Simutaneously vanishing higher derived limits without large cardinals
aa r X i v : . [ m a t h . L O ] F e b SIMULTANEOUSLY VANISHING HIGHER DERIVED LIMITS WITHOUTLARGE CARDINALS
JEFFREY BERGFALK, MICHAEL HRUˇS ´AK, AND CHRIS LAMBIE-HANSON
Abstract.
A question dating to Sibe Mardeˇsi´c and Andrei Prasolov’s 1988 work [12], and mo-tivating a considerable amount of set theoretic work in the ensuing years, is that of whether itis consistent with the
ZFC axioms for the higher derived limits lim n ( n >
0) of a certain inversesystem A indexed by ω ω to simultaneously vanish. An equivalent formulation of this question isthat of whether it is consistent for all n -coherent families of functions indexed by ω ω to be trivial.In this paper, we prove that, in any forcing extension given by adjoining i ω -many Cohen reals,lim n A vanishes for all n >
0. Our proof involves a detailed combinatorial analysis of the forcingextension and repeated applications of higher dimensional ∆-system lemmas. This work removesall large cardinal hypotheses from the main result of [6] and substantially reduces the least valueof the continuum known to be compatible with the simultaneous vanishing of lim n A for all n > Introduction
The set theoretic study of higher derived limits traces principally to Sibe Mardeˇsi´c and AndreiPrasolov’s 1988 work [12]; it was in this paper that a relationship between(1) the continuity properties of strong homology,(2) the behavior of the derived limits of inverse systems indexed by functions from ω to ω , and(3) infinitary combinatorics and assumptions supplementary to ZFC ,was first perceived. The most elementary of the systems as in (2) was denoted A by Mardeˇsi´c andPrasolov in [12], and the works which followed would show the behavior of its higher limits sensitiveto a variety of set theoretic hypotheses; additional interest in these behaviors derived from theirconnection to the broader set theoretic theme of nontrivial coherence (main works in this line were[8, 17, 10, 18, 9, 14, 3, 1]; see [6, Introduction] for a brief research history). The outstanding questiontracing to [12] was whether the statement “lim n A = 0 for all n >
0” is consistent with the
ZFC axioms; this was affirmatively answered in [6] in 2019, under the assumption of the existence of aweakly compact cardinal. Several immediately ensuing questions are listed in the conclusion of [6].The first of these, that of the consistency strength of this statement, is answered by our main result:
Main Theorem.
The statement “ lim n A = 0 for all n > ” holds in the extension of V by theforcing Add( ω, i ω ) for adjoining i ω -many Cohen reals. In particular, the statement “lim n A = 0 for all n >
0” carries no large cardinal strength what-soever. The second of the questions listed in [6] is that of the minimum value of the continuum
Mathematics Subject Classification.
Key words and phrases.
Cohen forcing, derived limit, nontrivial coherence, Delta system lemma, strong homology.The first author was partially supported by Austrian Science Foundation (FWF) Grant Number Y1012-N35.The second author was partially supported by a CONACyT grant A1-S-16164 and PAPIIT grant IN104220.The work in this paper began during a visit by the third author to the Centro de Ciencias Matem´aticas at UNAMMorelia while the first author was a postdoc at that institution. Both authors would like to thank the CCM for itshospitality and support. compatible with this statement. The works [12] and [6] established for that value lower and upperbounds of ℵ and a weakly inaccessible cardinal, respectively; the gap between them was substantial. Main Corollary.
It is consistent relative to the
ZFC axioms that lim n A = 0 for all n > and ℵ = ℵ ω +1 . As noted in [6], plausible scenarios exist in which ℵ ω +1 is optimal, a point we return to in ourconclusion below.In order to describe the structure of our paper, we should first say a few words about our overallargument and, in particular, about how it both builds on and departs from that of [6]. In both thatwork and this one, the idea is to argue in a given forcing extension that an arbitrary n -coherentfamily of functions Φ is trivial. In both cases, this is achieved in two steps: • First, a trivialization of the restricted family Φ ↾ A is found for some A ⊆ ω ω . • Second, trivializations of Φ ↾ A are shown to extend to trivializations of all of Φ.The requirements of these two steps are in tension; what’s wanted is an A which is at once “sufficientlysmall” and “sufficiently large” to effect the first and second steps, respectively. This is a tension whichthe large cardinal assumption of [6] may be viewed as resolving: there, the weak compactness of acardinal κ manifests as multidimensional ∆-system relationships among large families of conditions ina finite support forcing iteration of length κ . These families’ homogeneities lend them a “smallness”of the sort called for in step one; this being an iteration of Hechler forcings ensures that, nevertheless,the associated sets A are ≤ ∗ -cofinal in ω ω , from which step two follows easily.In the present work, cardinal arithmetic and inductive hypotheses on n together take the placeof the large cardinal assumption in [6]. Here again, higher-dimensional ∆-systems lie at the heartof step one, and we draw on [11] for their description and analysis. Observe, however, that withoutlarge cardinal assumptions, such systems can only appear, in general, together with some drop incardinality. In consequence, the set A associated to such a system in step one of our argument issmall in a much stronger sense than in [6]. Nevertheless, in the context of Cohen forcing, genericityarguments coupled, at each stage n , with inductive hypotheses on the triviality of k -coherent familiesof functions for k < n allow us to propagate the triviality of Φ ↾ A to all of Φ as desired.Our account of this argument is structured as follows: in Section 2, we record our basic conven-tions, some results on higher-dimensional ∆-systems, and the conversion of assertions about lim n A to assertions about the triviality of n -coherent families of functions. This section includes a briefhomological interlude that set theorists, for example, may safely ignore: from Section 2.3, only Defi-nition 2.7, Fact 2.9, and Proposition 2.13 are needed in the remainder of the paper. In Section 3, weprove a strong form of the n = 1 instance of our main theorem. In Section 4, we describe the “steptwo” portion of our argument outlined above, showing how, at each stage n >
1, our trivializationsof Φ ↾ A will extend to all of Φ. In Section 5, we record a framework deriving from [6] for definingtrivializations of families like Φ ↾ A . As the reader may have surmised, the paper from this pointforward is fairly technical, and we therefore close this section with several heuristic comments. InSection 6, we prove strong forms of the n > Preliminaries
Notational conventions. If X is a set and κ is a cardinal, then [ X ] κ = { Y ⊆ X | | Y | = κ } and [ X ] <κ = { Y ⊆ X | | Y | < κ } . If κ and λ are cardinals, then we say that λ is <κ -inaccessible if ν <κ < λ for all ν < λ . If X is a set of ordinals, then otp( X ) denotes the order-type of X . We IMULTANEOUSLY VANISHING HIGHER DERIVED LIMITS WITHOUT LARGE CARDINALS 3 will often view finite sets of ordinals as finite increasing sequences of ordinals, and vice versa. Forexample, if a ∈ [On] <ω and ℓ < otp( a ), then a ( ℓ ) is the unique α ∈ a such that | a ∩ α | = ℓ . If m ⊆ a , then a [ m ] = { a ( ℓ ) | ℓ ∈ m } . For any set X of ordinals and natural number n , the notation( α , . . . , α n − ) ∈ [ X ] n will denote the conjunction of the statements { α , . . . , α n − } ∈ [ X ] n and α < . . . < α n − . Frequently in what follows we index objects by finite sets, either of ordinals or offunctions or of other finite sets. Our use of commas or curly brackets in the associated subscriptsor superscripts is, in general, according to no other principle than readability. We will often, forexample, write expressions like q αβ for expressions like q { α,β } ; we handle these matters with somegreater care, however, in the context of the more technical Section 6.The forcings appearing herein will all be of the form P = Add( ω, χ ), where χ is an uncountablecardinal. We think of the conditions of P as finite partial functions from χ × ω to ω , ordered byreverse inclusion. Forcing with P produces a generic function F : χ × ω → ω . For a fixed α < χ wecall the function F ( α, · ) : ω → ω the α th Cohen real added by P , and we will typically denote thisfunction by f α ; we denote the canonical P -name in V for f α by ˙ f α . If G is P -generic over V and W ⊆ χ then G W denotes { p ∈ G | dom( p ) ⊆ W × ω } . For any condition p in P let u ( p ) denote theset { α < χ | dom( p ) ∩ ( { α } × ω ) = ∅} and let ¯ p denote the finite partial function from otp( u ( p )) × ω to ω defined as follows: for all i < otp( u ( p )) and all m < ω , define ( i, j ) to be in the domain of ¯ p if and only if ( u ( p )( i ) , j ) ∈ dom( p ); if so, let ¯ p ( i, j ) = p ( u ( p )( i ) , m ). Intuitively, ¯ p is a “collapsed”version of p . Notice that the set { ¯ p | p ∈ P } is a subset of the set of finite partial functions from ω × ω to ω and is therefore countable.For notational conventions pertaining more directly to coherent families of functions, see Section2.3 below.2.2. Higher-dimensional ∆ -systems. Our proofs will make repeated use of multidimensional∆-system lemmas. For this purpose we recall some relevant definitions and results from [11].
Definition 2.1.
Suppose that a and b are sets of ordinals.(1) We say that a and b are aligned if otp( a ) = otp( b ) and if otp( a ∩ γ ) = otp( b ∩ γ ) for all γ ∈ a ∩ b . In other words, if γ is a common element of two aligned sets a and b , then itoccupies the same relative position in both a and b .(2) If a and b are aligned then we let r ( a, b ) := { i < otp( a ) | a ( i ) = b ( i ) } . Notice that, in thiscase, a ∩ b = a [ r ( a, b )] = b [ r ( a, b )]. Definition 2.2.
Suppose that H is a set of ordinals, n is a positive integer, and u b is a set ofordinals for all b ∈ [ H ] n . We call h u b | b ∈ [ H ] n i a uniform n -dimensional ∆ -system if there is anordinal ρ and, for each m ⊆ n , a set r m ⊆ ρ satisfying the following statements.(1) otp( u b ) = ρ for all b ∈ [ H ] n .(2) For all a, b ∈ [ H ] n and m ⊆ n , if a and b are aligned with r ( a, b ) = m , then u a and u b arealigned with r ( u a , u b ) = r m .(3) For all m , m ⊆ n , we have r m ∩ m = r m ∩ r m .The following is a crucial feature of these ∆-systems: Lemma 2.3.
