Testing for projectivity and transfinite extensions of simple artinian rings Jan Trlifaj, Charles University Functor Categories, Model Theory, and Constructive Category Theory ∗ University of Tartu, P¨ arnu College Jan Trlifaj (MFF UK) Testing for projectivity & transfinite ext’s ... Functor cat’s ... 1 / 1
An overview I. Baer’s Criterion for injectivity and Faith’s Problem on its dual. II. Existence/non-existence of sets of epimorphisms testing for projectivity. III. Transfinite extensions of simple artinian rings. IV. Dual Baer Criterion for small transfinite extensions. Jan Trlifaj (MFF UK) Testing for projectivity & transfinite ext’s ... Functor cat’s ... 2 / 1
I. Baer’s Criterion for injectivity and Faith’s Problem on its dual Jan Trlifaj (MFF UK) Testing for projectivity & transfinite ext’s ... Functor cat’s ... 3 / 1
� � � � � � Testing for injectivity Baer’s Criterion ’40 Injectivity coincides with R -injectivity for any ring R and any module M . M is R -injective, if for each right ideal I , all f ∈ Hom R ( I , M ) extend to R : M � � f � � � � � � ⊆ � I � 0 R / I 0 R Ext 1 Equivalently: R ( R / I , M ) = 0 for each right ideal I of R . So there is always an i -test set of monomorphisms { f i | i ∈ I } : M is injective, iff Hom R ( f i , M ) is surjective for each i ∈ I . One morphism suffices: M is injective, iff Hom R ( ⊕ i ∈ I f i , M ) is surjective. Jan Trlifaj (MFF UK) Testing for projectivity & transfinite ext’s ... Functor cat’s ... 4 / 1
� � � � � Testing for projectivity M is R -projective, if for each right ideal I , all f ∈ Hom R ( M , R / I ) factorize through π I : M � � � � f � � � � ⊆ π I � R / I � I � R � 0 0 If Ext 1 R ( M , I ) = 0 for each right ideal I of R , then M is R -projective. The converse holds when R is right self-injective, but not in general. The Dual Baer Criterion (DBC for short) holds for a ring R , in case projectivity coincides with R -projectivity for any module M . Faith’ Problem ’76 For what kind of rings R does DBC hold? Jan Trlifaj (MFF UK) Testing for projectivity & transfinite ext’s ... Functor cat’s ... 5 / 1
Partial positive answers For any ring R , projectivity = R -projectivity for any finitely generated module M , i.e., DBC holds for all finitely generated modules over any ring . Sandomierski’64, Ketkar-Vanaja’81: DBC holds for all modules over any right perfect ring . Let K be a skew-field, κ an infinite cardinal, and R the endomorphism ring of a κ -dimensional left vector space over K . Then DBC holds for all ≤ κ -generated modules. In particular, if R is right perfect, then there is always a p -test set of epimorphisms { g j | j ∈ J } : M is projective, iff Hom R ( M , g j ) is surjective for each j ∈ J . One morphism suffices: M is projective, iff Hom R ( M , � j ∈ J g j ) is surjective. Jan Trlifaj (MFF UK) Testing for projectivity & transfinite ext’s ... Functor cat’s ... 6 / 1
Partial negative answers Hamsher’67: If R is a commutative noetherian, but not artinian, ring then there exists a countably generated R -projective module which is not projective. So DBC fails for countably generated modules. Puninski et al.’17: If R is a semilocal right noetherian ring. Then DBC holds, iff R is right artinian. Jan Trlifaj (MFF UK) Testing for projectivity & transfinite ext’s ... Functor cat’s ... 7 / 1
II. Existence/non-existence of sets of epimorphisms testing for projectivity Jan Trlifaj (MFF UK) Testing for projectivity & transfinite ext’s ... Functor cat’s ... 8 / 1
The set-theoretic barrier Assume Shelah’s Uniformization Principle. Let κ be an uncountable cardinal of cofinality ω . Then for each non-right perfect ring R of cardinality ≤ κ there exists a κ + -generated module M of projective dimension 1 such that Ext 1 R ( M , N ) = 0 for each module N of cardinality < κ . Corollary It is consistent with ZFC + GCH that there is no p -test set of epimorphisms for any non-right perfect ring R . In particular, it is consistent with ZFC + GCH that DBC fails for each non-right perfect ring R . Jan Trlifaj (MFF UK) Testing for projectivity & transfinite ext’s ... Functor cat’s ... 9 / 1
Refinements of Faith’s Problem Are the consistency results above actually provable in ZFC? If not, what is the border line between those non-right perfect rings, for which there is no p -test set of epimophisms in ZFC, and those, for which the existence of such set is independent of ZFC? What is the border line between those non-right perfect rings, for which DBC fails in ZFC, and those, for which it is independent of ZFC? Jan Trlifaj (MFF UK) Testing for projectivity & transfinite ext’s ... Functor cat’s ... 10 / 1
