INJECTIVE STABILIZATION OF ADDITIVE FUNCTORS. III. ASYMPTOTIC STABILIZATION OF THE TENSOR PRODUCT 7 1 ALEXMARTSINKOVSKYANDJEREMYRUSSELL 0 2 b Abstract. The injective stabilization of the tensor product is subjected to e an iterative procedure that utilizes its bifunctor property. The limit of this F procedure, calledtheasymptotic stabilizationofthetensor product, provides a homological counterpart of Buchweitz’s asymptotic construction of stable 0 cohomology. The resulting connected sequence of functors is isomorphic to 1 Triulzi’sJ-completion ofthe Tor functor. A comparisonmap fromVogel ho- mology to the asymptotic stabilization of the tensor product is constructed ] T andshowntobealwaysepic. R . h t a Contents m 1. Stable cohomology 1 [ 1.1. Vogel cohomology 2 2 1.2. Buchweitz cohomology 2 v 1.3. Mislin’s construction 3 8 2. Stable homology 3 6 2.1. Vogel homology 3 2 0 2.2. A homological analog of Mislin’s construction 4 0 2.3. Summary 4 . 3. The asymptotic stabilization of the tensor product 4 1 0 3.1. The first construction 5 7 3.2. The second construction 9 1 3.3. The third construction 12 : v 4. Two lemmas on connecting homomorphisms 15 i 4.1. Front, bottom, and right-hand faces 15 X 4.2. Top, back, and left-hand faces 17 r a 5. The asymptotic stabilization as a connected sequence of functors 18 5.1. The first construction 19 5.2. The second construction 21 6. Comparison homomorphisms 24 6.1. Comparing Vogel homology with the asymptotic stabilization 24 References 28 1. Stable cohomology Around1950,JohnTatenoticedthatthetrivialmodulekoverthegroupringkG (where k is a field or the ring of rational integers) of a finite group G admits a Date:February13,2017, 8h13min. 1 2 ALEXMARTSINKOVSKYANDJEREMYRUSSELL projective coresolution. Splicing it together with a projective resolution of the same module, he obtained a doubly infinite exact complex of projectives, called a complete resolution of k. Using it in place of the projective resolution of k, he modified the usual notion of group cohomology, obtaining what is now known as Tate cohomology. For more details, the reader is referred to [1] and [3]. In 1977, F. T. Farrell [4] constructed a cohomology theory for groups of finite virtualcohomologicaldimensionthat,forfinitegroups,gavethesameresultasTate cohomology. In the mid-1980s,R.-O. Buchweitz [2] constructed a generalizationof Tate (and Farrell) cohomology that workedover arbitrary Gorenstein rings. 1.1. Vogel cohomology. At about the same time, Pierre Vogel [5] came up with hisowngeneralizationofTatecohomology,andwhilehewasinterestedinarbitrary group rings, his approach actually worked over any ring. We now review that construction. Let Λ be a (unital) ring and M and N (left) Λ-modules. Choose projective resolutions (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) of abelian groups. The latter has Hom(P ,Q ) as its degree n component. It i i i+n contains the subdiagram (P,Q) of bounded maps, whose degree n component is b Q Hom(P ,Q ). Passing to the quotient, we have a short exact sequence of i i i+n diagrams ` 0−→(P,Q) −→(P,Q)−→(P[,Q)−→0 b Theusualformula,D(f):=∂◦f−(−1)degff◦∂,defines adifferentialonthe mid- dle diagram,which clearly restricts to a differential on the subdiagramof bounded maps. Thus the inclusion map is actually an inclusion of complexes, and the cor- responding quotient becomes the quotient complex. By construction, the maps in this short exact sequence are chain maps between the constructed complexes. The nthVogelcohomologyofM with coefficientsin N,wheren∈Z, isthen definedas the nth cohomology of the complex (P[,Q). We denote it by Vn(M,N). 