Mittag-Leffler Conditions, Gorenstein Modules and Homological Invariants

Abstract: In this paper, we investigate certain properties on the Mittag-Leffler conditions via set-theoretic methods. We establish that a strongly $\aleph_1$-presented module $M$ satisfying Ext$_R^{\ge 1}(M,D<sup>{(\aleph_1)})=0$ belongs to the left orthogonal class of $\overline{D}$, where $\overline{D}$ is the definable of $D$. This yields the consequence that every $\aleph_1$-generated strongly Gorenstein projective module is Gorenstein flat. Furthermore, we investigate when a flat Gorenstein projective module is projective. Finally, for any two-sided $\aleph_1$-coherent ring $R$, we prove the identity $$\operatorname{silp}R+\operatorname{silp}R<sup>{\mathrm{op}}=\operatorname{spli}R+\operatorname{spli}R<sup>{\mathrm{op}}$$.

• The paper establishes an Ext-vanishing criterion for strongly aleph_1-presented modules using definable closures, stationary-set arguments, and set-theoretic Mittag-Leffler methods.
• The results show that many aleph_1-generated or strongly aleph_1-presented Gorenstein projective modules are Gorenstein flat, while flat Gorenstein projective modules indexed below aleph_omega are projective.
• The paper proves a symmetry identity for silp and spli over two-sided aleph_1-coherent rings and constructs aleph_1-coherent examples that are neither weakly nor generalized coherent.

This paper develops set-theoretic techniques for Mittag-Leffler conditions on modules and applies them to two long-standing problems in Gorenstein homological algebra: whether every Gorenstein projective module is Gorenstein flat (a question of Holm), and whether every flat Gorenstein projective module is projective (a question of Bazzoni, Cortés-Izurdiaga, and Estrada). The main technical contribution is a criterion under which a strongly $\aleph_1$-presented module lies in the left orthogonal of a definable closure, which then yields partial answers to both problems and a new symmetry result for the homological invariants $\operatorname{silp}$ and $\operatorname{spli}$ over $\aleph_1$-coherent rings.

Set-theoretic machinery for Mittag-Leffler conditions

The paper's foundational result concerns the interplay between Ext-vanishing and stationarity. Recall that a module is strict $\mathcal{B}$-Mittag-Leffler if a direct system of finitely presented modules with colimit $M$ induces inverse systems $\operatorname{Hom}_R(\mathcal{F}, B)$ satisfying the Mittag-Leffler condition for each $B \in \mathcal{B}$. The key result states that if $M$ is strongly $\aleph_1$-presented (i.e., admits a projective resolution by $\operatorname{silp}$0-generated projectives) and $\operatorname{silp}$1 for a module $\operatorname{silp}$2, then $\operatorname{silp}$3 belongs to the left orthogonal class $\operatorname{silp}$4, where $\operatorname{silp}$5 is the definable closure of $\operatorname{silp}$6 — the smallest class closed under products, direct limits, and pure submodules.

The proof combines several ingredients. First, a construction lifted from Šaroch and Šťovíček's work on $\operatorname{silp}$7-continuous direct systems decomposes a short exact sequence $\operatorname{silp}$8 into an $\operatorname{silp}$9-continuous directed system of short exact sequences of $\operatorname{spli}$0-presented modules, using Kaplansky's theorem to handle the projective middle term. Applying $\operatorname{spli}$1 produces an inverse system whose transition maps are epimorphisms. The critical step is a stationary-set argument: assuming a certain set $\operatorname{spli}$2 of ordinals at which the dual transition maps fail to be epic is stationary, one inductively constructs a compatible family of morphisms into $\operatorname{spli}$3 whose images are eventually trapped in $\operatorname{spli}$4 for a club of $\operatorname{spli}$5; intersecting this club with the (nonempty, by hypothesis) set of limit points of $\operatorname{spli}$6 produces a contradiction, since a chosen non-factorizing map is forced to factorize. Hence $\operatorname{spli}$7 is non-stationary and a club $\operatorname{spli}$8 exists on which the inverse system is inner-continuous.