A positive consistency result Assume the Axiom of Constructibility. Let R be a non-right perfect ring, κ = 2card ( R ) , F be the free module of rank κ , and M be a module of finite projective dimension. Then M is projective, iff Ext i R ( M , F ) = 0 for all i > 0. Corollary It is consistent with ZFC + GCH that there is a p -test set of epimorphisms for any ring of finite global dimension. The assertion ‘For each non-right perfect ring of finite global dimension, there exists a p -test set of epimorphisms’ is independent of ZFC + GCH. Jan Trlifaj (MFF UK) Testing for projectivity & transfinite ext’s ... Functor cat’s ... 11 / 1
Further positive consistency results Flatness can always be expressed by vanishing of Ext, in ZFC. So we have Corollary Let R be a ring such that each flat module has finite projective dimension. Then the existence of a p-test set of epimorphisms is consistent with ZFC + GCH. Corollary The existence of a p-test set of epimorphisms is independent of ZFC + GCH whenever R is a ring which is either n -Iwanaga-Gorenstein, for n > 0, or commutative noetherian with 0 < Kdim( R ) < ∞ , or almost perfect, but not perfect. Jan Trlifaj (MFF UK) Testing for projectivity & transfinite ext’s ... Functor cat’s ... 12 / 1
What about validity of the Dual Baer Criterion? By Hamsher’67, DBC fails in ZFC already for all hereditary (= Dedekind) domains. Let’s explore other hereditary rings ... Jan Trlifaj (MFF UK) Testing for projectivity & transfinite ext’s ... Functor cat’s ... 13 / 1
III. Transfinite extensions of simple artinian rings Jan Trlifaj (MFF UK) Testing for projectivity & transfinite ext’s ... Functor cat’s ... 14 / 1
Semiartinian rings A ring R is right semiartinian, if R is the last term of the right Loewy sequence of R , i.e., there are an ordinal σ and a strictly increasing sequence ( S α | α ≤ σ + 1), such that S 0 = 0, S α +1 / S α = Soc( R / S α ) for all α ≤ σ , S α = � β<α S β for all limit ordinals α ≤ σ , and S σ +1 = R . R is von Neumann regular, if all (right R -) modules are flat. R has right primitive factors artinian (has right pfa for short) in case R / P is right artinian for each right primitive ideal P of R . Let R be a regular ring. R is right semiartinian, iff it is left semiartinian, and the right and left Loewy sequences of R coincide. R has right pfa, iff it has left pfa, iff all homogenous completely reducible (left or right) modules are injective. Jan Trlifaj (MFF UK) Testing for projectivity & transfinite ext’s ... Functor cat’s ... 15 / 1
Structure of semiartinian regular rings with pfa Let R be a right semiartinian ring and ( S α | α ≤ σ + 1) be the right Loewy sequence of R with σ ≥ 1. The following conditions are equivalent: R is regular with pfa. for each α ≤ σ there are a cardinal λ α , positive integers n αβ ( β < λ α ) and skew-fields K αβ ( β < λ α ) such that S α +1 / S α ∼ = � β<λ α M n αβ ( K αβ ), as rings without unit. Moreover, λ α is infinite iff α < σ . The pre-image of M n αβ ( K αβ ) in this isomorphism coincides with the β th homogenous component of Soc( R / S α ), and it is finitely generated as right R / S α -module for all β < λ α . P αβ := a representative of simple modules in the β th homogenous component of S α +1 / S α . Zg ( R ) := { P αβ | α ≤ σ, β < λ α } is a set of representatives of all simple modules, and also the Ziegler spectrum of R . The Cantor-Bendixson rank of Zg ( R ) is σ . Jan Trlifaj (MFF UK) Testing for projectivity & transfinite ext’s ... Functor cat’s ... 16 / 1
Transfinite extensions of simple artinian rings R / S σ M n σ 0 ( K σ 0 ) ⊕ ... ⊕ M n σ,λσ − 1 ( K σ,λ σ − 1 ) ... ... ... ... S α + 1 / S α M n α 0 ( K α 0 ) ⊕ ... ⊕ M n αβ ( K αβ ) ⊕ ... β < λ α ... ... ... ... S 2 / S 1 M n 10 ( K 10 ) ⊕ ... ⊕ M n 1 β ( K 1 β ) ⊕ ... β < λ 1 S 1 = Soc ( R ) M n 00 ( K 00 ) ⊕ ... ⊕ M n 0 β ( K 0 β ) ⊕ ... β < λ 0 The ordinal σ , the cardinals λ α for α ≤ σ , and the natural numbers n αβ and skew-fields K αβ for β < λ α are invariants of the ring R . Some limitations: R is a subring of � β<λ 0 M n 0 β ( K 0 β ), and similarly R / S α is a subring of � β<λ α M n αβ ( K αβ ) for each α ≤ σ . The exact possible values of, and relations among, these parameters are not clear. Jan Trlifaj (MFF UK) Testing for projectivity & transfinite ext’s ... Functor cat’s ... 17 / 1
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