Injective stabilization of additive functors. III. Asymptotic stabilization of the tensor product
aa r X i v : . [ m a t h . R T ] S e p INJECTIVE STABILIZATION OF ADDITIVE FUNCTORS, III.ASYMPTOTIC STABILIZATION OF THE TENSOR PRODUCT
ALEX MARTSINKOVSKY AND JEREMY RUSSELL
Abstract.
The injective stabilization of the tensor product is subjected toan iterative procedure that utilizes its bifunctor property. The limit of thisprocedure, called the asymptotic stabilization of the tensor product, providesa homological counterpart of Buchweitz’s asymptotic construction of stablecohomology. The resulting connected sequence of functors is isomorphic toTriulzi’s J -completion of the Tor functor. A comparison map from Vogel ho-mology to the asymptotic stabilization of the tensor product is constructedand shown to be always epic. Contents
1. Stable cohomology 21.1. Vogel cohomology 21.2. Buchweitz cohomology 21.3. Mislin’s construction 32. Stable homology 32.1. Vogel homology 32.2. A homological analog of Mislin’s construction 42.3. Summary 43. The asymptotic stabilization of the tensor product 53.1. The first construction 53.2. The second construction 93.3. The third construction 124. Two lemmas on connecting homomorphisms 154.1. Front, bottom, and right-hand faces 154.2. Top, back, and left-hand faces 175. The asymptotic stabilization as a connected sequence of functors 185.1. The first construction 195.2. The second construction 216. Comparison homomorphisms 246.1. Comparing Vogel homology with the asymptotic stabilization 24References 28
Date : September 13, 2018, 3 h 29 min.2010
Mathematics Subject Classification.
Primary: 16 E30 ; Secondary:
Key words and phrases.
Additive functor, injective stabilization of the tensor product, asymp-totic stabilization of the tensor product, Tate Cohomology, Tate homology, Vogel cohomology,Vogel homology, Buchweitz cohomology, connected sequence of functors, Mislin’s P-completion,Triulzi’s J-completion, connecting homomorphism, comparison homomorphism. Stable cohomology
Around 1950, John Tate noticed that the trivial module k over the group ring k G (where k is a field or the ring of rational integers) of a finite group G admits aprojective coresolution. Splicing it together with a projective resolution of thesame module, he obtained a doubly infinite exact complex of projectives, calleda complete resolution of k . Using it in place of the projective resolution of k , hemodified the usual notion of group cohomology, obtaining what is now known asTate cohomology. For more details, the reader is referred to [1] and [3].In 1977, F. T. Farrell [5] constructed a cohomology theory for groups of finitevirtual cohomological dimension that, for finite groups, gave the same result as Tatecohomology.In the mid-1980s, R.-O. Buchweitz [2] constructed a generalization of Tate (andFarrell) cohomology that worked over arbitrary Gorenstein commutative rings.1.1. Vogel cohomology.
At about the same time, Pierre Vogel [6] came up withhis own generalization of Tate cohomology, and while he was interested in arbitrarygroup rings, his approach actually worked over any ring. We now review thatconstruction.Let Λ be a (unital) ring and M and N (left) Λ-modules. Choose projectiveresolutions ( P , ∂ ) −→ M and ( Q , ∂ ) −→ N . Forgetting the differentials, we have Z -diagrams P and Q of left Λ-modules, together with a Z -diagram ( P , Q ) ofabelian groups. The latter has Q i Hom ( P i , Q i + n ) as its degree n component. Itcontains the subdiagram ( P , Q ) b of bounded maps, whose degree n component is ` i Hom ( P i , Q i + n ). Passing to the quotient, we have a short exact sequence ofdiagrams 0 −→ ( P , Q ) b −→ ( P , Q ) −→ ( [ P , Q ) −→ D ( f ) := ∂ ◦ f − ( − deg f f ◦ ∂ , yields a differential onthe middle diagram, which clearly restricts to a differential on the subdiagramof bounded maps. Thus the inclusion map is actually an inclusion of complexes,and the corresponding quotient becomes the quotient complex. By construction,the maps in this short exact sequence are chain maps between the constructedcomplexes. The n th Vogel cohomology group of M with coefficients in N , where n ∈ Z , is then defined as the n th cohomology group of the complex ( [ P , Q ). Wedenote it by V n ( M, N ).1.2.
Buchweitz cohomology.
As we mentioned before, Buchweitz was interestedin a generalized Tate cohomology over Gorenstein rings, but his construction (ac-tually, one of two proposed) turned out to work for any ring. We now describe hisapproach. Again, let Λ be an arbitrary (unital) ring, M and N (left) Λ-modules,and Λ-Mod the category of left Λ-modules and homomorphisms. First, we passto the category Λ-Mod of modules modulo projectives, which has the same ob-jects as Λ-Mod, but whose morphisms ( M, N ) are defined as the quotient groups(
M, N ) /P ( M, N ), where P ( M, N ) is the subgroup of all maps that can be fac-tored though a projective module. The composition of classes of homomorphismsis defined as the class of the composition of representatives. One of the advantagesof this new category is that the syzygy operation Ω on Λ-Mod becomes an addi-tive endofunctor on Λ-Mod. In particular, for M and N we have a sequence of homomorphisms of abelian groups( M, N ) −→ (Ω M, Ω N ) −→ (Ω M, Ω N ) −→ . . . The n th Buchweitz cohomology group B n ( M, N ), n ∈ Z is defined aslim −→ n + k,k ≥ (Ω n + k M, Ω k N ) . Mislin’s construction.
Yet another generalization of Tate cohomology wasgiven by G. Mislin [8] in 1994. It is a special case of a considerably more generalconstruct. For a cohomological (or, more generally, connected) sequence of functors { F i } , i ∈ Z Mislin uses a sequence of natural transformations F i −→ S ( F i +1 ) −→ S ( F i +2 ) −→ . . . , where S j denotes the j th left satellite, and defines what he calls the P -completionof { F i } as lim −→ k ≥ S k ( F i + k ) =: M i F. Evaluating the colimit on the group cohomology (viewed as a cohomological func-tor of the coefficients), he gets a new cohomological (or connected if the originalsequence is connected but not necessarily cohomological) sequence of functors. Hethen proves that, for groups of finite virtual cohomological dimension, the new co-homology is isomorphic to Farrell cohomology. Moreover, he also establishes, forarbitrary groups, an isomorphism between his construction and Buchweitz’s coho-mology (called in the paper the Benson-Carlson cohomology, after the two authors,who independently found Buchweitz’s cohomology in 1992). It should be clear,however, that Mislin’s construction is completely general and applies, in particular,to the Ext functor over any ring.2.
Stable homology
At this point, one may ask if there are homological analogs of the variouscohomology theories discussed above. The answer to this question is less clear.First, there was no “Tate homology” in Tate’s original work: only the Hom functorwas used with complete resolutions. However, at the same time when P. Vogelconstructed his cohomology, he also constructed a homology theory. We begin byreviewing his construction.2.1.
Vogel homology.
Let Λ be a ring, M a left Λ-module and N a right Λ-module. Choose a projective resolution ( P , ∂ ) −→ M and an injective resolution N −→ ( I , ∂ ). Forgetting the differentials, we have Z -diagrams P and I of left and,respectively, right Λ-modules, together with a Z -diagram P b ⊗ I of abelian groups.The latter has Q i ( P i ⊗ I i − n ) as its degree n component. It contains the subdiagram P ⊗ I , whose degree n component is ` i ( P i ⊗ I i − n ). Passing to the quotient, wehave a short exact sequence of diagrams0 −→ P ⊗ I −→ P b ⊗ I −→ P ∨ ⊗ I −→ D ( a ⊗ b ) := ∂ P ( a ) ⊗ b + ( − deg a a ⊗ ∂ I ( b )= (cid:0) ∂ P ⊗ − deg ( ) ⊗ ∂ I (cid:1) ( a ⊗ b ) , ALEX MARTSINKOVSKY AND JEREMY RUSSELL where a and b are homogeneous elements of P and, respectively, I , and deg ( )picks the degree of the first factor of a decomposable tensor, gives rise to a differ-ential on P ⊗ I . It is easy to check that it extends to a differential, denoted by D again, on P b ⊗ I . Indeed, if s ∈ ( P b ⊗ I ) n is a degree n element, then s = ( s i ) i ∈ Z ,where each s i ∈ P i ⊗ I i − n is just a finite sum of decomposable tensors. For each k ∈ Z , define D : Y i ( P i ⊗ I i − n ) −→ ( P k ⊗ I k +1 − n ) : s ( ∂ ⊗ s k +1 ) + ( − k (1 ⊗ ∂ )( s k )Now, we obtain the desired differential by the universal property of direct product.As a consequence, the third term in the short exact sequence above becomes acomplex, and Vogel homology is now defined by setting(2.2) V n ( M, N ) := H n +1 ( P ∨ ⊗ I ) . Remark 2.1.1.
Because of the shift in the subscript, the connecting homomor-phism in the long homology exact sequence is a map V n ( M, N ) −→ Tor n ( M, N ). Remark 2.1.2.
The choice of the projective and injective resolutions can beflipped. By choosing an injective resolution of M and a projective resolution of N ,one obtains another homological functor, which in general is different from theoriginal one. This can be seen by choosing M to be projective. In that case, theoriginal functor evaluates to zero, whereas the alternative construction produces,in general, a nonzero result.2.2. A homological analog of Mislin’s construction.
A homological analogof Mislin’s cohomological P -completion, called the J -completion, was defined byM. Triulzi in his PhD thesis [10] . Like its cohomological prototype, it is defined onconnected sequences of functors, but even if the original sequence is cohomological,the result doesn’t seem to be cohomological ; one can only claim that the resultingsequence is connected. For reference, we denote it by M i F .2.3. Summary.
