GI, i.e.R is virtually Gorenstein (Beligiannis, Krause ’08).

Theorem

Corollary

How PGF relates to GF?

Main theorem on Gorenstein flat modules

Let M be a module. Then the following conditions are equivalent:

1 M is Gorenstein flat.

2 There is a short exact sequence 0→ F → P → M → 0 with F flatand P ∈ PGF which remains exact after applying the functorHomR(−,C ) for any (flat) cotorsion module C . In particular, M isa pure-epimorphic image of a PGF-module.

3 Ext1R (M,C ) = 0 for all cotorsion modules C ∈ PGF⊥.

4 There is a short exact sequence 0→ M → F → P → 0 with F flatand P ∈ PGF .

5 There is a short exact sequence 0→ P → M ′ → F → 0 withP ∈ PGF and F flat such that M is a direct summand in M ′.

In particular, we get that GF is closed under extensions by (3), andGF ∩ PGF⊥ = F0 from (4). Finally, GF = (GF ,PGF⊥ ∩ EC).

In particular, we get that GF is closed under extensions by (3), andGF ∩ PGF⊥ = F0 from (4).

Finally, GF = (GF ,PGF⊥ ∩ EC).

The case of GI

Theorem

Let κ be the least infinite cardinal such that κ = κ|R|+ℵ0 . Then thecotorsion pair C generated by κ-presented modules from ⊥GI is theperfect hereditary cotorsion pair GI = (⊥GI,GI). Consequently, GI isan enveloping class.

The key step is to show that the left-hand class of C is coresolving (andresolving). This is done by using the same property of ⊥GI, and somemore-or-less standard reasoning about transfinite extensions.

Corollary (GF-test)

Let R be left coherent. There exists a left R-module T such thatGF = Ker TorR1 (−,T ).

The case of GI

Theorem

Corollary (GF-test)

The case of GI

Theorem

The key step is to show that the left-hand class of C is coresolving (andresolving).

This is done by using the same property of ⊥GI, and somemore-or-less standard reasoning about transfinite extensions.

Corollary (GF-test)

The case of GI

Theorem

Corollary (GF-test)

The case of GI

Theorem

Corollary (GF-test)

Miscellaneous results and open questions

Let R be an arbitrary ring.

We obtained three new Hovey triples(PGF ,PGF⊥,Mod-R), (GF ,PGF⊥, EC) and (Mod-R,⊥GI,GI).The first two define Quillen equivalent stable model structures onMod-R.

There is a perfect hereditary cotorsion pair B = (B,Ctac(I0)) in thecategory Ch(R) of complexes of right R-modules such that B iscoresolving.

What remains open

1 Are GP-modules necessarily Gorenstein flat? If not, does GP format least a precovering class of modules?

2 Is the GF-test Corollary true over a left non-coherent ring? If not, isGF at least closed under pure-epimorphic images?

3 Give an example of R left coherent with GF not closed under directproducts (or show that there is none).

What remains open

1 Are GP-modules necessarily Gorenstein flat?

If not, does GP format least a precovering class of modules?

What remains open

2 Is the GF-test Corollary true over a left non-coherent ring?

If not, isGF at least closed under pure-epimorphic images?

What remains open

Bibliography

H. Holm, Gorenstein homological dimensions, J. Pure Appl. Alg. 189(2004), 167–193.

J. S., J. St’ovıcek, Singular compactness and definability forΣ-cotorsion and Gorenstein modules, arxiv.org/abs/1804.09080.

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