1.2. Buchweitz cohomology. Aswementionedbefore,Buchweitzwasinterested in 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 his approach. 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 pass to 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 homomorphisms is defined as the class ofthe compositionof representatives. One ofthe advantages of 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)−→(ΩM,ΩN)−→(Ω2M,Ω2N)−→... The nth Buchweitz cohomology Bn(M,N), n∈Z is defined as lim (Ωn+kM,ΩkN). −→ n+k,k≥0 3 1.3. Mislin’s construction. Yet another generalization of Tate cohomology was given by G. Mislin [7] in 1994. It came as a special case of a considerably more generalconstruct. For a cohomological(or, more generally, connected) sequence of functors {Fi}, i∈Z Mislin constructs a sequence of natural transformations Fi −→S (Fi+1)−→S (Fi+2)−→..., 1 2 where S denotes the jth left satellite, and defines what he calls the P-completion j of {Fi} as limS (Fi+k)=:MiF. −→ k k≥0 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 original sequence is connected but not necessarily cohomological)sequence of functors. He then proves that, for groups of finite virtual cohomologicaldimension, the new co- homology is isomorphic to Farrell cohomology. Moreover, he also establishes, for arbitrary groups, an isomorphism between his construction and Buchweitz’s coho- mology(calledinthepapertheBenson-Carlsoncohomology,afterthetwoauthors, who independently found Buchweitz’s cohomology in 1992). It should be clear, however,thatMislin’sconstructioniscompletelygeneralandapplies,inparticular, to the Ext functor over any ring. 2. Stable homology At this point, one may ask if there are homological analogs of the various cohomology theories discussed above. The answer to this question is less clear. First,therewasno“Tatehomology”inTate’soriginalwork: onlytheHomfunctor was used with complete resolutions. However, at the same time when P. Vogel constructed his cohomology, he also constructed a homology theory. We begin by reviewing 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⊗I of abelian groups. Thelatterhas (P ⊗Ii−n)asitsdegreencomponent. Itcontainsthesubdiagram i i P⊗I, whose dQegree n component is i(Pi ⊗Ii−n). Passingb to the quotient, we have a short exact sequence of diagrams ` ∨ 0−→P⊗I−→P⊗I−→P⊗I−→0 The standard definition b D(a⊗b):=∂ (a)⊗b+(−1)degaa⊗∂ (b) P I (2.1) = ∂P ⊗1+(−1)deg1( )1⊗∂I (a⊗b), where a and b are homogene(cid:0)ous elements of P and, resp(cid:1)ectively, I, and deg ( ) 1 picks the degree of the first factor of a decomposable tensor, gives rise to a differ- entialonP⊗I. Itis notdifficult to seethatitextends to adifferential,denotedby D again, on P⊗I. Indeed, if s ∈ (P⊗I)n is a degree n element, then s = (si)i∈Z, b b 4 ALEXMARTSINKOVSKYANDJEREMYRUSSELL where each s ∈ P ⊗Ii−n is just a finite sum of decomposable tensors. For each i i k ∈Z, define D : (P ⊗Ii−n)−→(P ⊗Ik+1−n):s7→(∂⊗1)(s )+(−1)k(1⊗∂)(s ) i k k+1 k i Y Now,weobtainthe desireddifferentialby the universalpropertyofdirectproduct. As a consequence, the third term in the short exact sequence above becomes a complex, and Vogel homology is now defined by setting ∨ (2.2) V (M,N):=H (P⊗I). n n+1 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 (M,N)−→Tor (M,N). n n Remark 2.1.2. The choice of the projective and injective resolutions can be flipped. By choosingan injective resolutionof M and a projective resolutionof N, one obtains another homological functor, which in general is different from the original one. This can be seen by choosing M to be projective. In that case, the original 