The consequence $\operatorname{spli}$9 is then upgraded to $\aleph_1$0 by observing that $\aleph_1$1 is the class of pure-epimorphic images of modules in $\aleph_1$2 and verifying the required factorization property for arbitrary pure-epimorphisms. This is the engine driving all subsequent applications.

Gorenstein projective versus Gorenstein flat modules

Holm's question asks whether $\aleph_1$3. The paper gives a positive answer under cardinality restrictions, for arbitrary rings.

Main result: every $\aleph_1$4-generated strongly Gorenstein projective module is Gorenstein flat. The proof applies the orthogonal result with $\aleph_1$5 — noting that $\aleph_1$6 — to obtain strict $\aleph_1$7-stationarity, and then invokes Emmanouil's criterion relating stationarity to Gorenstein flatness. This strengthens the previously known case of strongly countably presented Gorenstein projective modules.

The result is extended in two directions. First, every strongly $\aleph_1$8-presented Gorenstein projective module is Gorenstein flat: the argument builds a complete projective resolution by $\aleph_1$9-generated projectives, shows $\mathcal{B}$0 is a direct summand of a strongly $\mathcal{B}$1-presented strongly Gorenstein projective module $\mathcal{B}$2, and uses the fact that all modules in $\mathcal{B}$3 are Gorenstein flat. Consequently, over any left $\mathcal{B}$4-coherent ring (where every $\mathcal{B}$5-presented module is strongly $\mathcal{B}$6-presented), every $\mathcal{B}$7-generated Gorenstein projective module is Gorenstein flat. These are strictly partial answers: the general case of Holm's question remains open, and the restriction to modules of size at most $\mathcal{B}$8 is essential to the method, which relies on the regularity of $\mathcal{B}$9 and on club-stationary combinatorics.

On the second question, the paper proves that a flat Gorenstein projective module expressible as a direct limit of projectives over an index set of cardinality $M$0 is projective. This uses a result of Šaroch and Šťovíček producing a finite projective resolution of the limit module with terms in $M$1 of the system, forcing projectivity. Additionally, a conditional equivalence is established: if $M$2 is closed under direct sums and $M$3, then every Gorenstein projective module is Gorenstein flat, and hence every flat Gorenstein projective module is projective. The proof of the latter proceeds by showing $M$4 via a dichotomy argument: a module in $M$5 would force all free modules to be $M$6-cotorsion, hence all flat modules cotorsion, hence $M$7 perfect — contradicting the non-perfect case under consideration.

Homological invariants over $M$8-coherent rings

The invariants $M$9 (supremum of projective lengths of injectives) and $\operatorname{Hom}_R(\mathcal{F}, B)$0 (supremum of injective lengths of projectives) are known to be equal when both are finite, but the general relationship between them is open; for Artin algebras the equality $\operatorname{Hom}_R(\mathcal{F}, B)$1 is equivalent to the Gorenstein Symmetry Conjecture. The paper proves that for a two-sided $\operatorname{Hom}_R(\mathcal{F}, B)$2-coherent ring, $\operatorname{Hom}_R(\mathcal{F}, B)$3.

The proof proceeds through an inequality $\operatorname{Hom}_R(\mathcal{F}, B)$4 for right $\operatorname{Hom}_R(\mathcal{F}, B)$5-coherent $\operatorname{Hom}_R(\mathcal{F}, B)$6. Given $\operatorname{Hom}_R(\mathcal{F}, B)$7, the syzygies $\operatorname{Hom}_R(\mathcal{F}, B)$8 of finitely presented modules $\operatorname{Hom}_R(\mathcal{F}, B)$9 are strongly $B \in \mathcal{B}$0-presented and right orthogonal to $B \in \mathcal{B}$1; the orthogonal result then gives vanishing of Ext against all of $B \in \mathcal{B}$2, and the strict Mittag-Leffler property together with the injectivity of the natural transformation $B \in \mathcal{B}$3 yields vanishing of $B \in \mathcal{B}$4. Since every injective right module is a summand of some $B \in \mathcal{B}$5, the flat lengths of injectives are bounded by $B \in \mathcal{B}$6. Combining this with the standard inequalities $B \in \mathcal{B}$7 and $B \in \mathcal{B}$8 gives the two-sided finiteness equivalence, and hence the sum identity. As corollaries, $B \in \mathcal{B}$9 for any $M$0-coherent ring isomorphic to its opposite, in particular for commutative $M$1-coherent rings. This strictly generalizes prior results of Emmanouil–Talelli ($M$2-Noetherian rings), Chatzistavridis–Emmanouil (weakly coherent rings), and Wang–Yang (generalized coherent rings), since $M$3-coherent rings need not belong to any of these classes — as the final section demonstrates.