Suppose that ≤ n < ω and h u b | b ∈ [ H ] n i is a uniform n -dimensional ∆ -system,as witnessed by ρ and h r m | m ⊆ n i , and assume for simplicity that H has no largest element. Foreach m < n and each a ∈ [ H ] m , define a set u a by choosing b ∈ [ H ] n such that b [ m ] = a and setting u a = u b [ r m ] . (Here and in similar places later in the paper, m denotes the set of natural numbersless than m , so, for instance, b [ m ] = { b ( ℓ ) | ℓ < m } , r m = r { ℓ | ℓ Definition 2.4. Suppose that λ is an infinite regular cardinal. Recursively define cardinals σ ( λ, n )for 1 ≤ n < ω by letting σ ( λ, 1) = λ and, given 1 ≤ n < ω , letting σ ( λ, n + 1) = (2 <σ ( λ,n ) ) + . Fact 2.5. [11, Theorem 2.10] Suppose that • ≤ n < ω ; • κ < λ are infinite cardinals, λ is regular and <κ -inaccessible, and µ = σ ( λ, n ) ; • c : [ µ ] n → <κ ; • for all b ∈ [ µ ] n , we are given a set u b ∈ [On] <κ .Then there are an H ∈ [ µ ] λ and k < <κ such that(1) c ( b ) = k for all b ∈ [ H ] n ;(2) h u b | b ∈ [ H ] n i is a uniform n -dimensional ∆ -system. In order to motivate these definitions, let us highlight a way in which Fact 2.5 will be employed. Lemma 2.6. Suppose that n is a positive integer, H is a set of ordinals, and h p b | b ∈ [ H ] n i is asequence of conditions in some P = Add( ω, χ ) such that • there is a fixed ¯ p such that ¯ p = ¯ p b for all b ∈ [ H ] n , and • h u ( p b ) | b ∈ [ H ] n i is a uniform n -dimensional ∆ -system.Then for all a, a ′ ∈ [ H ] n , if a and a ′ are aligned, then p a and p a ′ are compatible in P .Proof. Fix a, a ′ ∈ [ H ] n such that a and a ′ are aligned. To show that p a and p a ′ are compatible,it suffices to show that, for every ( α, m ) ∈ dom( p a ) ∩ dom( p a ′ ), we have p a ( α, m ) = p a ′ ( α, m ). Tothis end, fix ( α, m ) ∈ dom( p a ) ∩ dom( p a ′ ), so α ∈ u ( p a ) ∩ u ( p a ′ ). Since h p b | b ∈ [ H ] n i is a uniform IMULTANEOUSLY VANISHING HIGHER DERIVED LIMITS WITHOUT LARGE CARDINALS 5 n -dimensional ∆-system and a, a ′ ∈ [ H ] n are aligned, it follows that u ( p a ) and u ( p a ′ ) are aligned.There is therefore a single i < ω such that α = u ( p α )( i ) = u ( p α ′ )( i ). But then, since ¯ p = ¯ p α = ¯ p α ′ ,we have p a ( α, m ) = ¯ p ( i, m ) = p a ′ ( α, m ), as desired. (cid:3) Higher-dimensional coherence and triviality. Our main theorem is an assertion about thederived limits of an inverse system A ; just as in [12] and [6] (and, indeed, as in all the interveningworks cited in the latter), our analysis of these limits will be via their reformulation in terms ofmultidimensionally coherent indexed families of functions from subsets of ω to Z . Readers arereferred to the second of the aforementioned works for fuller details of this reformulation. We turnnow to the relevant conventions and definitions.Given functions f, g : ω → ω , let f ≤ g if and only if f ( j ) ≤ g ( j ) for all j ∈ ω . Let I ( f )denote the set (cid:8) ( j, k ) ∈ ω (cid:12)(cid:12) k ≤ f ( j ) (cid:9) ; visually, this is the region below the graph of f . Givena sequence ~f = ( f , . . . , f n ) of elements of ω ω , let ∧ ~f denote the greatest lower ≤ -bound of thefunctions f , . . . , f n . For any such sequence and i ≤ n , let ~f i denote the sequence of length n obtained by removing the i th entry of ~f ; in symbols, ~f i = ( f , . . . , f i − , f i +1 , . . . , f n ), sometimeswritten as ( f , . . . , ˆ f i , . . . , f n ). If π is a permutation of (0 , . . . , n ), then sgn ( π ) denotes the sign or parity of π , recorded as a 1 or − 1. The notation π ( ~f ) denotes the sequence ( f π (0) , . . . , f π ( n ) ).If ϕ and ψ are partial functions from ω to Z , then the expression ϕ = ∗ ψ will mean that the set { ( j, k ) ∈ dom( ϕ ) ∩ dom( ψ ) | ϕ ( j, k ) = ψ ( j, k ) } is finite. Implicit in this expression, in other words,are restrictions of ϕ and ψ to their shared domain; a similar convention will apply to sums of suchfunctions below. Definition 2.7. Fix an X ⊆ ω ω and a positive integer n and suppose thatΦ = D ϕ ~f : I ( ∧ ~f ) → Z (cid:12)(cid:12)(cid:12) ~f ∈ X n E is an indexed family of functions. • Φ is alternating if ϕ π ( ~f ) = sgn ( π ) ϕ ~f for every ~f ∈ X n and every permutation π of (0 , . . . , n − • Φ is n -coherent if it is alternating and if n X i =0 ( − i ϕ ~f i = ∗ ~f ∈ X n +1 . (As indicated, for readability, here we omit the restrictions of the functionsin the above expression to the intersection of their domains; formally, each ϕ ~f i should be ϕ ~f i ↾ I ( ∧ ~f ). We will continue this practice below.) • If n = 1, then Φ is n -trivial (i.e., ) if there exists a ψ : ω → Z such that ψ = ∗ ϕ f for all f ∈ X . If n > 1, then Φ is n -trivial if there exists an alternating familyΨ = D ψ ~f : I ( ∧ ~f ) → Z (cid:12)(cid:12)(cid:12) ~f ∈ X n − E such that n − X i =0 ( − i ψ ~f i = ∗ ϕ ~f JEFFREY BERGFALK, MICHAEL HRUˇS ´AK, AND CHRIS LAMBIE-HANSON for all ~f ∈ X n . We term such a ψ or Ψ an n -trivialization of Φ.When it is clear from context, we will frequently omit the prefix n - when speaking of triviality.Lastly, if A ⊆ X , then Φ ↾ A denotes h ϕ ~f | ~f ∈ A n i . Observation 2.8. Fix an ℓ ∈ ω and a family of functions Φ = h ϕ ~f : I ( ∧ ~f ) → Z | ~f ∈ X n i . Let˜Φ = h ˜ ϕ ~f | ~f ∈ X n i where ˜ ϕ ~f ( j, k ) = ( j ≤ ℓϕ ~f ( j, k ) if j > ℓ for all ( j, k ) ∈ I ( ∧ ~f ). Then Φ is trivial if and only if ˜Φ is.By the following equivalence, our main theorem is, equivalently, a statement about the n -trivialityof all n -coherent families of functions Φ = h ϕ ~f : I ( ∧ ~f ) → Z | ~f ∈ ( ω ω ) n i . Fact 2.9 ([3, 6]) . For all positive integers n , lim n A = 0 if and only if every n -coherent family offunctions h ϕ ~f : I ( ∧ ~f ) → Z | ~f ∈ ( ω ω ) n i is trivial. Our overall argument’s strategy is to arrange this fact’s latter condition; in such an approach,homological algebraic considerations appear as essentially external, or preliminary, to our main work.Instrumental in our forcing arguments, however, will be a more locally finitary characterization of n -triviality, one connecting to that of Definition 2.7 via a long exact sequence of higher derivedlimits of A -related inverse systems. As noted above, readers primarily interested in those argumentsmay proceed without danger or delay to Proposition 2.13 and continue their reading from there. Itsargument is straightforward, algebraic, and occupies roughly the next two pages. Definition 2.10. The inverse systems • A = ( A f , p fg , ω ω ) • B = ( B f , q fg , ω ω ) • B / A = (( B/A ) f , r fg , ω ω )are defined as follows: A f = L I ( f ) Z , B f = Q I ( f ) Z , and ( B/A ) f = B f /A f , for all f in ω ω . Forall f ≤ g in ω ω , the bonding maps p fg : A g → A f are simply the projection maps; similarly for thebonding maps of B and B / A . These systems assemble in a short exact sequence0 −→ A −→ B −→ B / A −→ −→ lim A −→ lim B −→ lim B / A ∂ −→ lim A −→ lim B −→ lim B / A ∂ −→ · · · (2)For any X ⊆ ω ω we write A ↾ X for the restriction of A to the index-set X ; similarly for B and B / A . Observe that for any such X , short and long exact sequences just as above exist for A ↾ X , B ↾ X , and B / A ↾ X .The expressions lim n denote the derived limits of the inverse limit functor lim; they are functorstaking, in our context, inverse systems to the category of abelian groups. A standard heuristicfor these functors is that, in aggregate, they at least potentially recover the data of an inversesystem that the lim functor alone might lose. The exact sequences above are a main instance of thisdynamic: lim alone applied to the sequence (1) may fail to conserve its exactness, but when appliedin combination with the higher derived limits lim n , as in (2), it does transmit exactness, as desired.Our characterizations of the vanishing of lim n A derive from the isomorphism lim n B / A ∼ =lim n +1 A for all n > 0; the mod finite relations of Definition 2.7, for example, are an artifact of IMULTANEOUSLY VANISHING HIGHER DERIVED LIMITS WITHOUT LARGE CARDINALS 7 the modulus A on the left-hand side of this isomorphism. This isomorphism, in turn, is an effect offollowing fact within the long exact sequence (2). Lemma 2.11. lim n ( B ↾ X ) = 0 for all X ⊆ ω ω and n > . We argue this fact via more concrete characterizations of lim n ( B ↾ X ); these are essentially thosegiven by [6, pages 10-11]. Definition 2.12. For n ≥ 0, the group lim n ( B ↾ X ) is the cohomology of the cochain complex · · · d n − −−−→ K n ( B ↾ X ) d n −→ K n +1 ( B ↾ X ) d n +1 −−−→ · · · where K n ( B ↾ X ) denotes the subgroup of Y ~f ∈ X n +1 B ∧ ~f whose elements c satisfy c ( π ( ~f )) = sgn ( π ) c ( ~f )for all ~f ∈ X n +1 and permutations π of (0 , . . . , n ). The differentials d n are defined as usual: for any c ∈ K n ( B ↾ X ), d n c ( ~f ) = n +1 X i =0 ( − i (cid:16) c ( ~f i ) ↾ I ( ∧ ~f ) (cid:17) (3)for each ~f ∈ X n +2 .lim n ( A ↾ X ) and lim n ( B / A ↾ X ) are defined analogously. Proof of Lemma 2.11. Fix an X ⊆ ω ω and an n > c ∈ K n ( B ↾ X ) for which d n c = 0.We will define a b ∈ K n − ( B ↾ X ) with d n − b = c . To that end, for each x ∈ S f ∈ X I ( f ) fix an f x ∈ X such that x ∈ I ( f x ). Then, for each ~f ∈ X n and x ∈ I ( ∧ ~f ), let b ( ~f )( x ) = ( − n c ( f , . . . , f n − , f x )( x ).For all ~f ∈ X n +1 and x ∈ I ( ∧ ~f ), we then have d n − b ( ~f )( x ) = n X i =0 ( − i b ( ~f i )( x )= n X i =0 ( − n + i c ( f , . . . , ˆ f i , . . . , f n , f x )( x )= ( − n d n c ( f , . . . , f n , f x )( x ) − ( − n +1 c ( ~f )( x )= c ( ~f )( x ),as desired. (cid:3) As noted, together with the X -indexed variants of the long exact sequence (2), Lemma 2.11 hasas consequences isomorphisms ∂ n : lim n ( B / A ↾ X ) ∼ = −−→ lim n +1 ( A ↾ X ) JEFFREY BERGFALK, MICHAEL HRUˇS ´AK, AND CHRIS LAMBIE-HANSON for each n > 0, as well as the isomorphism ∂ : lim( B / A ↾ X )im(lim( B ↾ X )) ∼ = −−→ lim ( A ↾ X ).As shown in [6], for all n ≥ ∂ n may be defined via the following procedure:(1) To define ∂ n [ c ], fix a cocycle c ∈ K n ( B / A ↾ X ) representing the cohomology class [ c ].(2) Fix then a Φ = h ϕ ~f : I ( ∧ ~f ) → Z | ~f ∈ X n +1 i representing c in the sense that each ϕ ~f falls inthe A ~f -coset c ( ~f ). Observe that such a Φ may be chosen to be alternating, and that in thiscase it will be an ( n +1)-coherent family of functions. (In fact, for all n ≥ n + 1)-coherent familiesof functions indexed by X by the ( n + 1)-trivial families of functions indexed by X , as thereader may verify.)