We summarize the existing constructions in the following table:
Cohomology Homology V i ( M, N ) V i ( M, N )B i ( M, N ) ?M i F M i F One of the goals of this paper is to replace the question mark by a homologicalanalog of Buchweitz’s construction.In this paper, we follow the terminology and notation established in [7]. Thereader may benefit from reviewing that source.Some results contained in the present paper overlap with some results obtainedby the second author in his PhD thesis [9]. The authors are grateful to Lucho Avramov for bringing this work to our attention and toLars Christensen for sending us a copy of it This is related to the fact that the inverse limit is not an exact functor. The asymptotic stabilization of the tensor product
Our next goal is to introduce what we shall call the asymptotic stabilization of the tensor product, which is a limit of a sequence of maps between injectivestabilizations of tensor products of iterated syzygy and cosyzygy modules. In thissection, this will be done in three equivalent ways.
Blanket assumption.
Whenever we deal with a connecting homomorphismin the snake lemma, we automatically assume that the homomorphism was con-structed by pushing and pulling the elements along a staircase path, as in thetraditional proof of the lemma.3.1.
The first construction.
We begin with constructing a homomorphismΩ A ⇁ ⊗ Σ B −→ A ⇁ ⊗ B of abelian groups, where A is a right Λ-module and B is a left Λ-module.Given a left Λ-module B , choose an injective resolution(3.1) 0 −→ B −→ I −→ I −→ . . . Similarly, given a right Λ-module A choose a projective resolution(3.2) . . . −→ P −→ P −→ A −→ −→ Ω A −→ P −→ A −→ −→ B −→ I −→ Σ B −→ & & ▼▼▼▼▼▼▼▼▼▼▼ (cid:15) (cid:15) Ω A ⇁ ⊗ Σ B ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ / / / / Tor ( A, Σ B ) (cid:15) (cid:15) Ω A ⊗ B (cid:15) (cid:15) / / Ω A ⊗ I (cid:15) (cid:15) / / Ω A ⊗ Σ B (cid:15) (cid:15) / / ' ' ◆◆◆◆◆◆◆◆◆◆◆ / / P ⊗ B / / (cid:15) (cid:15) P ⊗ I / / / / (cid:15) (cid:15) P ⊗ Σ B (cid:15) (cid:15) ' ' ' ' ◆◆◆◆◆◆◆◆◆◆◆ Ω A ⊗ I (cid:15) (cid:15) . . . / / A ⊗ B (cid:15) (cid:15) / / A ⊗ I / / / / (cid:15) (cid:15) A ⊗ Σ B (cid:15) (cid:15) P ⊗ I P ⊗ is an exact functor and the snake lemma, we have Lemma 3.1.1.
The above solid diagram induces an exact sequence
Tor ( A, B ) −→ Tor ( A, I ) −→ Tor ( A, Σ B ) δ −→ A ⇁ ⊗ B −→ , where δ is (the corestriction of ) the connecting homomorphism. If the injective I is projective, then δ is an isomorphism. (cid:3) ALEX MARTSINKOVSKY AND JEREMY RUSSELL As P ⊗ is an exact functor, the bottom southeast map is monic. The com-position of this map withΩ A ⇁ ⊗ Σ B / / Ω A ⊗ Σ B / / P ⊗ Σ B is obviously zero and, by the universal property of kernels, we have the dotted mapin the above diagram, making the top triangle commute. Notice that this mapis monic. Applying the snake lemma yields the following diagram with an exactbottom row Ω A ⇁ ⊗ Σ B (cid:15) (cid:15) Tor ( A, Σ B ) / / A ⊗ B / / A ⊗ I This diagram embeds in the commutative diagramΩ A ⇁ ⊗ Σ B (cid:15) (cid:15) Tor ( A, Σ B ) / / δ (cid:15) (cid:15) A ⊗ B / / A ⊗ I / / A ⇁ ⊗ B / / A ⊗ B / / A ⊗ I which produces a homomorphismΩ A ⇁ ⊗ Σ B ∆ / / A ⇁ ⊗ B Iteration of this process yields a sequence(3.4) . . . / / Ω A ⇁ ⊗ Σ B ∆ / / Ω A ⇁ ⊗ Σ B ∆ / / A ⇁ ⊗ B. Now we want to show that any two choices for ∆ , and hence for any other ∆ i ,are isomorphic. In addition to the resolutions (3.1) and (3.2), let(3.5) 0 −→ B −→ I ′ −→ I ′ −→ . . . be another injective resolution of B , and(3.6) . . . −→ P ′ −→ P ′ −→ A −→ A . Lifting the identity map on A , extending theidentity map on B , and taking the tensor product results in a commutative 3Dversion of diagram (3.3). By the naturality of the connecting homomorphism in the snake lemma, we have a commutative diagram with exact rows . . . / / Tor ( A, Σ ′ B ) / / (cid:15) (cid:15) & & & & ▲▲▲▲▲▲▲▲▲▲ A ⊗ B / / A ⊗ I ′ / / (cid:15) (cid:15) . . .A ⇁ ⊗ ′ B ∼ = α (cid:15) (cid:15) ; ; ; ; ✈✈✈✈✈✈✈✈ . . . / / Tor ( A, Σ B ) / / & & & & ▼▼▼▼▼▼▼▼▼▼ A ⊗ B / / A ⊗ I / / . . .A ⇁ ⊗ B : : : : ✈✈✈✈✈✈✈✈✈ By [7, Lemma 4.1)], α is an equality. On the other hand, the right-hand side of the3D-version of (3.3) yields a commutative diagram of solid arrowsTor ( A, Σ ′ B ) (cid:15) (cid:15) (cid:15) (cid:15) v v ♠♠♠♠♠♠♠♠ Tor ( A, Σ B ) (cid:15) (cid:15) (cid:15) (cid:15) / / Ω ′ A ⇁ ⊗ Σ ′ B - - / / β ∼ = x x ♣♣♣♣♣♣♣ Ω ′ A ⊗ Σ ′ B / / w w ♥♥♥♥♥♥♥♥♥♥ (cid:15) (cid:15) Ω ′ A ⊗ I ′ y y ssssssss (cid:15) (cid:15) / / Ω A ⇁ ⊗ Σ B - - / / Ω A ⊗ Σ B / / (cid:15) (cid:15) Ω A ⊗ I (cid:15) (cid:15) P ′ ⊗ Σ ′ B / / / / v v ♠♠♠♠♠♠♠♠♠ P ′ ⊗ I ′ x x qqqqqqq P ⊗ Σ B / / / / P ⊗ I with exact rows and columns. The dotted arrows also come from diagram (3.3) andmake the triangles containing them commute. By [7, Lemmas 9.1 and 9.2], β is thecanonical isomorphism. Using the fact that the map Tor ( A, Σ B ) −→ Ω A ⊗ Σ B is amonomorphism, we have that the curved square also commutes. Splicing it with theleft-hand square containing α from the preceding diagram, we have a commutativesquare Ω ′ A ⇁ ⊗ Σ ′ B ∆ ′ / / β ∼ = (cid:15) (cid:15) A ⇁ ⊗ ′ B α ∼ = (cid:15) (cid:15) Ω A ⇁ ⊗ Σ B ∆ / / A ⇁ ⊗ B with the vertical maps being canonical isomorphisms. This proves Proposition 3.1.2.
Any two choices for ∆ , and hence for any ∆ i , based on thediagram (3.3) are canonically isomorphic. (cid:3) Arguments very similar to the ones just used yield
ALEX MARTSINKOVSKY AND JEREMY RUSSELL
Proposition 3.1.3.
The homomorphism ∆ : Ω A ⇁ ⊗ Σ B −→ A ⇁ ⊗ B , and henceany ∆ i , is functorial in both A and B . (cid:3) For any integer n ∈ Z (including negative values), the process of constructingthe sequence (3.4) may be repeated with Ω k + n A in place of Ω k A , yielding sequences(3.7) M n ( A, B ) := Ω k + n A ⇁ ⊗ Σ k B, k, k + n ≥ Definition 3.1.4.
The asymptotic stabilization T n ( A, ) of the left tensor productin degree n with coefficients in the right Λ -module A is T n ( A, )( B ) := T n ( A, B ):= lim ←− k,k + n ≥ Ω k + n A ⇁ ⊗ Σ k B = lim ←− M n ( A, B )(3.8)It is easy to see that the T n ( A, ) : Λ-Mod −→ Ab , n ∈ Z are covariantadditive functors from the category of left Λ-modules to the category of abeliangroups. It is plain that the T n ( A, ) are injectively stable. The next result showsthat we also have dimension shifts, including the fixed argument. Lemma 3.1.5.
For all n ∈ Z , k ∈ Z ≥ , an j ∈ Z ≥ there are canonical isomor-phisms of functors T n ( A, Σ k ) ∼ = T n − k ( A, ) and T n (Ω j A, ) ∼ = T n + j ( A, ) Proof.
The sequences (including the structure maps) for the components of theformer (respectively, latter) pair of functors at any right Λ-module can be obviouslychosen to be shifts of each other. (cid:3)
Now we want to discuss the vanishing of the functors T • ( A, ). The first resultis an an immediate consequence of the definitions. Proposition 3.1.6.
If the right global dimension of Λ is finite then T n ( A, ) = 0 for all integers n . (cid:3) Proposition 3.1.7.
If the flat dimension of A is finite, then T n ( A, ) = 0 for allintegers n .Proof. As the diagram (3.3) shows, we have an injection Ω A ⇁ ⊗ Σ B −→ Tor ( A, Σ B ).In particular, Ω n + k A ⇁ ⊗ Σ k B , n + k, k ≥ (Ω n + k − A, Σ k B ). Butthe latter vanishes for n + k − ≥ fl. dim A . (cid:3) It is known that the vanishing of stable cohomology in one degree implies itsvanishing in all degrees. We do not know if a similar statement is true for T • ( A, ).A partial answer is provided by Proposition 3.1.8. If T n ( A, ) = 0 for some integer n , then T m ( A, ) = 0 forall m < n . If, in addition, Λ is quasi-Frobenius, then T m ( A, ) = 0 for all m ∈ Z .Proof. The first assertion is an immediate consequence of the first isomorphism ofLemma 3.1.5. Suppose now that Λ is quasi-Frobenius. Since projective modulesare injective, for any positive integer k , any right Λ-module B is a k th cosyzygymodule in an injective resolution of Ω k B , i.e., B ≃ Σ k Ω k B . Therefore,T n + k ( A, B ) ∼ = T n + k ( A, Σ k Ω k B ) ∼ = T n ( A, Ω k B ) = 0 . (cid:3) The second construction.
Next we want to show that Proposition 3.1.7 fol-lows from a more general result, namely, that the asymptotic stabilization T • ( A, B )can be computed via the Tor functors. Our goal is to construct a commutative di-agram . . . / / Tor (Ω A, Σ B ) / / % % % % ❑❑❑❑❑❑❑❑ Tor ( A, Σ B ) " " " " ❊❊❊❊❊❊❊ / / A ⊗ B. . . / / < < < < ③③③③③③③③③③ Ω A ⇁ ⊗ Σ B / / : : : : ✉✉✉✉✉✉✉✉ A ⇁ ⊗ B @ @ @ @ ✁✁✁✁✁✁✁ where the bottom sequence is given by (3.4), and the arrows in the top sequence areconnecting homomorphisms. Clearly, once such a diagram has been constructed,the limits of the two horizontal sequences will be isomorphic, showing that the as-ymptotic stabilization can indeed be constructed using the Tor functors. Moreover,we shall also show that all northeast arrows are monic and all southeast arrows areepic. This immediately implies that all stages in the original construction of theasymptotic stabilization can be recovered via the epi-mono factorizations of the toparrows.The construction requires explicit choices, so for a right Λ-module A we choosea projective resolution . . . −→ P −→ P −→ A −→ ( A, ) via the exact sequence(3.9) 0 −→ Tor ( A, ) −→ Ω A ⊗ −→ P ⊗ −→ A ⊗ −→ . For a projective resolution of Ω A we choose the projective resolution of A truncatedin degree 1. This allows us to claim that Tor i +1 ( A, ) = Tor i (Ω A, ), where wedo mean an equality rather than an abstract isomorphism.For a short exact sequence(3.10) 0 −→ C −→ D −→ E → A -modules, recall the construction the connecting homomorphismsTor i +1 ( A, E ) −→ Tor i ( A, C )in the corresponding long exact sequence of the Tor functors. The case i = 0consists of evaluating the sequence (3.9) on the short exact sequence above andthen using the snake lemma. For positive values of i , we describe the constructionwhen i = 1 and then use the dimension shift. To this end, we replace A with Ω A and build a snake diagram as in the case i = 0. The new diagram and the originalone have a common row,Ω A ⊗ C −→ Ω A ⊗ D −→ Ω A ⊗ E −→ , which allows to glue the two diagrams together: Tor (Ω A, C ) (cid:3) (cid:3) ✞✞✞✞✞✞✞✞✞✞✞✞✞✞ / / Tor (Ω A, D ) (cid:3) (cid:3) ✞✞✞✞✞✞✞✞✞✞✞✞✞✞ / / Tor (Ω A, E ) (cid:4) (cid:4) ✠✠✠✠✠✠✠✠✠✠✠✠✠✠ Ω A ⊗ C (cid:5) (cid:5) ☞☞☞☞☞☞☞☞☞☞☞☞☞ / / Ω A ⊗ D (cid:3) (cid:3) ✞✞✞✞✞✞✞✞✞✞✞✞✞✞ / / Ω A ⊗ E (cid:4) (cid:4) ✠✠✠✠✠✠✠✠✠✠✠✠✠✠ P ⊗ C (cid:7) (cid:7) ✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍ / / P ⊗ D (cid:4) (cid:4) ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ / / P ⊗ E (cid:4) (cid:4) ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ Tor ( A, C ) / / (cid:15) (cid:15) (cid:15) (cid:15) Tor ( A, D ) / / (cid:15) (cid:15) (cid:15) (cid:15) Tor ( A, E ) (cid:15) (cid:15) (cid:15) (cid:15) Ω A ⊗ C γ / / α (cid:15) (cid:15) T Ω A ⊗ D / / / / δ (cid:15) (cid:15) Ω A ⊗ E (cid:15) (cid:15) P ⊗ C / / β / / (cid:15) (cid:15) P ⊗ D / / / / (cid:15) (cid:15) P ⊗ E (cid:15) (cid:15) A ⊗ C / / A ⊗ D / / / / A ⊗ E Notice that the connecting homomorphismTor ( A, E ) = Tor (Ω A, E ) ǫ −→ Ω A ⊗ C in the horizontal part of the diagram factors through Ker α = Tor ( A, C ). In-deed, the commutativity of the square T shows that βαǫ = δγǫ = 0. Since β ismonic, αǫ = 0 and ǫ factors through Tor ( A, C ). As a result, we have a connectinghomomorphism Tor ( A, E ) −→ Tor ( A, C ) and the desired long exact sequence.Returning to the left Λ-module B , we specialize the short exact sequence (3.10)to the cosyzygy sequence(3.11) 0 −→ Σ B −→ I −→ Σ B −→ . The foregoing argument then yields a commutative squareTor ( A, Σ B ) / / ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ Tor ( A, Σ B ) (cid:15) (cid:15) (cid:15) (cid:15) Tor (Ω A, Σ B ) / / Ω A ⊗ Σ B where the diagonal map is the connecting homomorphism in the horizontal part ofthe diagram on page 10. The composition of this map with γ : Ω A ⊗ Σ B → Ω A ⊗ I is zero, hence it factors through the kernel of γ , which is by definition Ω A ⇁ ⊗ Σ B . We now have a commutative diagram of solid arrowsTor ( A, Σ B ) / / ' ' PPPPPPPPPP
Tor ( A, Σ B ) (cid:15) (cid:15) (cid:15) (cid:15) Ω A ⇁ ⊗ Σ B ' ' ◆◆◆◆◆◆◆◆◆◆◆◆◆ Tor (Ω A, Σ B ) / / Ω A ⊗ Σ B / / (cid:15) (cid:15) T Ω A ⊗ I (cid:15) (cid:15) P ⊗ Σ B / / / / P ⊗ I The existence of the dotted arrow and the fact that it is monic was established whenwe discussed the diagram (3.3); it makes the triangle on the right commute. Sincethe vertical map in that triangle is monic, the top triangle is also commutative. Byconstruction, the horizontal map in that triangle is the connecting homomorphismin the long exact sequence of the functors Tor i ( A, ) corresponding to the cosyzygysequence (3.11). Colloquially, the connecting homomorphism in the long exactTor-sequence factors through the injective stabilization. We view these connectinghomomorphisms as the structure maps in the sequence . . . −→ Tor (Ω i A, Σ i +1 B ) −→ Tor (Ω i − A, Σ i B ) −→ . . . On the other hand, as (3.3) showed, the structure map Ω A ⇁ ⊗ Σ B −→ A ⇁ ⊗ B factors through Tor ( A, Σ B ). Combining these two observations, we have a com-mutative diagram(3.12) . . . / / Tor (Ω A, Σ B ) / / % % % % ❑❑❑❑❑❑❑❑ Tor ( A, Σ B ) " " " " ❊❊❊❊❊❊❊ / / A ⊗ B. . . / / < < < < ③③③③③③③③③③ Ω A ⇁ ⊗ Σ B / / : : : : ✉✉✉✉✉✉✉✉ A ⇁ ⊗ B @ @ @ @ ✁✁✁✁✁✁✁ “intertwining” the two sequences. Taking into account the dimension shift, we nowhave Theorem 3.2.1.
The sequence of the
Tor functors in the above diagram is func-torial in both arguments. For any integer n , the two families of parallel arrows inthe (suitably shifted) above diagram induce mutually inverse isomorphisms T n ( A, )( B ) = lim ←− k,k + n ≥ Ω k + n A ⇁ ⊗ Σ k B ≃ lim ←− k,k + n ≥ Tor (Ω k + n A, Σ k +1 B ) Proof.
The first assertion follows from the functoriality of the connecting homo-morphism. The second assertion has already been established. (cid:3)
Remark 3.2.2.
In the above diagram, A and B can be replaced by their arbitrarysyzygy and, respectively, cosyzygy modules. With each map in the sequences, thepowers of syzygy and cosyzygy modules simultaneously go down by one. If the The reader has probably noticed that this theorem implies Proposition 3.1.7. powers of Ω run out first, then the last term on the right will be a tensor product.If the powers of Σ run out first, then the last term will be a Tor . Remark 3.2.3.
We have actually proved more. Since all southeast maps are epic,all northeast maps are monic, and since an epi-mono factorization of a morphismin an abelian category is determined uniquely up to an isomorphism, the lowersequence is determined uniquely up to an isomorphism by the maps in the uppersequence. In particular, this yields new equivalent definitions of both the injectivestabilization and the asymptotic stabilization of the tensor product.
Example 3.2.4.
Suppose Λ is quasi-Frobenius. Then, by Lemma 3.1.1, the south-east maps are all isomorphisms, making the two systems isomorphic. The nextexample shows that the two systems may be isomorphic over other types of rings.
Example 3.2.5.
Let Λ := Z , A := Z /p Z , where p is a prime number, and B := Z . Then, taking Σ B ≃ Q / Z , we have, since the injective stabilizationvanishes on injectives ([7, Lemma 4.5]), Ω A ⇁ ⊗ Σ B = 0. To compute A ⇁ ⊗ B , weapply the functor Z /p Z ⊗ to the injective envelope Z → Q of Z . The kernel ofthe resulting map Z /p Z → Z /p Z ⊗ Q is Z /p Z . Finally, we compute Tor ( A, Σ B )by using a projective resolution of A = Z /p Z . The result is the kernel of the map Q / Z .p −→ Q / Z , which is the subgroup Z /p Z = { , /p, . . . , ( p − /p } . Moreover,the diagram (3.3) shows that, in this case, the map Tor ( A, Σ B ) → A ⇁ ⊗ B is anisomorphism. Since all the remaining terms in the two directed systems vanish, wehave that the southeast maps in (3.12) make the two systems isomorphic.3.3. The third construction.
Recall that the right-hand side of the functorialisomorphism S Tor ( A, B ) ∼ = A ⇁ ⊗ B [7, Proposition 9.3] is the initial term of thesequence (3.4): . . . / / Ω A ⇁ ⊗ Σ B ∆ / / Ω A ⇁ ⊗ Σ B ∆ / / A ⇁ ⊗ B. Our next goal is to construct an isomorphic sequence starting with S Tor ( A, B ).Of course, this simply means constructing structure maps. First, we need yetanother observation about connecting homomorphisms. In the diagram (3.3), wehave two copies of Tor ( A, Σ B ), one at the top of the rightmost vertical exactsequence, the other (not shown) as the next term in the long exact sequence in thebottom row. Both map into A ⇁ ⊗ B , and we wish to make a commutative triangle byconstructing an isomorphism between the two copies of Tor. This will be a generalobservation. More precisely, let0 −→ B −→ C −→ D −→ −→ F −→ P −→ A −→ P projective. Tensoring them together, we have acommutative diagram with exact rows and columns Alternatively, since Ω A is projective, one can use [7, Lemma 4.8]). (3.13) 0 (cid:15) (cid:15) Tor ( A, D ) (cid:15) (cid:15) Ω A ⊗ B (cid:15) (cid:15) / / Ω A ⊗ C (cid:15) (cid:15) / / F ⊗ D (cid:15) (cid:15) / / / / P ⊗ B / / (cid:15) (cid:15) P ⊗ C / / (cid:15) (cid:15) P ⊗ D (cid:15) (cid:15) / / ( A, D ) α / / A ⊗ B (cid:15) (cid:15) β / / A ⊗ C / / (cid:15) (cid:15) A ⊗ D (cid:15) (cid:15) / /
00 0 0where the bottom row and the rightmost column are fragments of the correspondinglong homology exact sequences.
Lemma 3.3.1.
Let δ : Tor ( A, D ) → A ⊗ B be the connecting homomorphism inthe above diagram. Then there is an isomorphism γ : Tor ( A, D ) −→ Tor ( A, D ) such that αγ = δ .Proof. If C is projective, then we are immediately done by the snake lemma. More-over, the construction of the isomorphism is explicit – it is given by the connectinghomomorphism. In general, choose an epimorphism Q −→ D with Q projectiveand lift the identity map on D to obtain a commutative diagram with exact rows0 / / Ω D / / (cid:15) (cid:15) Q / / (cid:15) (cid:15) D / / / / B / / C / / D / / −→ F α −→ P β −→ A −→
0, we have aspatial commutative diagram with bottom face0 / / Tor ( A, D ) / / A ⊗ Ω D / / (cid:15) (cid:15) A ⊗ Q / / (cid:15) (cid:15) A ⊗ D / / ( A, C ) / / Tor ( A, D ) / / A ⊗ B / / A ⊗ C / / A ⊗ D / / (cid:3) Now we can start building structure maps∆ i : S Tor (Ω i +1 A, )(Σ i +1 B ) −→ S Tor (Ω i A, )(Σ i B ) . Clearly, it suffices to do this for i = 0, and we shall again use the diagram (3.3).This yields a commutative diagram of solid arrowsTor (Ω A, I ) ǫ ' ' ❖❖❖❖❖❖❖❖❖❖ (cid:15) (cid:15) Tor (Ω A, Σ B ) α ' ' PPPPPPPPPP / / Tor ( A, Σ B ) (cid:15) (cid:15) Ω A ⊗ B (cid:15) (cid:15) / / Ω A ⊗ I (cid:15) (cid:15) / / Ω A ⊗ Σ B (cid:15) (cid:15) / / & & ▲▲▲▲▲▲▲▲▲ / / P ⊗ B / / (cid:15) (cid:15) P ⊗ I / / (cid:15) (cid:15) P ⊗ Σ B (cid:15) (cid:15) % % % % ▲▲▲▲▲▲▲▲ Ω A ⊗ I (cid:15) (cid:15) Tor ( A, I ) ǫ / / Tor ( A, Σ B ) α / / A ⊗ B (cid:15) (cid:15) / / A ⊗ I / / (cid:15) (cid:15) A ⊗ Σ B (cid:15) (cid:15) P ⊗ I A and with Σ B replaced by Σ B . The diagramshows that α factors through Tor ( A, Σ B ), giving rise to a unique dotted mapmaking a commutative triangle. As Tor ( A, Σ B ) −→ Ω A ⊗ Σ B is monic, thedotted map composed with ǫ is zero, and therefore gives rise to a unique mapCoker ǫ = S Tor (Ω A, )(Σ B ) −→ Tor ( A, Σ B )Composing it with the isomorphism γ : Tor ( A, Σ B ) −→ Tor ( A, Σ B ) constructedin Lemma 3.3.1 and with the canonical epimorphismTor ( A, Σ B ) −→ Coker ǫ = S Tor ( A, )( B ) , we declare the resulting composition to be the structure map∆ : S Tor (Ω A, )(Σ B ) −→ S Tor ( A, )( B ) . By Lemma 3.3.1, it is compatible with the isomorphisms of [7, Proposition 9.3] (forfixed A and Ω A ). Similar arguments yield maps ∆ i for all natural i . We have thusproved Theorem 3.3.2.
The connecting homomorphism in the diagram (3.3) induces afunctor isomorphism of sequences ( S Tor (Ω i A, ) ◦ Σ i ( ) , ∆ i ) ≃ (Ω i A ⇁ ⊗ Σ i ( ) , ∆ i ) . This isomorphism is natural in A . (cid:3) In summary, all three constructions of the asymptotic stabilization of the tensorproduct yield isomorphic results. In particular,
Corollary 3.3.3.
The asymptotic stabilization T( A, ) and the J -completion ofthe connected sequence Tor ∗ ( A, ) are isomorphic as connected sequences of func-tors. Proof.
This follows from the fact that the directed system involved in Triulzi’s con-struction of the J -completion [10] and the directed system used in the constructionof the asymptotic stabilization are isomorphic. The isomorphism is precisely thatappearing in Theorem 3.3.2 (cid:3) Two lemmas on connecting homomorphisms
In this section we shall establish two results, stated and proved in a greatergenerality than is needed for this paper, helping us understand how to composeconnecting homomorphisms running in spatial diagrams. For an exact 3 × Front, bottom, and right-hand faces.
Lemma 4.1.1.
Let L ′ / / (cid:15) (cid:15) (cid:8) (cid:8) ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ M ′ / / / / (cid:15) (cid:15) (cid:8) (cid:8) ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ N ′ (cid:15) (cid:15) (cid:8) (cid:8) ✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑ L ′ / / (cid:15) (cid:15) (cid:15) (cid:15) ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ (cid:8) (cid:8) ✏✏✏✏✏✏✏ M ′ / / / / (cid:15) (cid:15) (cid:15) (cid:15) ✏✏✏✏✏✏✏✏✏✏✏✏✏✏ (cid:8) (cid:8) ✏✏✏✏✏✏✏ N ′ (cid:15) (cid:15) (cid:15) (cid:15) (cid:8) (cid:8) ✑✑✑✑✑✑✑✑✑✑✑ (cid:8) (cid:8) ✑✑✑✑✑✑✑✑✑✑✑ L ′ / / ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ (cid:8) (cid:8) ✏✏✏✏✏✏✏ M ′ / / / / (cid:7) (cid:7) ✏✏✏✏✏✏✏ ✏✏✏✏✏✏ (cid:7) (cid:7) ✏✏✏✏✏✏✏ N ′ (cid:8) (cid:8) ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ L / / (cid:15) (cid:15) (cid:8) (cid:8) (cid:8) (cid:8) ✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑ M '&%$ !" / / / / '&%$ !" (cid:15) (cid:15) (cid:7) (cid:7) (cid:7) (cid:7) ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ N '&%$ !" (cid:15) (cid:15) '&%$ !" ✏✏✏✏✏✏✏✏✏✏✏ (cid:8) (cid:8) (cid:8) (cid:8) ✏✏✏✏✏✏✏✏✏✏ L / / (cid:15) (cid:15) (cid:15) (cid:15) ✑✑✑✑✑✑✑✑✑✑✑✑✑✑ (cid:8) (cid:8) (cid:8) (cid:8) ✑✑✑✑✑✑✑ M '&%$ !" / / / / (cid:15) (cid:15) (cid:15) (cid:15) '&%$ !" ✏✏✏✏✏✏✏✏✏✏✏✏✏✏ (cid:7) (cid:7) (cid:7) (cid:7) ✏✏✏✏✏✏✏ N (cid:15) (cid:15) (cid:15) (cid:15) (cid:8) (cid:8) (cid:8) (cid:8) ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ L / / ✑✑✑✑✑✑✑ ✑✑✑✑✑✑ (cid:8) (cid:8) (cid:8) (cid:8) ✑✑✑✑✑✑✑ M / / / / ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ (cid:8) (cid:8) (cid:8) (cid:8) ✏✏✏✏✏✏✏ N (cid:8) (cid:8) (cid:8) (cid:8) ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ L ′′ / / (cid:15) (cid:15) M ′′ '&%$ !" / / / / '&%$ !" (cid:15) (cid:15) N ′′ '&%$ !" α (cid:15) (cid:15) L ′′ / / '&%$ !" / / (cid:15) (cid:15) (cid:15) (cid:15) M ′′ / / / / (cid:15) (cid:15) (cid:15) (cid:15) N ′′ (cid:15) (cid:15) (cid:15) (cid:15) L ′′ /.-,()*+ β / / M ′′ / / / / N ′′ be a commutative diagram subject to the following conditions: (1) any three-term sequence with arrows running in the same direction is exact(i.e., exact at the middle term); (2) each arrow preceded by an arrow in the same direction is epic; (3) the three middle three-term sequences L ′′ M ′′ N ′′ , M ′ M M ′′ , and N ′ N N ′′ on the front, the bottom and the right-hand faces of this cube are short-exact, i.e., each sequence is exact in the middle, the first map is monic,and the second map is epic.Then the image of the connecting homomorphism Ker α −→ L ′′ (in the front face)is in Ker β , and the composition of the connecting homomorphisms Ker α −→ L ′′ and Ker β −→ N ′ (in the bottom face) equals the negative of the connecting homo-morphism Ker α −→ N ′ (in the right-hand face).Proof. First of all, because of the assumptions on the three short exact sequences,the connecting homomorphisms mentioned in the statement are indeed defined. Thefirst assertion is now immediate. To prove the second assertion, pick an element n ′′ ∈ Ker α ⊂ N ′′ . By the commutativity of the diagram, m ′′ := 2 ◦ − ( n ′′ ) = 6 ◦ ◦ − ◦ − ( n ′′ ) . Along the way, we have an element m ∈ M such that7( m ) = 8 ◦ − ( n ′′ ) =: n. Let µ := (14) ◦ (12) − ◦ − ◦ m ) . Since 15( m ) = 0, we have m := 15( µ − m ) = (13) ◦ (11) − ◦ (10) ◦ − ( m ′′ )Since 7( µ ) = 0, we have 7( µ − m ) = − ◦ − ( n ′′ ) . By construction, 6( µ ) = 6( m ), and therefore µ − m = 16( m ′ ) for some m ′ ∈ M ′ .Now (19) ◦ (17)( m ′ ) = − n . Therefore, (18) ◦ (17)( m ′ ) is negative the value of theconnecting homomorphism Ker α −→ N ′ on n ′′ . Using the commutativity of thediagram again, we have the same value for (20) ◦ (21) − ( m ), which is the value ofthe composition of the other two connecting homomorphisms on the same element. (cid:3) Top, back, and left-hand faces.
Now we look at the composition of con-necting homomorphisms in the three remaining planes of the cube.
Lemma 4.2.1.
Let L ′ / / (cid:15) (cid:15) (cid:8) (cid:8) ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ M ′ '&%$ !" / / / / '&%$ !" (cid:15) (cid:15) (cid:8) (cid:8) '&%$ !" ✏✏✏✏✏✏✏✏✏✏✏ (cid:8) (cid:8) ✏✏✏✏✏✏✏✏✏✏✏ N ′ /.-,()*+ β (cid:15) (cid:15) (cid:8) (cid:8) ✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑ L ′ / / '&%$ !" / / '&%$ !" (cid:15) (cid:15) (cid:15) (cid:15) (cid:8) (cid:8) '&%$ !" ✏✏✏✏✏✏✏✏✏✏✏✏✏✏ (cid:8) (cid:8) ✏✏✏✏✏✏✏ M ′ ?>=<89:; / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:8) (cid:8) ?>=<89:; ✏✏✏✏✏✏✏✏✏✏✏✏✏✏ (cid:8) (cid:8) ✏✏✏✏✏✏✏ N ′ (cid:15) (cid:15) (cid:15) (cid:15) (cid:8) (cid:8) ✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑ L ′ / / ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ (cid:8) (cid:8) ✏✏✏✏✏✏✏ M ′ / / / / ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ (cid:7) (cid:7) ✏✏✏✏✏✏✏ N ′ (cid:8) (cid:8) ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ L '&%$ !" / / '&%$ !" (cid:15) (cid:15) '&%$ !" ✑✑✑✑✑✑✑✑✑✑ (cid:8) (cid:8) (cid:8) (cid:8) ✑✑✑✑✑✑✑✑✑✑ M / / / / ?>=<89:; (cid:15) (cid:15) (cid:7) (cid:7) (cid:7) (cid:7) ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ N (cid:15) (cid:15) (cid:8) (cid:8) (cid:8) (cid:8) ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ L / / ?>=<89:; / / (cid:15) (cid:15) (cid:15) (cid:15) ✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑ (cid:8) (cid:8) (cid:8) (cid:8) ✑✑✑✑✑✑✑ M / / / / (cid:15) (cid:15) (cid:15) (cid:15) ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ (cid:7) (cid:7) (cid:7) (cid:7) ✏✏✏✏✏✏✏ N (cid:15) (cid:15) (cid:15) (cid:15) (cid:8) (cid:8) (cid:8) (cid:8) ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ L / / ✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑ (cid:8) (cid:8) (cid:8) (cid:8) ✑✑✑✑✑✑✑ M / / / / ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ (cid:8) (cid:8) (cid:8) (cid:8) ✏✏✏✏✏✏✏ N (cid:8) (cid:8) (cid:8) (cid:8) ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ L ′′ '&%$ !" α / / '&%$ !" γ (cid:15) (cid:15) M ′′ / / / / (cid:15) (cid:15) N ′′ (cid:15) (cid:15) L ′′ / / (cid:15) (cid:15) (cid:15) (cid:15) M ′′ / / / / (cid:15) (cid:15) (cid:15) (cid:15) N ′′ (cid:15) (cid:15) (cid:15) (cid:15) L ′′ / / M ′′ / / / / N ′′
28 ALEX MARTSINKOVSKY AND JEREMY RUSSELL be a commutative diagram subject to the following conditions: (1) any three-term sequence with arrows running in the same direction is exact; (2) each arrow preceded by an arrow in the same direction is epic; (3) the three middle three-term sequences M ′ M M ′′ , L ′ M ′ N ′ , and L ′ LL ′′ onthe top, the back, and the left-hand faces of this cube are short-exact, i.e.,each sequence is exact in the middle, the first map is monic, and the secondmap is epic. Moreover, the two horizontal sequences LM N and M ′ M M ′′ passing through the center of the cube are also short-exact.Then the image of Ker α ∩ Ker γ under the connecting homomorphism Ker α −→ N ′ (in the top face) is in Ker β , and on Ker α ∩ Ker γ the composition of the connectinghomomorphisms Ker α −→ N ′ and Ker β −→ L ′ (in the back face) coincides withthe connecting homomorphism Ker γ −→ L ′ (in the left-hand face).Proof. First, we need to show that the connecting homomorphism Ker α −→ N ′ maps Ker α ∩ Ker γ to Ker β . Pick an element l ′′ ∈ Ker α ∩ Ker γ ⊂ L ′′ and let m ′ := 6 ◦ − ◦ ◦ − ( l ′′ ) . We need to show that 11( m ′ ) = 0 or, equivalently, that m ′ is in the image of 7. Let l := 8 ◦ − ( l ′′ ). By the commutativity of the diagram,12( m ′ ) = 10( l ) . On the other hand, since l ′′ ∈ Ker γ , l = 9( l ′ ) for some l ′ ∈ L ′ , and therefore12 ◦ l ′ ) = 10( l ) = 12( m ′ ) . Since 12 is assumed to be a monomorphism, m ′ = 7( l ′ ), which is the desired claim.Now we can prove the second claim. Because each of the morphisms 1, 2, and 3belongs to two connecting homomorphisms, it suffices to show that7 − ◦ ◦ − ◦ − ◦ . But the two morphisms become equal when we precompose them with the monomor-phism 10 ◦ (cid:3) The asymptotic stabilization as a connected sequence of functors
Our next goal is to define, for each short exact sequence0 −→ B ′ −→ B −→ B ′′ −→ , of left Λ modules, connecting homomorphisms ω n : T n ( A, B ′′ ) −→ T n − ( A, B ′ )and show that (T • ( A, ) , ω • ) is a connected sequence of functors. We continue to assume that the connecting homomorphism in the snake lemmais defined by pushing and pulling elements along a staircase pattern, as in thestandard proof of the lemma. Any sequence of additive functors can be made connected by choosing the zero map as theconnecting homomorphism. Our choice will be nonzero. The first construction.
By Lemma 3.1.5, it suffices to define ω : T ( A, B ′′ ) −→ T ( A, B ′ ) . To this end, we use the horse-shoe lemma and construct a commutative diagram(5.1) 0 (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / B ′ / / (cid:15) (cid:15) I ′ / / (cid:15) (cid:15) Σ B ′ / / γ (cid:15) (cid:15) / / B / / (cid:15) (cid:15) I / / (cid:15) (cid:15) Σ B / / (cid:15) (cid:15) / / B ′′ / / (cid:15) (cid:15) I ′′ / / (cid:15) (cid:15) Σ B ′′ / / (cid:15) (cid:15)
00 0 0where the rows and columns are exact, and the middle column is a split-exactsequence of injective modules. We will also need an embedding 0 −→ Σ B ′ ǫ −→ I ′ of Σ B ′ into the next step of an injective resolution of B ′ .Tensoring this diagram with Ω A gives us another commutative diagram of solidarrows(5.2) 0 (cid:15) (cid:15) Ω A ⇁ ⊗ B ′′ (cid:15) (cid:15) Ω A ⊗ B ′ / / (cid:15) (cid:15) Ω A ⊗ B / / (cid:15) (cid:15) Ω A ⊗ B ′′ / / (cid:15) (cid:15) / / Ω A ⊗ I ′ / / (cid:15) (cid:15) Ω A ⊗ I / / (cid:15) (cid:15) Ω A ⊗ I ′′ / / (cid:15) (cid:15) A ⇁ ⊗ Σ B ′ ' ' ' ' ❖❖❖❖❖❖❖❖ Ω A ⊗ Σ B ′ ⊗ γ / / (cid:15) (cid:15) ⊗ ǫ ' ' ◆◆◆◆◆◆◆ Ω A ⊗ Σ B / / (cid:15) (cid:15) x x Ω A ⊗ Σ B ′′ / / (cid:15) (cid:15) A ⊗ I ′ κ : Ω A ⇁ ⊗ B ′′ −→ Ω A ⊗ Σ B ′ with (1 ⊗ γ ) ◦ κ = 0. On the other hand, since I ′ is injective, ǫ extends over γ andtherefore 1 ⊗ ǫ extends over 1 ⊗ γ . Hence (1 ⊗ ǫ ) ◦ κ = 0, and κ factors throughKer (1 ⊗ ǫ ) = Ω A ⇁ ⊗ Σ B ′ . We have thus constructed a map κ : Ω A ⇁ ⊗ B ′′ −→ Ω A ⇁ ⊗ Σ B ′ The same procedure yields maps κ i : Ω i A ⇁ ⊗ Σ i − B ′′ −→ Ω i A ⇁ ⊗ Σ i B ′ for each natural i .The next step, as one would expect, is to show that the maps κ i are compatiblewith the structure maps ∆. Actually, as we will see, this is not true since therequisite squares anticommute rather than commute. This motivates Definition 5.1.1.
For each integer i , set ω i := ( − i κ i . Notice that both the ∆ and the κ are connecting homomorphisms in suitablediagrams. Lemma 5.1.2.
Under the above assumptions and notation, the diagram Ω A ⇁ ⊗ Σ B ′′ ω / / ∆ (cid:15) (cid:15) Ω A ⇁ ⊗ Σ B ′ ∆ (cid:15) (cid:15) Ω A ⇁ ⊗ B ′′ ω / / Ω A ⇁ ⊗ Σ B ′ commutes.Proof. Tensoring the commutative diagram (5.1) with the exact sequence0 −→ Ω A −→ P −→ Ω A −→ , where P is a projective module, we have a spatial commutative diagram satisfyingall conditions of Lemma 4.1.1. In that diagram, the connecting homomorphism onthe front face equals ∆ , and the connecting homomorphism on the bottom faceequals κ = − ω . By the lemma, the composition ω ∆ equals the connectinghomomorphism on the right-hand vertical face.Now we shift all indices in the diagram (5.1) one step up and again tensor itwith the above short exact sequence. The resulting spatial diagram satisfies allconditions of Lemma 4.2.1. In that diagram, the connecting homomorphism on thetop is κ = ω , and the connecting homomorphism on the back equals ∆ . Bythe lemma, the composition ∆ ω equals the connecting homomorphism on theleft-hand vertical face. Since that face coincides with the right-hand vertical faceof the former diagram, we have ω ∆ = ∆ ω . (cid:3) Applying the foregoing lemma repeatedly and passing to the limit, we have thedesired homomorphism ω : T ( A, B ′′ ) −→ T ( A, B ′ ) . As we observed before, the same construction yields homomorphisms ω n : T n ( A, B ′′ ) −→ T n − ( A, B ′ ) . for all integers n . Theorem 5.1.3.
The pair (T • ( A, ) , ω • ) , is a connected sequence of functors.Proof. We already remarked that the asymptotic stabilization of the tensor productis an additive functor. Therefore, given an exact sequence of left Λ-modules0 −→ B ′ α −→ B β −→ B ′′ −→ , we have that the compositionT n ( A, B ′ ) T n ( A,α ) −→ T n ( A, B ) T n ( A,β ) −→ T n ( A, B ′′ ) . of the induced maps is zero. The fact that T n − ( A, α ) ◦ ω n = 0 follows from thesnake lemma applied to the diagram (5.2). For the same reason, ω n ◦ T n ( A, β ) = 0.Thus it remains to show that the ω n are functorial. But this follows from thefunctoriality of the connecting homomorphism in the snake lemma applied to thediagram (5.2). (cid:3) The second construction.
We continue to work with the right Λ-module A and the short exact sequence0 −→ B ′ −→ B −→ B ′′ −→ i are functorial in the second argument. This implies that eachrow of the injective stabilizations in the sequence [7, (9.1)] gives rise tomorphisms of the requisite directed systems.Now we want to build structure maps for each of the Tor terms in [7, (9.1)]. Infact, those maps have already been built in the second construction of the asymp-totic stabilization, see the diagram (3.12) and the diagram on page 10, where B (respectively, B ′ , B ′′ ) should be replaced with Σ B (respectively, Σ B ′ , Σ B ′′ ). AsTheorem 3.2.1 shows, the resulting system is functorial in each argument. There-fore, each row of the Tor functors in the sequence [7, (9.1)] gives rise to mor-phisms of the requisite directed systems.Now we claim that the term-wise maps between the directed systems constitutemorphisms of those systems. The foregoing discussion shows that we only have tocheck the commutativity of the squares lying over the connecting homomorphismsin the sequence [7, (9.1)]. There are three types of such homomorphisms: betweentwo copies of Tor, between Tor and the injective stabilization, and between twocopies of injective stabilization. It is clear that we only have to check one square ofeach type.We first examine the square(s) connecting Tor and the requisite injective sta-bilization. Lemma 5.2.1.
In the above notation, the square
Tor (Ω A, Σ B ′′ ) f (cid:15) (cid:15) h / / Ω A ⇁ ⊗ Σ B ′ k (cid:15) (cid:15) Tor ( A, B ′′ ) g / / A ⇁ ⊗ B ′ anti-commutes. Proof.
We begin by describing the coinitial maps f and h ; they are both connectinghomomorphisms in the following commutative 3D diagram Ω A ⊗ B ′ / / (cid:15) (cid:15) (cid:4) (cid:4) ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ Ω A ⊗ I ′ / / / / (cid:15) (cid:15) (cid:4) (cid:4) ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ Ω A ⊗ Σ B ′ (cid:15) (cid:15) (cid:4) (cid:4) ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ P ⊗ B ′ / / (cid:15) (cid:15) (cid:15) (cid:15) ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ (cid:4) (cid:4) ✠✠✠✠✠✠✠✠ P ⊗ I ′ / / / / (cid:15) (cid:15) (cid:15) (cid:15) ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ (cid:4) (cid:4) ✟✟✟✟✟✟✟✟ P ⊗ Σ B ′ (cid:15) (cid:15) (cid:15) (cid:15) (cid:3) (cid:3) (cid:3) (cid:3) ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ Ω A ⊗ B ′ / / ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ (cid:4) (cid:4) ✠✠✠✠✠✠✠✠ Ω A ⊗ I ′ / / / / (cid:4) (cid:4) ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ (cid:4) (cid:4) ✟✟✟✟✟✟✟✟ Ω A ⊗ Σ B ′ (cid:3) (cid:3) ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ Ω A ⊗ B / / (cid:15) (cid:15) (cid:4) (cid:4) (cid:4) (cid:4) ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ Ω A ⊗ I / / / / (cid:15) (cid:15) (cid:4) (cid:4) (cid:4) (cid:4) ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ Ω A ⊗ Σ B (cid:15) (cid:15) (cid:3) (cid:3) (cid:3) (cid:3) ✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞ P ⊗ B / / (cid:15) (cid:15) (cid:15) (cid:15) ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ (cid:4) (cid:4) (cid:4) (cid:4) ✡✡✡✡✡✡✡✡ P ⊗ I / / / / (cid:15) (cid:15) (cid:15) (cid:15) ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ (cid:4) (cid:4) (cid:4) (cid:4) ✠✠✠✠✠✠✠✠ P ⊗ Σ B (cid:15) (cid:15) (cid:15) (cid:15) (cid:3) (cid:3) (cid:3) (cid:3) ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ Ω A ⊗ B / / ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ (cid:4) (cid:4) (cid:4) (cid:4) ✡✡✡✡✡✡✡✡ Ω A ⊗ I / / / / ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ (cid:4) (cid:4) (cid:4) (cid:4) ✠✠✠✠✠✠✠ Ω A ⊗ Σ B (cid:3) (cid:3) (cid:3) (cid:3) ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ Ω A ⊗ B ′′ / / (cid:15) (cid:15) Ω A ⊗ I ′′ / / / / (cid:15) (cid:15) Ω A ⊗ Σ B ′′ (cid:15) (cid:15) P ⊗ B ′′ / / / / (cid:15) (cid:15) (cid:15) (cid:15) P ⊗ I ′′ / / / / (cid:15) (cid:15) (cid:15) (cid:15) P ⊗ Σ B ′′ (cid:15) (cid:15) (cid:15) (cid:15) Ω A ⊗ B ′′ / / Ω A ⊗ I ′′ / / / / Ω A ⊗ Σ B ′′ The map f is determined by the front face and starts from inside the framed term,and h is determined by the right-hand face of the cube, with its image inside theframed node in the back.Let us now describe the coterminal maps g and k . These maps are both con-necting homomorphisms in the following commutative 3D diagram Ω A ⊗ B ′ / / (cid:15) (cid:15) (cid:5) (cid:5) ☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛ Ω A ⊗ I ′ / / / / (cid:15) (cid:15) (cid:4) (cid:4) (cid:4) (cid:4) ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ Ω A ⊗ Σ B ′ (cid:15) (cid:15) (cid:4) (cid:4) ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ P ⊗ B ′ / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:5) (cid:5) ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ (cid:5) (cid:5) ✡✡✡✡✡✡✡✡ P ⊗ I ′ / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:4) (cid:4) ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ (cid:4) (cid:4) ✠✠✠✠✠✠✠ P ⊗ Σ B ′ (cid:15) (cid:15) (cid:15) (cid:15) (cid:4) (cid:4) ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ A ⊗ B ′ / / ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ (cid:5) (cid:5) ✡✡✡✡✡✡✡✡ A ⊗ I ′ / / / / ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ (cid:4) (cid:4) ✠✠✠✠✠✠✠ A ⊗ Σ B ′ (cid:4) (cid:4) ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ Ω A ⊗ B / / (cid:15) (cid:15) (cid:5) (cid:5) (cid:5) (cid:5) ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ Ω A ⊗ I / / / / (cid:15) (cid:15) (cid:4) (cid:4) (cid:4) (cid:4) ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ Ω A ⊗ Σ B (cid:15) (cid:15) (cid:4) (cid:4) (cid:4) (cid:4) ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ P ⊗ B / / / / (cid:15) (cid:15) (cid:15) (cid:15) ☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛ (cid:5) (cid:5) (cid:5) (cid:5) ☛☛☛☛☛☛☛☛ P ⊗ I / / / / (cid:15) (cid:15) (cid:15) (cid:15) ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ (cid:5) (cid:5) (cid:5) (cid:5) ✡✡✡✡✡✡✡ P ⊗ Σ B (cid:15) (cid:15) (cid:15) (cid:15) (cid:4) (cid:4) (cid:4) (cid:4) ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ A ⊗ B / / ☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛ (cid:5) (cid:5) (cid:5) (cid:5) ☛☛☛☛☛☛☛☛ A ⊗ I / / / / ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ (cid:5) (cid:5) (cid:5) (cid:5) ✡✡✡✡✡✡✡ A ⊗ Σ B (cid:4) (cid:4) (cid:4) (cid:4) ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ Ω A ⊗ B ′′ / / (cid:15) (cid:15) Ω A ⊗ I ′′ / / / / (cid:15) (cid:15) Ω A ⊗ Σ B ′′ (cid:15) (cid:15) P ⊗ B ′′ / / (cid:15) (cid:15) (cid:15) (cid:15) P ⊗ I ′′ / / / / (cid:15) (cid:15) (cid:15) (cid:15) P ⊗ Σ B ′′ (cid:15) (cid:15) (cid:15) (cid:15) A ⊗ B ′′ / / A ⊗ I ′′ / / / / A ⊗ Σ B ′′ The map g is determined by the left-hand face and starts from inside the framedterm, and k is determined by the back face of the cube, with its image inside theframed term in the back.One can easily check that the first cube satisfies the conditions of Lemma 4.1.1,and the second cube satisfies the conditions of Lemma 4.2.1. However, we cannotimmediately apply these results because neither gf nor kh is contained in a singlecube. To bypass this obstacle, notice that the bottom face of the first cube coincideswith the top face of the second cube. Let δ be the connecting homomorphism in thiscommon face (it starts inside Ω A ⊗ B ′′ and ends inside Ω A ⊗ Σ B ′ ). By Lemma 4.1.1, − h = δf and, by Lemma 4.2.1, g = kδ . Therefore gf = − kh , as claimed. (cid:3) Now we look at the square(s) connecting two consecutive Tor functors.
Lemma 5.2.2.
In the above notation, the square
Tor (Ω A, Σ B ′′ ) f (cid:15) (cid:15) h / / Tor (Ω A, Σ B ′ ) k (cid:15) (cid:15) Tor ( A, B ′′ ) g / / Tor ( A, B ′ ) anti-commutes.Proof. The argument in this case is identical to that of the previous lemma, exceptthat, in the two cubes, A has to be replaced by Ω A . The details are left to thereader. (cid:3) Finally, we examine the square(s) connecting two consecutive shifts of the injec-tive stabilizations.
Lemma 5.2.3.
In the above notation, the square Ω A ⇁ ⊗ Σ B ′′ f (cid:15) (cid:15) h / / Ω A ⇁ ⊗ Σ B ′ k (cid:15) (cid:15) A ⇁ ⊗ B ′′ g / / A ⇁ ⊗ B ′ anti-commutes.Proof. The argument is similar to those in the previous two lemmas. To describethe composition gf we use the first cube from the proof of Lemma 5.2.1, where welower the index of each copy of Ω by one. Let δ be the connecting homomorphismin the right-hand face of that cube. By Lemma 4.1.1, gf = − δ .To describe the composition kh we use the second cube from the proof ofLemma 5.2.1, where we raise the index of each copy of Σ by one (in particular, B becomes Σ B , etc). The left-hand face of this cube coincides with the right-hand face of the previous cube, so they share the connecting homomorphism δ . ByLemma 4.2.1, δ = kh , and therefore gf = − kh , as claimed. (cid:3) The just proved results show that, to obtain morphisms of the directed systems,we have to offset the sign in the squares that contain connecting homomorphisms.For example, we can leave the vertical directed systems unchanged, but modifythe horizontal long exact sequences by introducing an alternating (in the verticaldirection) sign for the connecting homomorphisms. In summary, we have
Theorem 5.2.4.
The pair (T • ( A, ) , ρ • ) , where ρ • is the limit of the connectinghomomorphisms modified as above, is a connected sequence of functors. Comparison homomorphisms
At the moment we have three constructions of stable homology: Vogel’s, Triulzi’s,and the asymptotic stabilization of the tensor product. Our next goal is to comparethem.6.1.
Comparing Vogel homology with the asymptotic stabilization.
Wewant to construct a natural transformation from Vogel homology to the asymptoticstabilization of the tensor product. This will be done in degree zero; all otherdegrees are treated similarly. Let A be a right Λ-module with a projective resolution( P, ∂ P ) −→ A , and B be a left Λ-module with an injective resolution B −→ ( I, ∂ I ).Recall that the differential on V • ( A, B ) is induced by ∂ P ⊗ − deg ( ) ⊗ ∂ I (see (2.1)). To simplify notation, we set d P := ∂ P ⊗ d I := 1 ⊗ ∂ I . A homology class in V ( A, B ) can be represented by an infinite sequence s = ( s i ) ∞ i =1 ∈ ( P ⊗ I ) × ( P ⊗ I ) × · · · which vanishes under the differential of V • ( A, B ). This means that D ( s ) = ( d P ( s ) , − d I ( s ) + d P ( s ) , d I ( s ) + d P ( s ) , − d I ( s ) + d P ( s ) , . . . )represents the zero class in V − ( A, B ) and therefore has only finitely many nonzerocomponents. Let k be the smallest index such that(6.1) d I ( s k ) = d P ( s k +1 ) d I ( s k +1 ) = − d P ( s k +2 ) d I ( s k +2 ) = d P ( s k +3 ) d I ( s k +3 ) = − d P ( s k +4 ) . . . In short, for all i ≥ d I ( s k + i ) = ( − i d P ( s k + i +1 )Observe that since s k +1 ∈ P k +1 ⊗ I k , d P ( s k +1 ) ∈ P k ⊗ I k . Denote d P ( s k +1 ) by • in the following commutative diagram with exact rows and columns:Ω k +1 A ⊗ Σ k B (cid:15) (cid:15) / / Ω k +1 A ⊗ I k ◦ (cid:15) (cid:15) / / Ω k +1 A ⊗ Σ k +1 B (cid:15) (cid:15) / / / / P k ⊗ Σ k B ✷ (cid:15) (cid:15) / / P k ⊗ I k • (cid:15) (cid:15) / / P k ⊗ Σ k +1 B (cid:15) (cid:15) / / k A ⊗ Σ k B (cid:15) (cid:15) / / Ω k A ⊗ I k (cid:15) (cid:15) / / Ω k A ⊗ Σ k +1 B (cid:15) (cid:15) Since • = d I ( s k ), pulls back to some element ✷ . Pushing it down, we produce ω k ∈ Ω k A ⊗ Σ k B . Since • = d P ( s k +1 ), the element • is the image of some element ◦ in Ω k +1 A ⊗ I k . By the commutativity of the diagram, the image of ω k in Ω k A ⊗ I k is zero, i.e., ω k ∈ Ω k A ⇁ ⊗ Σ k B , and we set ϕ k := ω k .This process is well-defined up to choice of sign. To see this, notice that theelement • goes to 0 when applying the horizontal map. Hence, by commutativityof the diagram, s k +1 also goes to 0 using the vertical top right map and hence isin the kernel of this map. Now one can apply the map from the snake lemma toproduce the exact same element ω k . Since we may also take the negative of thisconnecting homomorphism, we even have the freedom to choose ± ω k . Once thischoice is fixed, ω k will be well-defined.Next, apply the same procedure to d P ( s k +2 ) = − d I ( s k +1 ) ∈ P k +1 ⊗ I k +1 , pro-ducing ω k +1 ∈ Ω k +1 A ⇁ ⊗ Σ k +1 B . Flipping its sign, we set ϕ k +1 := − ω k +1 . Toobtain ϕ k +2 , perform the same procedure with d P ( s k +3 ) and set ϕ k +2 := − ω k +2 .To obtain ϕ k +3 , perform the same procedure with d P ( s k +4 ) and set ϕ k +3 := ω k +3 .Iterating this process, for any i ≥
0, we set ϕ k + i := ( ω k + i if i ≡ , − ω k + i if i ≡ , . We claim that the sequence ( ϕ k , ϕ k +1 , . . . ) is coherent, i.e., in the notation of (3.4),∆ n ( ϕ n ) = ϕ n − for any n ≥ k + 1. It suffices to check this claim for n = k + 1; theremaining cases are similar. To this end, we examine the commutative diagram P k +1 ⊗ I k '&%$ !" (cid:15) (cid:15) '&%$ !" / / /.-,()*+ T d I ) ) d P " " P k +1 ⊗ Σ k +1 B '&%$ !" (cid:15) (cid:15) '&%$ !" / / P k +1 ⊗ I k +1 • (cid:15) (cid:15) Ω k +1 A ⊗ I k '&%$ !" (cid:15) (cid:15) '&%$ !" / / Ω k +1 A ⊗ Σ k +1 B ϕ k +1 (cid:15) (cid:15) / / Ω k +1 A ⊗ I k +1 P k ⊗ Σ k B '&%$ !" (cid:15) (cid:15) '&%$ !" / / P k ⊗ I k • / / P k ⊗ Σ k +1 B Ω k A ⊗ Σ k B ϕ k Here (1) ◦ (5) = d I , (8) ◦ (6) = d P , and the bullets denote d P ( s k +2 ) = − d I ( s k +1 ) = − (1) ◦ (5)( s k +1 ) and d P ( s k +1 ) . The element ϕ k +1 is obtained from the upper bullet by applying (2) ◦ (1) − and ϕ k is obtained from the lower bullet by applying (4) ◦ (3) − . Since the square T commutes, ϕ k +1 = − (2) ◦ (1) − ( d P ( s k +2 )) = − (7) ◦ (6) ◦ (5) − ◦ (1) − ( d P ( s k +2 )) . On the other hand, recalling the construction of ∆ k +1 (this is just the restrictionof the connecting homomorphism in the diagram (3.3)) we have∆ k +1 ( ϕ k +1 ) = (4) ◦ (3) − ◦ (8) ◦ (7) − ( ϕ k +1 )= − (4) ◦ (3) − ◦ (8) ◦ (7) − ◦ (7) ◦ (6) ◦ (5) − ◦ (1) − ( d P ( s k +2 ))= (4) ◦ (3) − ◦ (8) ◦ (6) ◦ (5) − ◦ (1) − ( d I ( s k +1 ))= (4) ◦ (3) − ◦ (8) ◦ (6)( s k +1 )= (4) ◦ (3) − ( d P ( s k +1 ))= ϕ k . Thus we have shown that the sequence ( ϕ k , ϕ k +1 , . . . ) is coherent. It uniquelyextends to a coherent sequence ( ϕ i ) ∞ i =0 , and we set κ ( s ) := ( ϕ i ) ∞ i =0 . A similarargument yields κ l : V l ( A, ) −→ T l ( A, ) for each integer l . Theorem 6.1.1.
Let A be a right Λ -module. For each l ∈ Z , κ l : V l ( A, ) −→ T l ( A, ) is a natural transformation.Proof. We only need to show the naturality of each κ l . But this follows from thenaturality of the connecting homomorphism. (cid:3) Theorem 6.1.2.
In the above notation, for each l ∈ Z , the natural transformation κ l : V l ( A, ) −→ T l ( A, ) is an epimorphism.Proof. The proof is primarily diagram chase and only a sketch will be given. Let( ϕ , ϕ , ϕ , . . . ) be a coherent sequence in the asymptotic stabilization of the tensorproduct. Then ϕ k ∈ Ω k A ⊗ Σ k B . We will construct an element in V ( A, B ) which maps onto this coherent sequence. One will benefit from the following diagram: P ⊗ I (cid:1) (cid:1) ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄ (cid:15) (cid:15) (cid:15) (cid:15) / / P ⊗ I (cid:15) (cid:15) (cid:15) (cid:15) / / P ⊗ Σ B (cid:15) (cid:15) (cid:15) (cid:15) / / A ⊗ Σ B / / ❴❴❴ (cid:15) (cid:15) ✤✤✤ Ω A ⊗ I (cid:15) (cid:15) ✤✤✤ / / ❴❴❴ Ω A ⊗ Σ B ϕ (cid:15) (cid:15) ✤✤ / / ❴❴❴ / / P ⊗ B / / (cid:15) (cid:15) (cid:15) (cid:15) P ⊗ I / / / / (cid:15) (cid:15) (cid:15) (cid:15) P ⊗ Σ B (cid:15) (cid:15) (cid:15) (cid:15) ✤✤✤ (cid:31) (cid:127) / / ❴❴❴❴ P ⊗ I / / ❴❴❴❴ (cid:15) (cid:15) ✤✤✤ P ⊗ Σ B (cid:15) (cid:15) ✤✤✤ / / ❴❴❴ A ⊗ B (cid:11) (cid:19) + Ω A ⊗ I (cid:11) (cid:19) + + Ω A ⊗ Σ ϕ B (cid:11) (cid:19) / / ❴❴❴ Ω A ⊗ I / / ❴❴❴ Ω A ⊗ Σ B / / ❴❴❴ + P ⊗ B (cid:11) (cid:19) + P ⊗ I (cid:11) (cid:19) + P ⊗ Σ B (cid:11) (cid:19) A ⊗ B ϕ (cid:11) (cid:19) + A ⊗ I (cid:11) (cid:19) + A ⊗ Σ B (cid:11) (cid:19) s ∈ P ⊗ I that maps onto ϕ . Then d P ( s ) will pullbackto ϕ . Now select t ∈ P ⊗ I that maps onto ϕ . By diagram chase we getthat there exists y ∈ P ⊗ I such that d I ( s ) − d P ( t ) = d P ( d I ( y )) which yields d I ( s ) = d P ( t − d I ( y )). Define s := t − d I ( y ). Then s still maps onto ϕ and d P ( s ) pulls back to ϕ .Now select t ∈ P ⊗ I that maps onto − ϕ . Then − d P ( t ) pulls back to ϕ as does d I ( s ). By diagram chasing, there exists y ∈ P ⊗ I such that d I ( s ) + d P ( t ) = d P ( d I ( y )). Define s := t − d I ( y ). Then s maps onto − ϕ so − d P ( s )pulls back to ϕ . Moreover d I ( s ) = − d P ( s ).If we continue this process paying attention to signs, we can construct an element( s k ) ∞ k =1 ∈ V ( A, B ) that maps onto the coherent sequence ( ϕ k ) ∞ k =1 . The details areleft to the reader. (cid:3) Let U be a connected sequence of functors and denote by M • ( U ) its J -completion(see 2.2). In [10, Proposition 6.1.2], Triulzi shows that there is a morphism ofconnected sequences of functors τ : M • ( U ) → U satisfying the following universalproperty. Given any morphism β : V → U , where V is a connected sequence offunctors that is injectively stable in all degrees, there exists a unique morphism φ : V → M • ( U ) such that φτ = β . From this, we can now establish a commutativediagram of comparison maps between Vogel homology, the asymptotic stabilizationof the tensor product, and Triulzi’s J -completion of the functor Tor. Proposition 6.1.3.
For any module A , there is a commutative diagram of con-nected sequences of functors V • ( A, ) κ / / / / θ (cid:15) (cid:15) T • ( A, ) ≃ w w ♥♥♥♥♥♥♥♥♥♥♥♥ λ (cid:15) (cid:15) M • (Tor( A, )) τ / / Tor( A, ) Proof.
The connected sequence of functors V • ( A, ) is J -complete, i.e., injec-tively stable in every degree. The natural transformation τ is the J -completionof Tor( A, ). The morphism θ is induced by the universal property of this ap-proximation applied to the morphism λκ : V • ( A, ) → Tor( A, ). As a result, wehave a commutative square. The diagonal isomorphism is precisely that appearingin Theorem 3.3.2. Under that isomorphism, λ is identified with τ , i.e., the lowertriangle commutes. Since κ is epic, the upper triangle also commutes. (cid:3) Corollary 6.1.4.
The comparison map from Vogel homology to the J -completionof Tor is epic in each degree. (cid:3)
In view of the foregoing result, it is natural to try and identify the kernel of κ : V • ( A, ) −→ T • ( A, ) or, equivalently, of θ : V • ( A, ) −→ M • (Tor( A, )).Driven by a formal analogy between κ and the natural transformation from Steenrod-Sitnikov homology to ˇCech homology, the first author conjectured in 2014 that thekernel of κ should be given by a derived limit. The following recent result of I. Em-manouil and P. Manousaki shows that this is indeed the case. Theorem 6.1.5 ([4], Theorem 2.2) . There is an exact sequence −→ lim ←− i Tor • + i +1 ( A, Σ i ) −→ V • ( A, ) −→ M • (Tor( A, )) −→ . References [1] K. S. Brown.
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A functorial approach to linkage and the asymptotic stabilization of the tensorproduct . ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)–Northeastern University.[10] M. E. Triulzi. Completion constructions in homological algebra and finiteness conditions.ProQuest LLC, Ann Arbor, MI, 1999. Thesis (Dr.sc.math.)–Eidgenoessische TechnischeHochschule Zuerich (Switzerland). Mathematics Department, Northeastern University, Boston, MA 02115, USA
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