A class of non-weakly coherent $M$4-Noetherian rings

To show the new coherence hypothesis is genuinely weaker than those in the literature, the paper constructs rings that are $M$5-Noetherian (hence $M$6-coherent) but neither weakly nor generalized coherent. Let $M$7 be commutative local with $M$8 and $M$9 infinite. The paper first establishes an explicit projective resolution of $\aleph_1$0 with $\aleph_1$1, where $\aleph_1$2, with differential induced by multiplication — a bar-type resolution valid because $\aleph_1$3.

For $\aleph_1$4 (product of $\aleph_1$5 copies of $\aleph_1$6, $\aleph_1$7 infinite), the tensor complex splits into a short exact sequence of complexes with zero differentials on two of the three terms, and the connecting homomorphism is $\aleph_1$8 where $\aleph_1$9 is the canonical map. Two claims drive the conclusion: $\operatorname{silp}$00 is injective (a nonzero vector annihilated by all coordinates must have all coordinates zero), and $\operatorname{silp}$01 (the sequence $\operatorname{silp}$02 of linearly independent vectors cannot lie in the span of finitely many rank-one tensors). Consequently $\operatorname{silp}$03 for all $\operatorname{silp}$04, so $\operatorname{silp}$05. Since $\operatorname{silp}$06 is a product of flat modules with infinite flat dimension, $\operatorname{silp}$07 is not weakly coherent, and hence not generalized coherent. Meanwhile $\operatorname{silp}$08, being the unique prime ideal, is $\operatorname{silp}$09-generated, so $\operatorname{silp}$10 is $\operatorname{silp}$11-Noetherian and $\operatorname{silp}$12-coherent. A concrete instance is $\operatorname{silp}$13, which is $\operatorname{silp}$14-coherent, not $\operatorname{silp}$15-Noetherian, and not weakly coherent. This confirms that the main theorem on $\operatorname{silp}$16 and $\operatorname{silp}$17 applies to rings outside all previously treated classes.

Limitations and open questions

The paper's methods are intrinsically tied to the cardinal $\operatorname{silp}$18: the club-stationary argument and the $\operatorname{silp}$19-continuity of the constructed direct systems do not obviously extend to higher regular cardinals, and the paper does not address whether the corresponding statements hold for $\operatorname{silp}$20. Holm's question in full generality — whether every Gorenstein projective module is Gorenstein flat — remains open, as does the Bazzoni–Cortés-Izurdiaga–Estrada question of whether every flat Gorenstein projective module is projective without cardinality or dimension restrictions. The conditional result reducing the latter to the closure of $\operatorname{silp}$21 under direct sums and the union decomposition $\operatorname{silp}$22 identifies concrete hypotheses whose verification would settle the problem, but neither hypothesis is established in general. Finally, whether $\operatorname{silp}$23 holds for arbitrary rings, or at least for all coherent rings, remains unresolved.

Conclusion

The paper demonstrates that Ext-vanishing against a single coproduct $\operatorname{silp}$24 of controlled size suffices, for strongly $\operatorname{silp}$25-presented modules, to force membership in left orthogonal classes of definable closures — a substantial refinement of the countable Mittag-Leffler technology. This yields Gorenstein flatness for $\operatorname{silp}$26-generated strongly Gorenstein projective modules over arbitrary rings, projectivity of flat Gorenstein projective modules under an index-set cardinality bound, the sum identity $\operatorname{silp}$27 for two-sided $\operatorname{silp}$28-coherent rings, and an explicit family of $\operatorname{silp}$29-coherent rings failing weak coherence. The results collectively extend the reach of set-theoretic homological algebra to a strictly larger class of rings than previously covered, while leaving the unrestricted forms of both Gorenstein questions open.