(3) Let ∂ n [ c ] be the cohomology class of the cocycle d n Φ ∈ K n +1 ( A ↾ X ), where d n Φ( ~f ) = n +1 X i =0 ( − i ϕ ~f i for all ~f ∈ X n +2 .That this procedure defines an isomorphism implies that an ( n + 1)-coherent family Φ is ( n + 1)-trivial if and only if d n Φ is a coboundary in K n +1 ( A ↾ X ), i.e., if and only if there exists a familyof finitely supported functions h ψ ~f : I ( ∧ ~f ) → Z | ~f ∈ X n +1 i such that d n Φ( ~f ) = n +1 X i =0 ( − i ψ ~f i for all ~f ∈ X n +2 . By way of these observations, together with Observation 2.8, we arrive to oursecond criterion for n -triviality: Proposition 2.13. Fix X ⊆ ω ω and a positive integer n and let Φ = h ϕ ~f | ~f ∈ X n i be an n -coherentfamily of functions. Then Φ is trivial if and only if there exists an ℓ < ω and an alternating familyof finitely supported functions Ψ = h ψ ~f : I ( ∧ ~f ) → Z | ~f ∈ X n i such that n X i =0 ( − i ϕ ~f i ( j, k ) = n X i =0 ( − i ψ ~f i ( j, k ) for all ~f ∈ X n +1 and all ( j, k ) ∈ I ( ∧ ~f ) with j > ℓ . When there is a possibility of confusion, we will refer to a Ψ as above as a type II trivialization and a Ψ or ψ as in Definition 2.7 as a type I trivialization . By and large, however, these two sorts oftrivializations correspond to two distinct phases of our argument; in particular, the trivializationsunder discussion in Sections 3 and 4 are all of type I, while those under discussion in Sections 5 and6 are of type II. 3. The case of n = 1We now argue the base case of our main theorem. The main result of this section is essentiallydue to Kamo [10]. In fact, the result in [10] is superior to the one presented here in that Kamoproves that lim A = 0 in any extension obtained by adding ω -many Cohen reals, whereas ourhypothesis is that we have added at least ( i +1 ) V -many Cohen reals. Our reason for presenting this IMULTANEOUSLY VANISHING HIGHER DERIVED LIMITS WITHOUT LARGE CARDINALS 9 slightly suboptimal proof is simply that many of the ideas of the proof of the general case appearhere in a significantly simplified setting; this section therefore serves as an introduction to some ofthe techniques and ideas that will make an appearance in a more complicated guise later in thepaper. Theorem 3.1. Let P = Add( ω, χ ) for a cardinal χ > i . The following then holds in V P : For anyset X ⊆ ω ω containing at least ( i +1 ) V -many of the Cohen reals added by P , every -coherent family Φ = h ϕ f | f ∈ X i indexed by X is trivial.Proof. Fix a condition p ∈ P and P -names ˙ X and ˙Φ = h ˙ ϕ ˙ f | ˙ f ∈ ˙ X i such that • p (cid:13) “ |{ α < χ | ˙ f α ∈ ˙ X }| ≥ ( i +1 ) V ”, and • p (cid:13) “ ˙Φ is a 1-coherent family”.We will produce a condition q ≤ p forcing ˙Φ to be trivial.Begin by letting Y be the set of α < χ such that there is a condition p α ≤ p such that p α (cid:13) “ ˙ f α ∈ ˙ X ”; observe that | Y | ≥ i +1 by assumption. For each α ∈ Y , fix such a condition p α . Since P is i +1 -Knaster, there exists a set Y ′ ⊆ Y of size i +1 such that { p α | α ∈ Y ′ } consists of pairwisecompatible conditions.For each ( α, β ) ∈ [ Y ′ ] , fix a condition q α,β extending both p α and p β and deciding the value of { ( j, k ) ∈ I ( ˙ f α , ˙ f β ) | ˙ ϕ ˙ f α ( j, k ) = ˙ ϕ ˙ f β ( j, k ) } to be equal to some set e α,β ∈ [ ω × ω ] <ω . By extending q α,β if necessary, we may assume that { α, β } ⊆ u ( q α,β ). Let u α,β = u ( q α,β ).By Fact 2.5, there exists a set H ∈ [ Y ′ ] ℵ and a ¯ q , e , and i ∗ such that • h u α,β | ( α, β ) ∈ [ H ] i is a uniform 2-dimensional ∆-system; • (¯ q α,β , e α,β ) = (¯ q, e ) for all ( α, β ) ∈ [ H ] ; • β = u α,β ( i ∗ ) for all ( α, β ) ∈ [ H ] .By shrinking H if necessary, we may assume that otp( H ) = ω . Now let h r m | m ⊆ i witnessthat h u α,β | ( α, β ) ∈ [ H ] i is a uniform 2-dimensional ∆-system, and let h u α | α ∈ H i and u ∅ be asgiven by Lemma 2.3. For each α ∈ H , define a condition q α ∈ P by choosing a β ∈ H \ ( α + 1) andletting q α = q α,β ↾ ( u α × ω ). We claim that this definition is independent of our choice of β . Indeed,suppose that β < β ′ are elements of H \ ( α + 1). Then u α = u α,β ∩ u α,β ′ = u α,β [ r ] = u α,β ′ [ r ],hence if ( δ, m ) ∈ dom( q α,β ) ∩ ( u α × ω ) then there is an i ∈ r such that u α,β ( i ) = δ = u α,β ′ ( i ).Since ¯ q α,β = ¯ q = ¯ q α,β ′ , it follows that q α,β ( δ, m ) = ¯ q ( i, m ) = q α,β ′ ( δ, m ), so q α,β ↾ ( u α × ω ) ⊆ q α,β ′ ↾ ( u α × ω ). A symmetric argument yields the reverse inclusion, showing that our definition of q α isindeed independent of our choice of β . Observe that q α = T { q α,β | β ∈ H \ ( α + 1) } ; as each q α,β extends p α , it follows that q α ≤ p α and hence that q α (cid:13) “ ˙ f α ∈ ˙ X ”.Similarly, define a condition q ∅ by choosing ( α, β ) ∈ [ H ] and letting q ∅ = q α,β ↾ ( u ∅ × ω ). By anargument exactly as in the previous paragraph, this definition is independent of our choice of ( α, β ).Note that q ∅ = T { q α,β | ( α, β ) ∈ [ H ] } ; in consequence, since each q α,β extends p , we have q ∅ ≤ p .We claim that q ∅ forces that ˙Φ is trivial. Let ˙ A be a P -name for { α ∈ H | q α ∈ ˙ G } , where ˙ G isthe canonical P -name for the generic filter. Claim 3.2. q ∅ (cid:13) “ | ˙ A | = ℵ ” .Proof. It suffices to show for each η ∈ H that the set { q α | α ∈ H \ η } is pre-dense below q ∅ . Tothis end, fix such an η ∈ H and an r ≤ q ∅ . We desire an α ∈ H \ η such that q α is compatible with r . Since h u α | α ∈ H \ η i is an infinite ∆-system with root u ∅ , and since u r is finite, there existsan α ∈ H \ η such that u α \ u ∅ is disjoint from u r . Decompose q α as q ∅ ∪ ( q α ↾ ( u α \ u ∅ ) × ω ) andobserve that r ≤ q ∅ and dom( r ) ∩ (( u α \ u ∅ ) × ω ) = ∅ . It follows that r and q α are compatible, asdesired. (cid:3) Now recall that β = u α,β ( i ∗ ) for all ( α, β ) ∈ [ H ] . Notice that i ∗ / ∈ r , since the alternative wouldimply that β ∈ u α = u α,β [ r ] for all ( α, β ) ∈ [ H ] , contradicting the fact that u α is finite. Hence β ∈ u α,β \ u α for all ( α, β ) ∈ [ H ] . Let ℓ be the least natural number j such that • e ⊆ j × ω and; • { j ′ | ( i ∗ , j ′ ) ∈ dom(¯ q ) } ⊆ j .We then have { j ′ | ( β, j ′ ) ∈ dom( q α,β ) } ⊆ ℓ for each ( α, β ) ∈ [ H ] . Claim 3.3. q ∅ forces that { ( j, k ) ∈ I ( ˙ f α , ˙ f α ′ ) | ˙ ϕ ˙ f α ( j, k ) = ˙ ϕ ˙ f α ′ ( j, k ) } ⊆ ℓ × ω. for all α, α ′ ∈ ˙ A .Proof. If not, then there exist an r ≤ q ∅ , a pair of ordinals α < α ′ in H , and a ( j, k ) ∈ ω × ω suchthat • r ≤ q α , q α ′ ; • j ≥ ℓ ; • r forces that ( j, k ) is in I ( ˙ f α , ˙ f α ′ ) and that ˙ ϕ ˙ f α ( j, k ) = ˙ ϕ ˙ f α ′ ( j, k ).Both h u α,β | β ∈ H \ ( α + 1) i and h u α ′ ,β | β ∈ H \ ( α + 1) i are infinite ∆-systems with roots u α and u α ′ , respectively; as u r is finite, there therefore exists a β ∈ H such that both u α,β \ u α and u α ′ ,β \ u α ′ are disjoint from u r .By Lemma 2.6, the conditions q α,β and q α ′ ,β are compatible. Observe also that q α,β = q α ∪ ( q α,β ↾ ( u α,β \ u α ) × ω ). Since r ≤ q α and dom( r ) ∩ (( u α,β \ u α ) × ω ) = ∅ , the conditions r and q α,β arecompatible. Similarly, r and q α ′ ,β are compatible, and therefore r ∗ = r ∪ q α,β ∪ q α ′ ,β is a conditionin P . Notice also that β / ∈ u r . By the paragraph preceding Claim 3.3 and the fact that j ≥ ℓ , itfollows that ( β, j ) / ∈ dom( r ∗ ), so we may extend r ∗ to a condition r ∗∗ such that ( β, j ) ∈ dom( r ∗∗ )and r ∗∗ ( β, j ) = k . In particular, r ∗∗ will force that ( j, k ) is in I ( ˙ f β ).Recall that j ≥ ℓ implies ( j, k ) / ∈ e . Therefore, since it extends both q α,β and q α ′ ,β and forces( j, k ) to be in I ( ˙ f α , ˙ f α ′ , ˙ f β ), the condition r ∗∗ will force“ ˙ ϕ ˙ f α ( j, k ) = ˙ ϕ ˙ f β ( j, k ) = ˙ ϕ ˙ f α ′ ( j, k )” , contradicting the fact that r ∗∗ ≤ r and r forces “ ˙ ϕ ˙ f α ( j, k ) = ˙ ϕ ˙ f ′ α ( j, k )”. (cid:3) Now let G be P -generic over V with q ∅ ∈ G . Let Φ = h ϕ f | f ∈ X i and A denote the realizationsin V [ G ] of ˙Φ and ˙ A , respectively. Define a function ψ : ω × ω → Z as follows. For any ( j, k ) ∈ ω × ω with j ≥ ℓ , if ( j, k ) ∈ I ( f α ) for some α ∈ A then let ψ ( j, k ) = ϕ f α ( j, k ) (by Claim 3.3, this definitionis independent of our choice of α ). In all other cases, let ψ ( j, k ) = 0.We claim that ψ witnesses that Φ is trivial. Assume for contradiction that it does not, so thatfor some f ∈ X the set E f := { ( j, k ) ∈ I ( f ) | ϕ f ( j, k ) = ψ ( j, k ) } is infinite. Since I ( f ) ∩ ( ℓ × ω ) isfinite, the set E ∗ f = E f ∩ ([ ℓ, ω ) × ω ) is then infinite and there are infinitely many j < ω for which E ∗ f ∩ ( j × ω ) = ∅ . As P has the countable chain condition, E f ∈ V [ G W ] for some countable set W ⊆ χ in V . Fix α ∈ A \ W . By genericity, I ( f α ) ∩ E ∗ f is infinite. It follows from our definition of ψ that ψ ( j, k ) = ϕ f α ( j, k ) for all ( j, k ) ∈ I ( f α ) ∩ E ∗ f . Hence ϕ f ↾ ( I ( f α ) ∩ E ∗ f ) = ∗ ψ ↾ ( I ( f α ) ∩ E ∗ f ),by the coherence of Φ, contradicting the definition of E ∗ f and the fact that I f α ∩ E ∗ f is infinite. Thisshows that q ∅ forces ˙Φ to be trivial, concluding the proof. (cid:3) Already in the above proof the broader contours of our more general argument are legible. Inparticular, observe that comparatively few of the Cohen reals in X , namely just those in the size- ℵ set { f α | α ∈ A } , played any essential role in the definition of the trivialization ψ . Similarly, in whatfollows we will construct trivializations of higher-dimensional coherent families Φ n = h ϕ ~f | ~f ∈ X n i by first trivializing those families over small indexing subsets A of X . More precisely, as outlined inour introduction, the arguments for the higher- n versions of Theorem 3.1 will split into two mainphases: in the first, repeated application of principles like Fact 2.5 determine trivializations of smallsubfamilies Φ n ↾ A of Φ n ; in the second, inductive hypotheses propagate their triviality back out tothe entirety of Φ n . We describe the mechanics of these two phases in reverse order in the next twosections; we then apply these descriptions in the proof of our main theorem, Theorem 6.1.4. Propagating trivializations In this section we show how the triviality of restrictions of n -coherent families to domains con-staining sufficiently many Cohen reals implies the triviality of the entire families. We first introducea slight technical variation of n -coherent families. Definition 4.1. Fix an X ⊆ ω ω , a function h ∈ ω ω , and a positive integer n . We say that a familyof functions Φ = h ϕ ~f : I ( ∧ ~f ) ∩ I ( g ) → Z | ~f ∈ X n i is n -coherent below g if it satisfies the firsttwo bullet points of Definition 2.7, the only difference being that in this case the domain of ϕ ~f is I ( ∧ ~f ) ∩ I ( g ) rather than I ( ∧ ~f ). We say that such a family is n -trivial below g if there is a ψ or Ψ asin the third bullet point of Definition 2.7, again with the only difference being that, in case n = 1,we have ψ : I ( g ) → Z , and in case n > 1, we have ψ ~f : I ( ∧ ~f ) ∩ I ( g ) → Z for all ~f ∈ X n − .The following proposition is a simple observation but will be necessary in the arguments of thissection. Proposition 4.2. Suppose that X ⊆ ω ω , h ∈ ω ω , n is a positive integer, and every n -coherentfamily of functions indexed by X n is trivial. Then every n -coherent family of functions below g indexed by X n is trivial below g .Proof. Let Φ = h ϕ ~f : I ( ∧ ~f ) ∩ I ( g ) → Z | ~f ∈ X n i be n -coherent below g . Define an n -coherentfamily Φ ∗ = h ϕ ∗ ~f : I ( ∧ ~f ) → Z | ~f ∈ X n i by letting ϕ ∗ ~f ( j, k ) = ϕ ~f ( j, k ) for all ( j, k ) ∈ I ( ∧ ~f ) ∩ I ( g ) and ϕ ∗ ~f ( j, k ) = 0 for all ( j, k ) ∈ I ( ∧ ~f ) \ I ( g ). It is easily verified that Φ ∗ is n -coherent. By assumption,Φ ∗ is trivial, as witnessed by a single function ψ ∗ if n = 1 or a family Ψ ∗ = h ψ ∗ ~f | ~f ∈ X n − i if n > n = 1, then the function ψ := ψ ∗ ↾ I ( g ) witnesses that Φ is trivial below g , and if n > 1, then thefamily Ψ := h ψ ∗ ~f ↾ I ( ∧ ~f ) ∩ I ( g ) | ~f ∈ X n − i witnesses that Φ is trivial below g , as desired. (cid:3) For motivation, we now begin with the n = 2 case of Theorem 6.1; assume the aforementioned“first phase” of its proof completed. More precisely, this assumption takes the following form: let λ = i +1 and let λ = σ ( λ +1 , 5) (see again Definition 2.4 for the expression σ ( · , · )). By the argumentsof Section 6, for any χ ≥ λ , the following will hold in V Add( ω,χ ) : if • X ⊆ ω ω contains at least λ -many of the Cohen reals added by Add( ω, χ ) , and • Φ = h ϕ f,g | ( f, g ) ∈ X i is -coherent,then there exists an A ⊆ X such that • A contains λ -many of the Cohen reals added by Add( ω, χ ) , and • Φ ↾ A is trivial. Now let G be Add( ω, χ )-generic over V and work in V [ G ]. By assumption, there exists an A ⊆ χ indexing λ -many Cohen reals f α ( α ∈ A ) and a family h τ α : I ( f α ) → Z | α ∈ A i trivializing Φ ↾ A .We propagate this trivialization to all of Φ as follows: for all f ∈ X and α ∈ A , let ς fα = ϕ α,f + τ α . (Here and below, in subscripts we will tend to abbreviate Cohen reals by their indices; the secondΦ ↾ A just above, which denotes h ϕ α,β | ( α, β ) ∈ A i , is a related minor abuse. We note lastlythat, in keeping with our restriction-conventions, the domain of ς fα should be understood to bedom( ϕ α,f ) ∩ dom( τ α ).) Claim 4.3. For each f ∈ X the family C f := h ς fα : I ( f ∧ f α ) → Z | α ∈ A i is coherent below f .Proof. This follows from the fact that, for all α and β in A , we have ς fβ − ς fα = ϕ β,f + τ β − ϕ α,f − τ α = ∗ ϕ β,f − ϕ α,f + ϕ α,β = ∗ , where the first = ∗ follows from the fact that h τ α | α ∈ A i trivializes Φ ↾ A and hence τ β − τ α = ∗ ϕ α,β ,the second = ∗ follows from the 2-coherence of Φ. (cid:3) As A contains more than i -many Cohen reals, Theorem 3.1 and Proposition 4.2 imply that eachsuch family C f admits a trivialization τ f : I ( f ) → Z . Claim 4.4. The family T := h τ f | f ∈ X i trivializes Φ .Proof. Suppose for contradiction that it did not. Then for some f, g ∈ X and infinite E ⊆ I ( f ∧ g ), τ g ( j, k ) − τ f ( j, k ) = ϕ f,g ( j, k )for all ( j, k ) ∈ E . As Add( ω, χ ) has the countable chain condition, there exists a W ∈ [ χ ] ℵ suchthat E ∈ V [ G W ]. By genericity, for any β ∈ A \ W the domain I ( f β ) then has infinite intersectionwith E . However, τ g − τ f = ∗ ς gβ − ς fβ = ϕ β,g + τ β − ϕ β,f − τ β = ∗ ϕ f,g , where, as indicated, the equalities should each be read as applying over the restricted domain I ( f ∧ g ∧ f β ). It follows that τ g ( j, k ) − τ f ( j, k ) = ϕ f,g ( j, k ) for only finitely many ( j, k ) ∈ E ∩ I ( f β ),contradicting our assumption. (cid:3) This family T is the propagation of the trivialization h τ α | α ∈ A i to all of X which we haddesired. The above technique generalizes, but entails, unsurprisingly, more steps in the cases ofhigher n . We precede its generalized description with a proof-free sketch of the n = 4 case, simply tobetter indicate these steps’ shape. All coherent and trivializing families below should be understoodto be alternating. Example 4.5. Structuring our argument is an increasing sequence of cardinals λ n ; see the paragraphpreceding Theorem 6.1 for their precise definition. The case of n = 4 begins with a forcing P =Add( ω, χ ) for some χ ≥ λ ; as before, we will work in V P . Let X ⊆ ω ω contain at least λ -many ofthe Cohen reals added by P and let Φ = h ϕ ~f | ~f ∈ X i be 4-coherent. The arguments of Section 6will furnish us with an A ∈ [ χ ] λ and a T A = h τ αβγ | ( α, β, γ ) ∈ A i trivializing Φ ↾ A . For each f ∈ X and ( α, β, γ ) ∈ A let ς fαβγ = ϕ αβγf + τ αβγ . IMULTANEOUSLY VANISHING HIGHER DERIVED LIMITS WITHOUT LARGE CARDINALS 13 This defines a 3-coherent family C f = h ς fαβγ | ( α, β, γ ) ∈ A i below f . By the n = 3 case of Theorem6.1 and Proposition 4.2, for each f ∈ X there exist trivializations T f = h τ fαβ | ( α, β ) ∈ A i of C f .Using these functions, define for each ( f, g ) ∈ X the 2-coherent family C fg = h ς fgαβ | ( α, β ) ∈ A i below f ∧ g via the assignments ς fgαβ = ϕ αβfg − τ gαβ + τ fαβ . By the n = 2 case of Theorem 6.1 and Proposition 4.2, for each ( f, g ) ∈ X there then existtrivializations T fg = h τ fgα | α ∈ A i of C fg . Using these functions, define for each ( f, g, h ) ∈ X the1-coherent family C fgh = h ς fghα | ( α, β ) ∈ A i below f ∧ g ∧ h via the assignments ς fghα = ϕ αfgh + τ ghα − τ fhα + τ fgα . By Theorem 3.1 and Proposition 4.2, for each ( f, g, h ) ∈ X there exists a trivialization τ fgh of C fgh . As above, we then conclude by observing that the collection h τ fgh | ( f, g, h ) ∈ X i trivializesΦ, as desired.Now, within the larger context of our inductive argument, we summarize the general case. Fix aninteger n > n th instance of our inductive hypothesis, namely, that for any j < n and cardinal χ , any j -coherent family of functions Φ = h ϕ ~f | ~f ∈ X j i in V Add( ω,χ ) whose index-set X contains at least λ j -many of the Cohen reals added by P = Add( ω, χ ) is trivial . We show how,combined with the arguments of Section 6, the n th instance of our inductive hypothesis implies the( n + 1) st instance. For χ < λ n this implication is trivial. Therefore fix χ ≥ λ n and, working in V Add( ω,χ ) , fix an n -coherent family Φ = h ϕ ~f | ~f ∈ X n i whose index-set X contains at least λ n -manyof the Cohen reals added by P = Add( ω, χ ). We introduce the organizing notations T nk and C nk andshow by the following sequence of steps that Φ is trivial:(1) Fix an n -coherent Φ as above. The arguments of Section 6 secure for us a set A ∈ [ χ ] λ n − and a T n such that T n trivializes Φ ↾ A .(2) T n is the first in a sequence of families of functions T nk = h τ ~f~α | ~f ∈ X k − and ~α ∈ A n − k i in which k ranges from 1 to n and T nn trivializes Φ. These families are inductively definedalongside a series of related families C nk , as described in items (3) and (4) below.(3) If k is less than n then T nk induces a family of ( n − k )-coherent families of functions C nk = h ς ~f~α | ~f ∈ X k and ~α ∈ A n − k i . To be precise, C nk is the union of the ( n − k )-coherent families of functions C ~fn − k = h ς ~f~α | ~α ∈ A n − k i as ~f ranges through X k .(4) Our inductive hypothesis ensures us trivializations T ~fn − k − of each C ~fn − k . These serve thento define T nk +1 := [ ~f ∈ X k T ~fn − k − , and repeated, alternating applications of this and the previous step cumulatively yield thesequence h T nk | ≤ k ≤ n i of item (2), as desired. Two points in the above scheme merit further discussion:(i) We must specify precisely how the ( n − k )-coherent families of functions C ~fn − k derive fromthe families T nk , and verify that they are in fact ( n − k )-coherent.(ii) We must verify that T nn does indeed trivialize Φ.We begin with item (i). The families C nk are inductively defined on positive integers k ≤ n . Thepattern when k = 1 is plain enough from the examples above: for each f ∈ X the subclass C fn − = h ς f~α | ~α ∈ A n − i of C n is defined from T n = h τ ~α | ~α ∈ A n − i by ς f~α = ϕ ~αf + ( − n τ ~α (4)for each ~α ∈ A n − . Observe then that for all ~α ∈ A n , n − X i =0 ( − i ς f~α i = n − X i =0 ( − i ϕ ~α i f + ( − n n − X i =0 ( − i τ ~α i = ∗ n − X i =0 ( − i ϕ ~α i f + ( − n ϕ ~α = ∗ , by the coherence of Φ. This shows that C fn − is ( n − f and hence, by the inductivehypothesis, ( n − C nk , suppose that the family T nk is defined; supposealso that the families C nj and T nj are defined for all j < k and that each exhibits the coherence andtrivialization features, respectively, described above. We then define C nk by letting ς ~f~α = ϕ ~α ~f + ( − n − k +1 k − X i =0 ( − i τ ~f i ~α (5)for each ~f ∈ X k and ~α ∈ A n − k . Observe that equation (4) identifies naturally with the case of k = 1. Claim 4.6. For each ~f ∈ X k the family C ~fn − k = h ς ~f~α | ~α ∈ A n − k i is ( n − k ) -coherent.Proof. The more formal statement of the claim is that for each ~f ∈ X k and ~α ∈ A n − k +1 , n − k X i =0 ( − i ς ~f~α i = ∗ . IMULTANEOUSLY VANISHING HIGHER DERIVED LIMITS WITHOUT LARGE CARDINALS 15 This is computationally verified as follows: n − k X i =0 ( − i ς ~f~α i = n − k X i =0 ( − i ϕ ~α i ~f + ( − n − k +1 n − k X i =0 ( − i k − X j =0 ( − j τ ~f j ~α i = ∗ n − k X i =0 ( − i ϕ ~α i ~f + ( − n − k +1 k − X j =0 ( − j ς ~f j ~α = ∗ n − k X i =0 ( − i ϕ ~α i ~f + ( − n − k +1 (cid:18) k − X j =0 ( − j (cid:16) ϕ ~α ~f j + ( − n − k k − X ℓ =0 ( − ℓ τ ( ~f j ) ℓ ~α (cid:17)(cid:19) = ∗ n X i =0 ( − i ϕ ( ~α ~f ) i + ( − n − k +1 (cid:18) k − X j =0 ( − j + n − k k − X ℓ =0 ( − ℓ τ ( ~f j ) ℓ ~α (cid:19) = ∗ − (cid:18) k − X j =0 ( − j k − X ℓ =0 ( − ℓ τ ( ~f j ) ℓ ~α (cid:19) = ∗ − (cid:18) X j ≤ ℓ ≤ k − ( − j + ℓ τ ( ~f j ) ℓ ~α + X ℓ Lemma 4.7. Fix n > and cardinals κ ≤ χ and let P = Add( ω, χ ) . The following then holds in V P : suppose that • Φ = h ϕ ~f | ~f ∈ X n i is an n -coherent family of functions; • A ⊆ X ⊆ ω ω contains at least κ -many of the Cohen reals added by P ; • Φ ↾ A is trivial; • Any j -coherent family Ψ = h ψ ~f | ~f ∈ Y j i in which ≤ j < n and Y contains at least κ -many of the Cohen reals added by P is trivial.Then Φ is trivial as well. Note that, by Proposition 4.2, the above lemma applies also to n -coherent families below anyfixed g ∈ ω ω . Defining trivializations To apply Lemma 4.7, we must first show in the appropriate models V P that n -coherent familiesΦ indexed by large numbers of Cohen reals always admit trivial restrictions Φ ↾ A to index-sets A which are large in the settings of lower dimensions. This we argue by defining type II (i.e., finitelysupported) trivializations of Φ ↾ A . These definitions require variations on the machinery of [6];describing this machinery is the object of this section. Already in [6], however, this apparatus takeson a certain opacity; as this is, if anything, even more the case for the variations listed here, weconclude this section with several heuristic remarks. Definition 5.1. Suppose that b is a finite set of ordinals. A subset-final segment of b of length m is a sequence ~a = h a i | ≤ i ≤ m i such that • m ≤ | b | , • a ⊆ · · · ⊆ a m = b , and • | a i | = | a | + i − i with 1 ≤ i ≤ m .If ~a is a subset-final segment of b and | a | = 1, then we say that ~a is a long string or a long stringfor b . Notice that in this case m = | b | . If ~a is not long, then it is short .Suppose now that X is a set of ordinals and we are working with an injective sequence h f α | α ∈ X i of elements of ω ω . (In the present context, this will always be a sequence of Cohen reals, but thatis not important for the results in this section.) For each nonempty ~α = h α k | k < n i in X <ω , let I ( ~α ) denote T k 1, then let d ~ε~a = a ∪ { ε a k | ≤ k ≤ m } . Note that, in this case, d ~ε~a ∈ [ X ] | b | +1 .For ~α ∈ X <ω of length at least two, let e ~ Φ ( ~α ) = X i< | ~α | ( − i ϕ ~α i . When the family ~ Φ is clear from context, we will omit it from the superscript; similarly for thesuperscript of d ~ε~a . Recall also our habit of viewing finite sets a ∈ [ X ] <ω as sequences enumeratedin increasing order; expressions like e ( a ) should be interpreted on this principle. Since each Φ n is n -coherent, e ( ~α ) is finitely supported for each ~α ∈ X <ω . Let e ( ~α ) denote the restriction of e ( ~α ) toits support.We will be interested in linear combinations L of the form X i<ℓ c i e ( ~α i ) , IMULTANEOUSLY VANISHING HIGHER DERIVED LIMITS WITHOUT LARGE CARDINALS 17 where ℓ < ω , each c i is an integer, and each ~α i is an element of X <ω of length at least two. Givensuch a linear combination L and an ordinal ε ∈ X , we let the expression L ∗ ε denote X i<ℓ c i e ( ~α i⌢ h ε i ) . For integers n ≥ 2, we now define interrelated • linear combinations A ~ Φ n ( a ), parametrized by a ∈ [ X ] n , and • linear combinations C ~ Φ n ( b ), parametrized by b ∈ [ X ] n +1 .We again omit the superscripts ~ Φ and ~ε and restriction-notations whenever they are contextuallyclear; as elsewhere, sums of functions in expressions like C n ( b ) below should always be understoodto be taken on the intersection of those functions’ domains.We begin our definitions by letting A ( a ) = e ( a ⌢ h ε a i ) . for each a ∈ [ X ] . Next, suppose that 2 ≤ n < ω and A n ( a ) has been defined for all a ∈ [ X ] n . Given b ∈ [ X ] n +1 , let C n ( b ) = e ( b ) − n X i =0 ( − i A n ( b i ) , and(6) A n +1 ( b ) = ( − n +1 C n ( b ) ∗ ε b . (7)The following lemma is easily verified by induction on n , so its proof is left to the reader. Lemma 5.2. For all b ∈ [ X ] , letting ~a = h b i , we have A ( b ) = e ( d ~a ) .For all n with ≤ n < ω and all b ∈ [ X ] n +1 , we have:(1) C n ( b ) is of the form e ( b ) + X i<ℓ c i e ( d ~a i ) , where ℓ < ω and, for each i < ℓ , c i is an integer and ~a i is a short subset-final segment ofsome element of [ b ] n .(2) A n +1 ( b ) is of the form X i<ℓ c i e ( d ~a i ) , where ℓ < ω and, for each i < ℓ , c i is an integer and ~a i is a short subset-final segment of b . One consequence of this lemma is that, while the expressions defining A n ( a ) and C n ( b ) are con-structed via a recursion involving ~ Φ, the actual values of A n and C n are only dependent on Φ n , sowe can meaningfully speak of them in situations in which we have only Φ n , and not Φ m for any m = n , before us. In addition, the values of A n ( a ) and C n ( b ) are only dependent on ordinals ε a fornonempty c ⊆ a or c ⊆ b , respectively, so, when working just with the expressions A n ( a ) or C n ( b ),we need not require that ε c is defined for any c that is not a subset of a or b , respectively. Finally,if b ∈ [ X ] n +1 and ( j, k ) ∈ ω × ω is an element of I ( f ε a ) for all nonempty a ⊆ b , then ( j, k ) is in thedomain of C n ( b ).The following is a consequence of [6, Lemma 6.4]. In settings like these, notational choices are simply of the lesser evil. We will write ~α i ( j ) for the j th element of ~α i . Lemma 5.3. Suppose that ≤ n < ω , b ∈ [ X ] n +1 , ( j, k ) ∈ ω × ω , and the following two statementshold. • There exists a single integer w such that e ( d ~a )( j, k ) = w for every long string ~a for b . • ( j, k ) ∈ I ( f ε a ) for all nonempty a ⊆ b .Then C n ( b )( j, k ) = 0 . Remark 5.4. We briefly describe how Lemma 5.3 follows from the argument of [6, Lemma 6.4].The set b here corresponds to τ in that result. The expression S n ( τ ) in [6] is an auxiliary expressionthat always equals 0. The two assumptions in Lemma 5.3 play the role of the statement u n ( τ ) from[6].As we will see, the significance of these definitions is the following: the expressions A n ( a ) willcorrespond to the elements of candidate type II trivializations of Φ ↾ A . Under the conditions ofLemma 5.3, the expressions C n ( b ) amount to verifications that these families of expressions A n ( a )do indeed trivialize Φ ↾ A . These relations, together with the necessity of working coordinatewise,as in Lemma 5.3, are points we expand on in the following remark. Though not strictly needed forthe continuation of our argument, it is hoped that it may be clarifying. Remark 5.5. The variability in | d ~a | noted above, depending on whether | a | = 1, underscores theunique status of the sets d ~a which are indexed by long strings; it is on the cancellations betweentheir associated terms e ( d ~a ) that the desired relations between the other terms in C n ( b ) depend.This we hope to illuminate by the following diagram and discussion.0 1010 202 12 122012 10120 01202 012 1201201012 w ww ww w Figure 1. The subdivision organizing the n = 2 case.The n -tuples ~f structuring our various coherent families Φ are naturally viewed as simplices, with( n − ~f i ( i < n ) as faces. Within this view, strings of increasing subsets · · · ( a i ( · · · of b = ~f correspond to simplices in the barycentric subdivision of the simplex b (see, e.g., [16, Chapter3.3]). For example, the white inner region of Figure 1 corresponds to the barycentric subdivisionof the triangle with vertices 0, 1, and 2 (we bend that region’s edges for reasons soon to be madeclear): the 2-faces of that subdivided region are { , , } , { , , } , etc., each correspondingto inclusion-increasing sequences of nonempty subsets of { , , } .Fix now a 2-coherent family of functions Φ. Let a = { α , α , α } ∈ [ X ] and suppose thatthe hypotheses of Lemma 5.3 hold. For readability, we let, for example, ε , denote ε α ,α (in IMULTANEOUSLY VANISHING HIGHER DERIVED LIMITS WITHOUT LARGE CARDINALS 19 particular, ε ℓ = α ℓ for ℓ < ϕ , for the function indexed by ( f ε , f ε ), and so on, whatare wanted are finitely supported functions ψ , , and so on, whose differences reproduce those amongthe corresponding functions of Φ, as described in Proposition 2.13. The idea of the above machineryis to derive these finitely supported functions from the coherence of Φ itself, as the differencesbetween carefully chosen families of functions ϕ ; at the stage n = 2, for example, we will have ψ , := A ( ε , ε ) = ϕ , − ϕ , + ϕ , . (8)Visually, this definition corresponds to the grey triangle at the base of Figure 1, under the nat-ural association of the functions ϕ , , ϕ , , and ϕ , with the edges { , } , { , } , and { , } ,respectively. Under this correspondence, the desired relation ϕ , − ϕ , + ϕ , = ψ , − ψ , + ψ , (9)may be viewed as asserting the equality of the oriented sum of the functions ϕ associated to theboundary of the triangle { , , } with that of the functions ϕ associated to the boundary of the greyregion. (As should be clear, this is a deliberately schematic discussion; we return to the questionof the argument of these functions below.) This holds precisely because of the first bulleted “longstring” condition listed in Lemma 5.3, which amounts in the present context to the boundary sumsassociated to the triangles { , , } , { , , } , etc., all equaling w . As these are oriented sums,they entail cancellations, so that first, the boundary sum associated to { , , } may be identifiedwith that associated to { , , } , and second, such identifications for each of the grey trianglescancel inside the triangle { , , } , leaving nothing summed but its boundary, just as equation 9requires.At the arithmetic level, all of this manifests (with only minor notational adjustments) as exactlythe two types of cancellations in the summed equations concluding Section 6 of [6]. The ensuingsimplification of that sum is an instance of what Lemma 5.3 records as C n ( b ) = 0, which translates,in turn, to the equation 9 we had desired.The subdivision perspective sketched above is valuable for returning sense to what appear here orin [6] as rather opaque and complicated algebraic identities: the meaning of those identities is that,in forcing extensions, higher-dimensional ∆-systems can determine trivializing structures within n -coherent families by uniformizing the boundary sums associated to the n -faces of the barycentricsubdivision of any ( n + 1)-tuple of indices, viewed as a simplex. This perspective clarifies the passagefrom one dimension to the next, as well; as the interested reader may verify, in the n = 3 case, alongwith with the face { , , } , the 2-faces pictured in Figure 1 play within a tetrahedron exactly therole that the boundaries of the grey faces had played within a triangle in the case of n = 2. Putdifferently, the way verification-expressions C n figure in the trivializing expressions A n +1 of the nextlevel, as in equation 7, amounts to little other than the fact that the restriction of the barycentricsubdivision of an ( n + 1)-simplex to any n -face is a barycentric subdivision of that face.Complicating the above considerations, however, is the issue of domain: for the right-hand sideof equations like (9) to truly be trivializing in the sense of Proposition 2.13, the domain of ψ , , likethat of ϕ , , must be I ( f ε ∧ f ε ), and similarly for the functions ψ , and ψ , . If ψ , is definedas in equation 8, then this amounts to a requirement that f ε ≥ f ε ∧ f ε , which, if the functionsindexed are Cohen reals, can never be the case. This is a requirement we can only meet locally,choosing for each ( j, k ) ∈ I ( f ε ∧ f ε ) an ε j,k such that ( j, k ) ∈ I ( f ε j,k ). This is the approach wetake, and this is the meaning of the parameter ( j, k ) appearing in Lemma 5.3. The good news inthis approach is that equations like (9) hold if and only if they hold coordinatewise, so that ourarithmetic is essentially unaffected. The bad news is that functions like ψ , may now fail to be finitely supported, and much of the work of the following section is towards ensuring that they willbe. 6. The cases of n > λ n appearing therein are defined by recursion on n ≥ λ = i +1 ; then, for all n > 1, let λ n = σ ( λ + n − , n + 1)(again see Definition 2.4 for the notation σ ( · , · )). Note that sup { λ n | n < ω } = i ω . It is alsoreadily verified that each λ n is < ℵ -inaccessible; in consequence, since λ n is a successor cardinal, λ + n is also < ℵ -inaccessible. Theorem 6.1. Let n be a positive integer, let χ ≥ λ n be a cardinal, and let P = Add( ω, χ ) . Thefollowing then holds in V P : For any set X ⊆ ω ω containing at least λ n -many of the Cohen realsadded by P , every n -coherent family Φ = h ϕ ~f | ~f ∈ X n i indexed by X is trivial.Proof. The proof is by induction on n . The case n = 1 was that of Theorem 3.1. Therefore fix an n > m < n . Also fix a cardinal χ ≥ λ n , P -names˙ X and ˙Φ = h ˙ ϕ ˙ ~f | ˙ ~f ∈ ˙ X n i , and a condition p ∈ P such that • p (cid:13) “ |{ α < χ | ˙ f α ∈ ˙ X }| ≥ λ n ”, and • p (cid:13) “ ˙Φ is an n -coherent family”.We will find a q ≤ p and a P -name ˙ A such that q forces the following statements: • | ˙ A | ≥ λ n − ; • { ˙ f α | α ∈ ˙ A } ⊆ ˙ X ; • ˙Φ ↾ ˙ A is trivial.It will be clear from our argument below that each induction step of our proof conserves the hy-potheses of Lemma 4.7. That lemma will therefore apply to show that q in fact forces that ˙Φ istrivial; this will conclude the induction step of the proof and, therefore, the proof itself.As before, begin by letting Y be the set of α < χ for which there is a condition p α ≤ p suchthat p α (cid:13) “ ˙ f α ∈ ˙ X ”; observe that | Y | ≥ λ n by assumption. For each α ∈ Y , fix such a condition p α . Since P is λ n -Knaster, there exists a set Y ′ ⊆ Y of size λ n such that { p α | α ∈ Y ′ } consists ofpairwise compatible conditions. Note that, for all a ∈ [ Y ′ ] <ω , we have S α ∈ a p α ∈ P .Given a P -name ˙ ~h = h ˙ h , . . . , ˙ h n i for an element of ( ω ω ) n +1 , let ˙ e ( ˙ ~h ) be a P -name that is forcedto be equal to n X i =0 ( − i ϕ ~h i if ˙ ~h ∈ ˙ X n +1 and is forced to be 0 otherwise. Since p forces that Φ is n -coherent, any extension of p will force that ˙ e ( ˙ ~h ) is a finitely-supported function from a subset of ω × ω into Z . Let ˙ e ( ˙ ~h ) be a P -name for the restriction of ˙ e ( ˙ ~h ) to its support. For all a ∈ [ Y ′ ] n +1 , let ˙ e ( a ) denote ˙ e ( h ˙ f α | α ∈ a i ).For each a ∈ [ Y ′ ] n +1 let h q a,ℓ | ℓ < ω i enumerate a maximal antichain A a of conditions in P below S α ∈ a p α such that each q a,ℓ decides the value of ˙ e ( a ) to be equal to some finite partial function e a,ℓ ∈ V . Recall that, for p ∈ P , u ( p ) is the set { α < χ | dom( p ) ∩ ( { α } × ω ) = ∅} . For readability,let u ( a, ℓ ) denote u ( q a,ℓ ). IMULTANEOUSLY VANISHING HIGHER DERIVED LIMITS WITHOUT LARGE CARDINALS 21 For each b ∈ [ Y ′ ] n +1 let v b = S { u ( a, ℓ ) | a ∈ [ b ] n +1 , ℓ < ω } . Define a “coding” function F : [ Y ′ ] n +1 → H ( ω ) as follows. First, for each b ∈ [ Y ′ ] n +1 , each m ∈ [2 n + 1] n +1 and each ℓ < ω ,let w b m ,ℓ = { η < otp( v b ) | v b ( η ) ∈ u ( b [ m ] , ℓ ) } . (Note that w b m ,ℓ ∈ [otp( v b )] <ω ). Then, for each b ∈ [ Y ′ ] n +1 , let F ( b ) = h (¯ q b [ m ] ,ℓ , w b m ,ℓ , e b [ m ] ,ℓ ) | m ∈ [2 n + 1] n +1 , ℓ < ω i . Recall that λ n = σ ( λ + n − , n + 1) and λ + n − is < ℵ -inaccessible. Therefore, by Fact 2.5, there exists H ∈ [ Y ′ ] λ + n − such that • F is constant on [ H ] n +1 , taking value h (¯ q m ,ℓ , w m ,ℓ , e m ,ℓ ) | m ∈ [2 n + 1] n +1 , ℓ < ω i , and • h v b | b ∈ [ H ] n +1 i is a uniform (2 n + 1)-dimensional ∆-system.By taking an initial segment of H if necessary, we can assume that otp( H ) = λ + n − . Let ρ and h r m | m ⊆ n + 1 i witness that h v b | b ∈ [ H ] n +1 i is a uniform (2 n + 1)-dimensional ∆-system.Let h v a | a ∈ [ H ] < n +1 i be given by Lemma 2.3 applied to h v b | b ∈ [ H ] n +1 i . We will actuallyneed slightly more than what Lemma 2.3 gives us. Given a ∈ [ H ] n , k ≤ n , and α ∈ H , we saythat α is k -addable for a if α / ∈ a and | a ∩ α | = k . In other words, α is k -addable to a if, letting a ′ = a ∪ { α } , we have | a ′ | = n + 1 and a ′ ( k ) = α . Given an a ∈ [ H ] n and a k ≤ n such that thereis at least one α ∈ H that is k -addable for a , define v a,k as follows. Let α ∈ H be such that α isaddable for a , let b ∈ [ H ] n +1 be such that b [ n + 1] = a ∪ { α } , and let v a,k = v b [ r ( n +1) \{ k } ]. Claim 6.2. For each a ∈ [ H ] n and k ≤ n for which v a,k is defined, the value of v a,k is independentof our choice of α and b .Proof. Suppose that α, α ′ ∈ H are both k -addable for a and b, b ′ ∈ [ H ] n +1 are such that b [ n + 1] = a ∪ { α } and b [ n + 1] = a ∪ { α ′ } . We will show that v b [ r ( n +1) \{ k } ] = v b ′ [ r ( n +1) \{ k } ].First, fix c ∈ [ H ] n such that min( c ) > max( b ∪ b ′ ), let d = a ∪ { α } ∪ c , and let d ′ = a ∪ { α ′ } ∪ c .Then b and d are aligned, with r ( b, d ) = n + 1, and b ′ and d ′ are aligned, with r ( b ′ , d ′ ) = n + 1. Also, d and d ′ are aligned, with either r ( d, d ′ ) = n + 1 (if α = α ′ ) or r ( d, d ′ ) = ( n + 1) \ { k } (if α = α ′ ).Altogether, it follows that v b [ r ( n +1) \{ k } ] = v d [ r ( n +1) \{ k } ] = v d ′ [ r ( n +1) \{ k } ] = v b ′ [ r ( n +1) \{ k } ] , as desired. (cid:3) Claim 6.3. Suppose that a ∈ [ H ] n and k ≤ n are such that v a,k is defined. Then the collection { v a ∪{ α } | α ∈ H is addable for a } is a (1-dimensional) ∆ -system, with root v a,k .Proof. Fix α < α ′ in H such that both α and α ′ are k -addable for a . Fix c ∈ [ H ] n such thatmin( c ) > max( a ∪ { α, α ′ } ). Let b = a ∪ { α } ∪ c and b ′ = a ∪ { α ′ } ∪ c . Then v a ∪{ α } = v b [ r n +1 ], v a ∪{ α ′ } = v b ′ [ r n +1 ], and v a,k = v b [( n + 1) \ { k } ] = v b ′ [( n + 1) \ { k } ]. Moreover, b and b ′ are alignedwith r ( b, b ′ ) = (2 n + 1) \ { k } , so v b ∩ v b ′ = v b [(2 n + 1) \ { k } ] = v b ′ [(2 n + 1) \ { k } ]. Altogether, thisimplies v a ∪{ α } ∩ v a ∪{ α ′ } = v b [ r n +1 ] ∩ v b ′ [ r n +1 ]= v b [ r n +1 ] ∩ v b ′ [ r n +1 ] ∩ v b [(2 n + 1) \ { k } ] ∩ v b ′ [(2 n + 1) \ { k } ]= v b [ r ( n +1) \{ k } ] ∩ v b ′ [ r ( n +1) \{ k } ]= v a,k ∩ v a,k = v a,k . Therefore, { v a ∪{ α } | α ∈ H is addable for a } is a ∆-system with root v a,k . (cid:3) Claim 6.4. Let a ∈ [ H ] n , k ≤ n , and ℓ < ω , and suppose that α, α ′ ∈ H are both k -addable for a .Then q a ∪{ α } ↾ ( v a,k × ω ) = q a ∪{ α ′ } ↾ ( v a,k × ω ) .Proof. We can assume that α = α ′ , as otherwise the claim is trivial. Fix c ∈ [ H ] n with min( c ) > max( a ∪ { α, α ′ } ), and let b = a ∪ { α } ∪ c and b ′ = a ∪ { α ′ } ∪ c . By Claim 6.2, we have v a,k = v b [ r ( n +1) \{ k } ] = v b ′ [ r ( n +1) \{ k } ].Now suppose that ( γ, j ) ∈ dom( q a ∪{ α } ) ∩ ( v a,k × ω ). Then there is η ∈ r ( n +1) \{ k } such that γ = v b ( η ); moreover, η ∈ w n +1 ,ℓ and, since b and b ′ are aligned with r ( b, b ′ ) ⊇ ( n + 1) \ { k } , we have γ = v b ′ ( η ). Let i < ω be such that η = w n +1 ,ℓ ( i ). Then, since ¯ q a ∪{ α } ,ℓ = ¯ q a ∪{ α ′ } ,ℓ = ¯ q n +1 ,ℓ , we have( γ, j ) ∈ dom( q a ∪{ α ′ } ) and q a ∪{ α ′ } ( γ, j ) = ¯ q n +1 ,ℓ ( i, j ) = q a ∪{ α } ( γ, j ) . A symmetric argument shows that, if ( γ, j ) ∈ dom( q a ∪{ α ′ } ) ∩ ( v a,k × ω ), then ( γ, j ) ∈ dom( q a ∪{ α } )and q a ∪{ α } ( γ, j ) = q a ∪{ α ′ } ( γ, j ). It follows that q a ∪{ α } ↾ ( v a,k × ω ) = q a ∪{ α ′ } ↾ ( v a,k × ω ). (cid:3) We next note that the values of ¯ q m ,ℓ and e m ,ℓ are independent of m ∈ [2 n + 1] n +1 . Indeed,suppose that b ∈ [ H ] n +1 , m ∈ [2 n + 1] n +1 , and a = b [ m ]. Then we can find a b ∗ ∈ [ H ] n +1 forwhich a = b ∗ [ n + 1] (i.e., a is an initial segment of b ∗ ). But then¯ q m ,ℓ = ¯ q b [ m ] ,ℓ = ¯ q a,ℓ = ¯ q b ∗ [ n +1] ,ℓ = ¯ q n +1 ,ℓ , and similarly for e m ,ℓ . Hence we may in fact fix a sequence h (¯ q ℓ , e ℓ ) | ℓ < ω i such that (¯ q m ,ℓ , e m ,ℓ ) =(¯ q ℓ , e ℓ ) for all m ∈ [2 n ] n +1 .Now let H , H , . . . , H n be subsets of H such that otp( H k ) = λ n − and H k < H k ′ for all k 1. First, if | a | = 1 (and hence if | b | = m ), then let d j,k~a = { ε j,ka i | ≤ i ≤ m } ∪ { δ i | m ≤ i ≤ n } . Notice that in this case | d j,k~a ∩ H i | = 1 for all i ≤ n .Next, if | a | > | b | ≤ n , then let d j,k~a = a ∪ { ε j,ka i | ≤ i ≤ m } ∪ { δ i | | a | + m − ≤ i < n } . Notice that in this case | d j,k~a ∩ H | = | a | , d j,k~a ∩ H i = ∅ for 0 < i < | a | − i = n , and | d j,k~a ∩ H i | = 1 for | a | − ≤ i < n . If | a | > | b | = n + 1, then leave d j,k~a undefined. In anycase, if d j,k~a is defined, then d j,k~a ∈ [ H ] n +1 .We may now state our final requirement for the ordinals ε j,ka :(4) For every nonempty b ∈ [ A ] ≤ n +1 , every subset-final segment ~a = h a i | ≤ i ≤ m i of b , andevery ( j, k ) ∈ I ( b ) >j ∗ ,(a) if | a | = 1, then q d j,k~a , ∈ G ;(b) if | a | > | b | ≤ n and j > j b , then q d j,k~a ,ℓ a ∈ G .A main resource for the construction of the ordinals ε j,ka will be Claim 6.6 below. To facilitate itsstatement, we introduce the following terminology and convention: if c ∈ [ H ] n +1 , i ≤ n , and α ∈ H ,then we say that α is i -possible for c if the following two statements hold: • if i > 0, then α > c ( i − • if i < n , then α < c ( i + 1).Intuitively, α is i -possible for c if c ( i ) can be replaced by α without changing the positions of theother elements of c within the set. If α is i -possible for c , then c [ i α ] denotes this replacement,i.e., it denotes the set ( c \ { c ( i ) } ) ∪ { α } . Claim 6.6. Suppose that c , c ∈ [ H ] n +1 , i , i ≤ n , and c ( i ) = c ( i ) . Suppose also that α ∈ H is i -possible for c and i -possible for c and ℓ , ℓ < ω are such that q c ,ℓ and q c ,ℓ are compatiblein P . Then q c [ i α ] ,ℓ and q c [ i α ] ,ℓ are also compatible in P .Proof. If c ( i ) = α , then there is nothing to prove, so assume that c ( i ) = α . Let c = c ∪ c .Since c and c share at least one element, we know that | c | ≤ n + 1. Let b ∈ [ H ] n +1 be a (possiblytrivial) end-extension of c such that every element of b \ c is greater than α . Let i ∗ ≤ n be such that b ( i ∗ ) = c ( i ) = c ( i ), and let b ∗ = ( b \ { c ( i ) } ) ∪ { α } . Since α is i -possible for c and i -possiblefor c , and since all elements of b \ c are greater than α , we know that b and b ∗ are aligned and r ( b, b ∗ ) = (2 n + 1) \ { i ∗ } =: m . Let m , m ∈ [2 n + 1] n +1 be such that c = b [ m ] and c = b [ m ],and hence such that c [ i α ] = b ∗ [ m ] and c [ i α ] = b ∗ [ m ].Suppose for the sake of contradiction that q c [ i α ] ,ℓ and q c [ i α ] ,ℓ are incompatible in P . Thenthere is a ( γ ∗ , j ) in the intersection of their domains such that q c [ i α ] ,ℓ ( γ ∗ , j ) = q c [ i α ] ,ℓ ( γ ∗ , j ) . Suppose that η < ρ is such that γ ∗ = u b ∗ ( η ). Let γ = u b ( η ). Since F ( b ∗ ) = F ( b ), and in particularsince w b m ,ℓ = w b ∗ m ,ℓ , w b m ,ℓ = w b ∗ m ,ℓ , ¯ q b [ m ] ,ℓ = ¯ q b ∗ [ m ] ,ℓ , and ¯ q b [ m ] ,ℓ = ¯ q b ∗ [ m ] ,ℓ , we knowthat ( γ, j ) is in the domain of both q c ,ℓ and q c ,ℓ and also that q c ,ℓ ( γ, j ) = q c [ i α ] ,ℓ ( γ ∗ , j )and q c ,ℓ ( γ, j ) = q c [ i α ] ,ℓ ( γ ∗ , j ). It follows that q c ,ℓ ( γ, j ) = q c ,ℓ ( γ, j ), but this contradicts ourassumption that q c ,ℓ and q c ,ℓ are compatible in P . (cid:3) We turn now more directly to the construction of the family of ordinals h ε j,ka | a ∈ [ A ] ≤ n +1 is nonempty and ( j, k ) ∈ I ( a ) >j ∗ i satisfying the requirements (1)–(4) listed above. The construction is by recursion on | a | . If α ∈ A and ( j, k ) ∈ I ( f α ) >j ∗ , then condition (2) dictates that ε j,k { α } = α . Conditions (1)–(3) are then triviallysatisfied. To see condition (4), observe that the only subset-final segment of { α } for any α ∈ A is ~a = h{ α }i ; by the definition of A , we then have q d j,k~a , = q d α , ∈ G , just as required by condition(4a). This concludes the cases in which | a | = 1.Next suppose that b ∈ [ A ] ≤ n and | b | ≥ 2; fix ( j, k ) ∈ I ( b ) >j ∗ and suppose also that we have defined ε j,ka for all nonempty a ( b . We will define an ε j,kb ∈ H | b |− . Suppose that ~a = h a i | ≤ i ≤ m i isa subset-final segment of b . If m = 1, then define d − ~a to be d b (see again the third paragraph afterClaim 6.5 for the definition of d b ). If m > 1, then notice that ~a − := h a i | ≤ i < m i is a subset-finalsegment of a m − , and define d − ~a to be d j,k~a − . Observe that q d − ~a ,ℓ a ∈ G in either case: in the m = 1case, this follows from the definition of ℓ a = ℓ b . In the m > a m − . If ~a is a long string (i.e., if | a | = 1),then let i ~a = | b − | , and if ~a is a short string (i.e., if | a | > i ~a = | b | . Notice that, in anycase, we have d − ~a ( i ~a ) = δ | b |− and, once we have defined ε j,kb , we will have d j,k~a = d − ~a [ i ~a ε j,kb ].Now, to see that we can find an ordinal ε j,kb satisfying conditions (1)–(4), move back to V and fixan arbitrary r ∈ P extending q ∗ := [ { q d − ~a ,ℓ a | ~a = h a i | i ≤ m i is a subset-final segment of b } , IMULTANEOUSLY VANISHING HIGHER DERIVED LIMITS WITHOUT LARGE CARDINALS 25 which we know to be in G . We will find a condition s ≤ r and an ordinal ε such that s forces ε to be a valid choice for ε j,kb . By the preceding paragraph and Claim 6.3, we know that for everysubset-final segment ~a of b , the collection { v d − ~a [( i ~a ) ε ] | ε ∈ H | b |− } is an infinite (1-dimensional)∆-system with root v d − ~a \{ δ | b |− } ,i ~a . Therefore we can fix an ε ∈ H | b |− such that ε / ∈ u ( r ) and suchthat v d − ~a [ i ~a ε ] \ v d − ~a \{ δ | b |− } ,i ~a is disjoint from u ( r ) for every subset-final segment ~a of b . For eachsuch ~a , Claim 6.4 implies that q d − ~a [ i ~a ε ] ,ℓ a ↾ ( v d − ~a \{ δ | b |− } ,i ~a × ω ) = q d − ~a ,ℓ a ↾ ( v d − ~a \{ δ | b |− } ,i ~a × ω ) , and we know that r extends q d − ~a ,ℓ a . Therefore, q d − ~a [ i ~a ε ] ,ℓ a is compatible with r . Moreover, for allpairs ~a and ~a ∗ of subset-final segments of b , since q d − ~a ,ℓ a and q d − ~a ∗ ,ℓ a ∗ are both in G and are thereforecompatible, Claim 6.6 implies that q d − ~a [ i ~a ε ] ,ℓ a and q d − ~a ∗ [ i ~a ∗ ε ] ,ℓ a ∗ are compatible.We now split into two cases. Suppose first that j ≤ j b , so that we need to satisfy requirement(4a) but not (4b). Let s = r ∪ [ { q d − ~a [ i ~a ε ] ,ℓ a | ~a is a subset-final segment of b and | a | = 1 } . By the previous paragraph, s is a condition in P . Notice also that ℓ a = 0 for all such ~a in theabove union, by our inductive condition (4a). By the definition of j ∗ , the fact that j > j ∗ , and thefact that ε / ∈ u ( r ), we know that ( ε, j ) / ∈ dom( s ). Therefore we can extend s to a condition s such that ( ε, j ) ∈ dom( s ) and s ( ε, j ) ≥ k , i.e., s (cid:13) “( j, k ) ∈ I ( ˙ f ε )”. This s in fact forces that letting ε j,ka = ε satisfies requirements (1)–(4), as the reader may easily verify.If, on the other hand, j > j b , then we need to satisfy both the conditions (4a) and (4b). Let s = r ∪ [ { q d − ~a [ i ~a ε ] ,ℓ a | ~a is a subset-final segment of b } . As in the previous case, s is a condition in P . By the definition of j b , the fact that j > j b , and thefact that ε / ∈ u ( r ), we know that ( ε, j ) / ∈ dom( s ). Just as in the previous case, we can extend s toa condition s such that ( ε, j ) ∈ dom( s ) and s ( ε, j ) ≥ k . Also as in the previous case, this s forcesthat letting ε j,ka = ε satisfies requirements (1)–(4), as desired.By genericity, our analysis in V shows that we may choose in V [ G ] an ε j,kb satisfying requirements(1)–(4), and thereby continue with our construction.Finally, suppose that b ∈ [ A ] n +1 and fix ( j, k ) ∈ I ( b ) >j ∗ , and suppose that we have defined ε j,ka for all nonempty a ( b . We will define an ε j,kb ∈ H n ; this will be similar to the previous case, butwe no longer need to satisfy requirement (4b) and can therefore focus exclusively on long strings.Suppose that ~a = h a i | ≤ i ≤ n i is a long string for b . Set ~a − := h a i | ≤ i < n i and d − ~a := d j,k~a − ,and note that ~a − is a long string for a n − . As in the previous case, we have q d − ~a , ∈ G , d − ~a ( n ) = δ n and, once we have defined ε j,kb , we will have d j,k~a = d − ~a [ n ε j,kb ].To see that we can find an ordinal ε j,kb satisfying requirements (1)–(4), move back to V and fixan arbitrary r ∈ P extending q ∗ := [ { q d − ~a , | ~a is a long string for b } , which we know to be in G . We will find s ≤ r and an ordinal ε such that s forces ε to be a validchoice for ε j,kb . By the preceding paragraph and Claim 6.3, we know that, for every long string ~a for b , the collection { v d − ~a [ n ε ] | ε ∈ H n } is an infinite ∆-system with root v d − ~a \{ δ n } ,n . Therefore, we canfix an ε ∈ H n such that ε / ∈ u ( r ) and such that v d − ~a [ n ε ] \ v d − ~a \{ δ n } ,n is disjoint from u ( r ) for every long string ~a for b . For each such ~a , Claim 6.4 implies that q d − ~a [ n ε ] , ↾ ( v d − ~a \{ δ n } ,n × ω ) = q d − ~a , ↾ ( v d − ~a \{ δ n } ,n × ω ) , and we know that r extends q d − ~a , . Therefore, q d − ~a [ n ε ] , is compatible with r . Moreover, for all pairs ~a and ~a ∗ of long strings for b , since q d − ~a , and q d − ~a ∗ , are both in G and are therefore compatible,Claim 6.6 implies that q d − ~a [ n ε ] , and q d − ~a ∗ [ n ε ] , are compatible.Now let s = r ∪ [ { q d − ~a [ n ε ] , | ~a is a long string for b } . By the previous paragraph, s is a condition in P . By the definition of j ∗ , the fact that j > j ∗ , andthe fact that ε / ∈ u ( r ), we know that ( ε, j ) / ∈ dom( s ). Therefore we can extend s to a condition s such that ( ε, j ) ∈ dom( s ) and s ( ε, j ) ≥ k , i.e., s (cid:13) “( j, k ) ∈ I ( ˙ f ε )”. This s in fact forces that letting ε j,ka = ε satisfies requirements (1)–(4), as the reader may easily verify.Suppose now that the construction of the ordinals h ε j,ka | a ∈ [ A ] ≤ n +1 is nonempty and ( j, k ) ∈ I ( a ) >j ∗ i is completed. For all a ∈ [ A ] n and all ( j, k ) ∈ I ( a ) >j ∗ , let A j,kn ( a ) be defined as in Section 5, usingthe n -coherent family Φ and the ordinals h ε j,ka ′ | a ′ ∈ [ A ] ≤ n i . Similarly define C j,kn ( b ) for b ∈ [ A ] n +1 and ( j, k ) ∈ I ( b ) >j ∗ . For a ∈ [ A ] n , define a function ψ a : I ( a ) → Z as follows. If ( j, k ) ∈ I ( a ) and j ≤ j ∗ , then let ψ a ( j, k ) = 0. If ( j, k ) ∈ I ( a ) >j ∗ , then let ψ a ( j, k ) = A j,kn ( a )( j, k ). Define ψ ~a fornon-increasing ~α ∈ A n in the unique way that renders Ψ = h ψ ~α | ~α ∈ A n i an alternating family. Weclaim that Ψ together with the natural number j ∗ witnesses the triviality of Φ ↾ A in the sense ofFact 2.13.We first show that each ψ a is finitely supported. To see this, fix an arbitrary a ∈ [ A ] n . For eachnonempty a ′ ⊆ a , we have a finite partial function e ℓ a ′ such that, if ~a is a subset-final segment of a with a = a ′ , then, for all ( j, k ) ∈ I ( a ) >j ∗ , if q d j,k~a ,ℓ a ′ ∈ G , then e ( d j,k~a ) = e ℓ a ′ . Fix a natural number j ∗ a ≥ j a such that, for all nonempty a ′ ⊆ a , we have dom( e ℓ a ′ ) ⊆ ( j ∗ a × ω ).We claim that ψ a ( j, k ) = 0 for all ( j, k ) ∈ I ( a ) >j ∗ a . To see this, fix such a pair ( j, k ). By thedefinition of ψ a ( j, k ), we know that ψ a ( j, k ) = A j,kn ( a )( j, k ). By Lemma 5.2, we know that A j,kn ( a )is of the form X i<ℓ c i e ( d j,k~a i ) , where ℓ < ω and each c i is an integer and each ~a i is a subset-final segment of a with | a i (1) | > ~a i (1) denotes the first element of ~a i ). Moreover, since j > j ∗ a ≥ j a , we know bycondition (4b) that q d j,k~ai ,ℓ ~ai (1) ∈ G , and hence that e ( d j,k~a i ) = e ℓ ~ai (1) , for all i < ℓ . In particular,dom( e ( d j,k~a i )) ⊆ ( j ∗ a × ω ), so e ( d j,k~a i )( j, k ) = 0. It follows that ψ a ( j, k ) = 0. In consequence, thesupport of ψ a is a subset of I ( a ) ∩ (( j ∗ a + 1) × ω ), which is a finite set.It now only remains to be shown that for all ~β ∈ A n +1 and all ( j, k ) ∈ I ( ~β ) >j ∗ , e ( ~β )( j, k ) = n X i =0 ( − i ψ ~β i ( j, k ) . Since Φ and Ψ are both alternating, it suffices to prove this for b ∈ [ A ] n +1 . Fix such a b anda coordinate-pair ( j, k ) ∈ I ( b ) >j ∗ . Notice that for every long string ~a for b , since ε j,k~a satisfiesrequirement (4a), we have q d j,k~a, ∈ G and hence e ( d j,k~a ) = e . Thus we have e ( d j,k~a )( j, k ) = e ( j, k ),where e : ω × ω → Z is the function whose restriction to its support is equal to e . Moreover, by IMULTANEOUSLY VANISHING HIGHER DERIVED LIMITS WITHOUT LARGE CARDINALS 27 the construction of ε j,ka for nonempty a ⊆ b and the assumption that ( j, k ) ∈ I ( b ) >j ∗ , we know that( j, k ) ∈ I ( f ε j,ka ) for all nonempty a ⊆ b . Therefore the hypotheses of Fact 5.3 hold, and consequently C j,kn ( b )( j, k ) = 0. By the definition of C j,kn ( b ), we then have0 = C j,kn ( b )( j, k ) = e ( b )( j, k ) − n X i =0 ( − i A j,kn ( b i )( j, k ) = e ( b )( j, k ) − n X i =0 ( − i ψ b i ( j, k ) , implying that e ( b )( j, k ) = n X i =0 ( − i ψ b i ( j, k ) , as desired.It follows that, in V [ G ], the restricted family Φ ↾ A is trivial. By our inductive hypotheses,together with the fact that | A | = λ n − , Lemma 4.7 applies, and we may conclude that Φ is trivial;this concludes the proof. (cid:3) Clearly the theorem stated in our introduction is a special case of Theorem 6.1; observe alsothat assuming, for example, the generalized continuum hypothesis in our ground model V yields thecorollary recorded there as well. 7. Conclusion As noted in our introduction, this work fully answers the first question, and partially or potentiallyaddresses the second question, appearing in [6]. We restate the latter: Question 7.1. What is the minimum value of the continuum compatible with the statement “ lim n A =0 for all n > ”? By our Main Corollary, this question is tantamount to the following: Question 7.2. Does ℵ < ℵ ω imply that lim k A = 0 for some k > ? Answering this question will entail answering the following (a revision, in light of present knowl-edge, of one appearing in [14]): Question 7.3. Does ℵ ≤ ℵ imply that either lim A = 0 or lim A = 0 ? Of interest in its own right, but all the more so in light of Question 7.2, is: Question 7.4. What is the behavior of the groups lim n A in the standard forcing extensions inwhich ℵ = ℵ ? By [3] , of particular interest among them will be those models in which b < d ;prominent among these is the Miller model. The fundamental reason that 2 ℵ = ℵ n implies lim n A = 0 when n = 1 is that the answer to thefollowing question is yes when n = 1 as well. Question 7.5. Is it a ZFC theorem that any F ⊆ ω ω of < ∗ -ordertype ω n indexes a nontrivial n -coherent family? Recently, Veliˇckovi´c and Vignati [19] have obtained a positive answer to Question 7.5 in thepresence of additional cardinal arithmetic assumptions. In particular, they prove that if 2 ω k < ω k +1 for all 1 ≤ k < n , then every F ⊆ ω ω of < ∗ -ordertype ω n indexes a nontrivial n -coherent family.A main way of seeing that the answer to Question 7.5 is yes when n = 1 applies walks techniquesto transfer large portions of a nontrivial coherent family on ω to any F as above [2, pp. 96-98]. Question 7.5 is more generally in large part a question about the combinatorics of the ordinals ω n ( n ∈ ω ). Here our researches link up with those of [5] and [4] in ways we may take the occasion toclarify. A central focus of both those works is nontrivial n -coherent families of functions indexedby ordinals ξ ; much as in the present work, such functions represent nonzero elements of lim n of aninverse system C ( ξ, Z ), which is defined as follows: for any ordinal ξ and abelian group A let C ( ξ, A )denote the inverse system ( ⊕ α A, p αβ , ξ ) in which the maps p αβ : ⊕ β A → ⊕ α A are projections forall α ≤ β < ξ . Highly relevant for Question 7.5 are the following facts: • lim m C ( ω n , Z ) = 0 for all n ≥ m ≥ L , as shown in [5]. • There exists (in ZFC ) an abelian group A such that lim n C ( ω n , A ) = 0 for all n ≥ 0, asshown in [4].Against this background, one of the most central of questions is surely the following: Question 7.6. Is it a ZFC theorem that lim n C ( ω n , Z ) = 0 for all n ≥ ? Put differently, do thereexist height- ω n nontrivial n -coherent families of functions mapping to Z for all n > in any modelof the ZFC axioms? Broadly speaking, the argument of [4] is that the fundamental content of a main result from[13] is the existence of higher-dimensional variants of the walks apparatus first appearing in [17].It seems likely that the answer to Question 7.6 will depend on a better understanding of thesehigher-dimensional walks, particularly if that answer is yes. Question 7.7. How much of the classical machinery of walks extends to the n -dimensional walkson ω n of [4] ? Question 7.5 may be viewed as a special case of Question 7.7. The prominence of classicalcoherence phenomena in infinitary combinatorics, as well as the growing prominence of their higher-dimensional variants, is partly explained in [5] by their connections both to the ˇCech cohomologygroups of the ordinals and to the broader set-theoretic theme of incompactness. The project ofunderstanding higher-dimensional coherence will in part entail understanding its relation to centralincompactness principles like (cid:3) ( κ ). Question 7.8. What are the behaviors of n -dimensional walks on cardinals κ > ω n , particularlyunder assumptions like (cid:3) ( κ ) ? Complementary to the ZFC focus of Questions 5–7 above, in other words, are consistency ques-tions. As the possible behaviors of lim n C ( ω , A ) and lim n C ( ω , A ) are either understood or sub-sumed by previous questions, the following is among the most immediate: Question 7.9. Is it consistent with the ZFC axioms that lim C ( ω , A ) = 0 for all abelian groups A ? Most of the above may be framed as questions about the possible “spectra” of nontrivial multi-dimensional coherence phenomena, or equivalently, of nonvanishing lim n , either of A or of C ( − , − ).Bound up with these questions seems to be that of the relation of these inverse systems’ higherlimits to each other. Several other families of inverse systems’ higher limits seem to be implicatedin these behaviors as well; among the more obvious generalizations of the system A , for example,are those which replace its index-set ω ω with κ λ for arbitrary cardinals κ and λ . As it happens,the vanishing of these systems’ higher limits carries implications within the framework of Scholze’s condensed mathematics [15, 7]. If κ is infinite and λ is uncountable, then lim of the associatedsystem is nonzero. The systems in which λ = ω , on the other hand, are denoted A κ in [3]; there it IMULTANEOUSLY VANISHING HIGHER DERIVED LIMITS WITHOUT LARGE CARDINALS 29 is shown that lim A = 0 if and only if lim A κ = 0 for all κ ≥ ω . Whether this holds for higherlim n is an interesting question, as is the following: Question 7.10. Let κ be an uncountable cardinal. Is it consistent that lim n A κ = 0 for all n > ? A second generalization of the system A retains the order ω ω , but varies the groups which itindexes, as well as the homomorphisms connecting them. The work [1] isolates a class of suchsystems significant in strong homology computations; it then shows that arguments applied to A in[6] in fact apply to this broader class of systems. This carries the consequence that it is consistentwith the ZFC axioms that strong homology is additive on the category of locally compact separablemetric spaces; notably, however, these arguments require the existence of a weakly compact cardinal.Somewhat surprisingly, and in contrast to [6] and [1], there is no straightforward adaptation of thepresent work’s argument to this wider class, for the simple reason that the equivalence of type I andtype II triviality so essential to this paper’s argument no longer holds in that more general setting. Question 7.11. What is the consistency strength of the statement “strong homology is additive onthe category of locally compact separable metric spaces”? A last context in which these questions are likely interesting is in the presence of determinacyhypotheses. Relatedly, one might ask how “definable” a nontrivial n -coherent family of functionsindexed by ω ω (viewed as a set of real numbers) can be. When n = 1, such a family is necessarilynonanalytic [18]; the following question was communicated to the first author by Justin Tatch Moorein 2014. Question 7.12. Fix n > . Can a nontrivial n -coherent family of functions indexed by ω ω beanalytic? References 1. Nathaniel Bannister, Jeffrey Bergfalk, and Justin Tatch Moore, On the additivity of strong homology for locallycompact separable metric spaces , (2020), Preprint. https://arxiv.org/abs/2008.13089.2. M. 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Stevo Todorcevic, The first derived limit and compactly F σ sets , J. Math. Soc. Japan (1998), no. 4, 831–836.19. Boban Veliˇckovi´c and Alessandro Vignati, Nontriviality of higher derived limits , 2021, in preparation. Universit¨at Wien, Institut f¨ur Mathematik, Kurt G¨odel Research Center, Kolingasse 14-16, 1010Wien, Austria Email address : [email protected] Centro de Ciencas Matem´aticas, UNAM, A.P. 61-3, Xangari, Morelia, Michoac´an, 58089, M´exico Email address : [email protected] Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond,VA 23284, United States Email address ::