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 analog of Mislin’s cohomological P-completion, called the J-completion, was defined by M. Triulzi in his PhD thesis [9]1. Like its cohomologicalprototype, it is defined on connectedsequencesoffunctors,butevenif theoriginalsequence iscohomological, the resultdoesn’t seemto be cohomological2; one canonly claimthatthe resulting sequence is connected. For reference, we denote it by M F. i 2.3. Summary. We summarize the existing constructions in the following table: Cohomology Homology Vi(M,N) V (M,N) i Bi(M,N) ? MiF M F i One of the goals of this paper is to replace the question mark by a homological analog of Buchweitz’s construction. In this paper, we follow the terminology and notation established in [6]. The reader may benefit from reviewing that source. Some results contained in the present paper overlap with some results obtained by the second author in his PhD thesis [8]. 3. The asymptotic stabilization of the tensor product Ournextgoalistointroducewhatweshallcalltheasymptotic stabilization of the tensor product, which is a limit of a sequence of maps between injective stabilizations of tensor products of iterated syzygy and cosyzygy modules. In this section, this will be done in three equivalent ways. 1The authors are grateful to Lucho Avramov for bringing this work to our attention and to LarsChristensenforsendingusacopyofit 2Thisisrelatedtothefactthattheinverselimitisnotanexactfunctor. 5 Blanket assumption. Whenever we deal with a connecting homomorphism in the snake lemma, we automatically assume that the homomorphism was con- structed by pushing and pulling the elements along a staircase path, as in the traditional proof of the lemma. 3.1. The first construction. We begin with constructing a homomorphism ⇁ ⇁ ΩA ⊗ ΣB −→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 −→I0 −→I1 −→... Similarly, given a right Λ-module A choose a projective resolution (3.2) ...−→P −→P −→A−→0 1 0 Tensoring the short exact sequences 0−→ΩA−→P −→A−→0 and 0−→B −→I0 −→ΣB −→0 0 we have the following commutative diagram of solid arrows whose rows, columns, and diagonal are exact: 0 0 ▼▼▼▼▼▼▼▼▼▼▼&& ⇁ (cid:15)(cid:15) ΩA ⊗ ΣB // //Tor (A,ΣB) ❖❖❖❖❖❖❖❖❖❖❖❖❖''1 (cid:15)(cid:15) ΩA⊗B //ΩA⊗I0 //ΩA⊗ΣB //0 (3.3) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) ◆◆◆◆◆◆◆◆◆◆◆'' 0 // P ⊗B // P ⊗I0 ////P ⊗ΣB ΩA⊗I1 0 (cid:15)(cid:15) 0 (cid:15)(cid:15) 0 (cid:15)(cid:15) ◆'' ◆◆◆◆◆◆◆◆◆◆'' (cid:15)(cid:15) ... // A⊗B //A⊗I0 //// A⊗ΣB P ⊗I1 0 (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) 0 0 0 Using the fact that P ⊗ is an exact functor and the snake lemma, we have 0 Lemma 3.1.1. The above solid diagram induces an exact sequence Tor (A,B)−→Tor (A,I0)−→Tor (A,ΣB)−δ→A⇁⊗B −→0, 1 1 1 where δ is (the corestriction of) the connecting homomorphism. If the injective I0 is projective, then δ is an isomorphism. (cid:3) As P ⊗ is an exact functor, the bottom southeast map is monic. The com- 0 position of this map with ⇁ ΩA ⊗ ΣB //ΩA⊗ΣB // P ⊗ΣB 0 6 ALEXMARTSINKOVSKYANDJEREMYRUSSELL is obviously zero and, by the universal property of kernels, we have the dotted mapinthe abovediagrammakingthe toptrianglecommute. Notice thatthis map is monic. Applying the snake lemma yields the following diagram with an exact bottom row ⇁ ΩA ⊗ ΣB (cid:15)(cid:15) Tor (A,ΣB) //A⊗B // A⊗I0 1 Because the composition of the bottom maps is zero, this diagram embeds in the commutative diagram ⇁ ΩA ⊗ ΣB (cid:15)(cid:15) Tor (A,ΣB) //A⊗B //A⊗I0 1 δ (cid:15)(cid:15) ⇁ 0 //A ⊗ B //A⊗B // A⊗I0 which produces a homomorphism ΩA ⇁⊗ ΣB ∆1 // A⇁⊗B Iteration of this process yields a directed system (3.4) ... //Ω2A ