Merging $\lim^1 \mathbf{A} \ne 0$ with other nonvanishing constructions

Abstract: We develop methods for forcing $\lim¹ \mathbf{A} \ne 0$, where $\mathbf{A}$ is a particular inverse system of abelian groups introduced by Mardešić and Prasolov in their computation of certain strong homology groups. These methods allow us to extend previous nonvanishing results of Casarosa and Lambie-Hanson for $\lim^k \mathbf{A}$ for $k \geq 2$. Specifically we show that, for a given $n$, it is relatively consistent with ZFC that $\mathfrak{b} = \mathfrak{d} = ω<em>n$ and $\lim^k \mathbf{A} \ne 0$ whenever $1 \leq k \leq n$ (previously established with $2 \leq k \leq n$). We also show it is relatively consistent with ZFC that $\mathfrak{b} = \mathfrak{d} = ω</em>{ω+2}$ and $\lim^k \mathbf{A} \ne 0$ for all $k \geq 1$ (previously established with $k \geq 2$). We also adapt proofs of Kamo to show that $\lim¹ \mathbf{A} = 0$ holds in many finite support iterated forcing extensions.

• The paper introduces novel forcing methods that ensure lim^1 A is nontrivial while establishing simultaneous nonvanishing for higher derived limits.
• It employs both nonlinear and linear Hechler-style iterations along with combinatorial principles like weak diamond and square to control forcing outcomes.
• The results connect cardinal characteristics with derived functor behavior, offering consistent models that advance research in homological algebra and set theory.

Forcing Nonvanishing of Derived Limits: Extending the Interplay of $\lim^1 \mathbf{A}$ with Combinatorial Set Theory

Introduction and Background

This paper, "Merging $\lim^1 \mathbf{A} \ne 0$ with other nonvanishing constructions" (2607.03995), advances the study of derived functors in the context of inverse systems of abelian groups, particularly the system $\mathbf{A}$ singled out by Mardešić and Prasolov in their analysis of strong homology. The vanishing or nonvanishing of the higher derived limits $\lim^n \mathbf{A}$ carries implications in homological algebra, strong shape theory, and the theory of condensed abelian groups. The paper addresses open consistency questions regarding the simultaneous nonvanishing of the first and higher derived limits of $\mathbf{A}$, developing new forcing constructions to achieve set-theoretic models exhibiting targeted patterns of nonvanishing.

Main Contributions

The principal achievements of the paper are as follows:

• New Forcings for Nonvanishing $\lim^1 \mathbf{A}$: The authors construct variants of Hechler-style (both nonlinear and linear) iterations that force $\lim^1 \mathbf{A} \neq 0$, supplementing prior constructions that only realized nonvanishing for higher derived limits ($\lim^k \mathbf{A}\neq 0$ for $k \geq 2$).
• Simultaneous Nonvanishing for Arbitrary Finite and Countable Intervals: By integrating these forcing notions with combinatorial hypotheses such as weak diamond ($w\diamondsuit$) and square ($\lim^1 \mathbf{A} \ne 0$0), the authors show it is consistent with ZFC (relative to the consistency of ZFC itself) that for any $\lim^1 \mathbf{A} \ne 0$1:
  • $\lim^1 \mathbf{A} \ne 0$2, and
  • $\lim^1 \mathbf{A} \ne 0$3 for all $\lim^1 \mathbf{A} \ne 0$4.
• Unbounded Nonvanishing: Extending the result, they exhibit models with $\lim^1 \mathbf{A} \ne 0$5 and $\lim^1 \mathbf{A} \ne 0$6 for all $\lim^1 \mathbf{A} \ne 0$7.
• Destructibility and Indestructibility Under Forcing: The paper analyzes the (in)destructibility of nontrivial 1-coherent families under finite support iterations of Knaster forcings, building protocols both for freezing (preserving) and trivializing such families depending on the iteration scheme.
• Interplay with Cardinal Characteristics: The results exhibit tight coupling between homological invariants ($\lim^1 \mathbf{A} \ne 0$8) and combinatorial cardinal characteristics of the continuum ($\lim^1 \mathbf{A} \ne 0$9).

Technical Developments

Forcing Frameworks

• Nonlinear Hechler-style Iterations: The authors introduce a recursive definition of posets $\mathbf{A}$0 indexed by a well-founded partial order $\mathbf{A}$1, culminating in a ccc forcing $\mathbf{A}$2 that codifies cofinal orders in $\mathbf{A}$3 and enforces the existence of a nontrivial 1-coherent family, thus forcing $\mathbf{A}$4.
• Simultaneous Nonvanishing via Product Forcings: By taking products of these posets with Cohen-type forcings adding subsets of large cardinals, together with combinatorial square and diamond principles, the paper realizes models where the aforementioned lim$\mathbf{A}$5 groups are simultaneously nonvanishing for arbitrarily long initial intervals.
• Linear Iterations and Fragments of Martin's Axiom: The construction is extended to linear iterations for forcing $\mathbf{A}$6 alongside $\mathbf{A}$7.

(In)Destructibility of Coherent Families

• The authors rigorously analyze conditions under which 1-coherent families remain nontrivial vs. become trivial under various finite-support iterations of ccc or Knaster posets.
• Lemmas due to Kunen and elaborated by Kamo are adapted, yielding equivalence between the ccc property of certain posets, destructibility via $\mathbf{A}$8-preserving forcing, and the internal combinatorics of the coherence relations.
• Further, by coding the process via generic sequences (e.g., Cohen reals via antichains), the paper shows that certain coherent families cannot be extended after long Knaster iterations, leading to the vanishing of $\mathbf{A}$9 in these models.

Numerical and Consistency Results

The paper's main theorems can be paraphrased as follows:

Model Construction	Main Consistency Result
Forcing with $\lim^n \mathbf{A}$0, products of Cohen-type	$\lim^n \mathbf{A}$1 and $\lim^n \mathbf{A}$2 for $\lim^n \mathbf{A}$3
Forcing with $\lim^n \mathbf{A}$4, wider product	$\lim^n \mathbf{A}$5 and $\lim^n \mathbf{A}$6 for all $\lim^n \mathbf{A}$7
Finite support iteration of Knaster forcings	$\lim^n \mathbf{A}$8 in the extension

All results are obtained by sophisticated iterations obeying ccc (ensuring preservation of $\lim^n \mathbf{A}$9) and careful synchronization with cardinal arithmetic.

Implications and Future Directions

The paper robustly bridges derived functor calculations with the combinatorial set theory of the continuum, further integrating insights from forcing theory (particularly iterations and the handling of ccc/Knaster properties) with homological considerations. The explicit construction of models exhibiting fine-tuned nonvanishing patterns for the derived functors broadens the toolkit for constructing counterexamples and establishing independence in problems originating in topology and homological algebra.

The authors' methods suggest several lines for future exploration:

• Refinement of Nonvanishing Patterns: Is it possible to achieve more arbitrary prescribed patterns of vanishing and nonvanishing for the sequence $\mathbf{A}$0?
• Extension To Other Systems: Analogous techniques might yield independence results for derived limits of other notable inverse systems in homological and condensed mathematics.
• Interaction With Other Cardinological Principles: Investigating similar phenomena under variations of large cardinal, determinacy, or even anti-large cardinal assumptions.

Conclusion

The paper delivers substantial progress in understanding the subtle connection between set-theoretic properties of the continuum and the vanishing behavior of higher derived limits in a key inverse system. Through highly technical iterations and combinatorial arguments, it enhances prior results by Casarosa, Lambie-Hanson, and others, particularly by demonstrating the consistency of nonvanishing $\mathbf{A}$1 together with arbitrary large finite intervals of higher nonvanishing derived limits. These developments are of theoretical significance for both set theory and homological algebra, providing new invariants and obstructions to additivity in strong homology and related contexts.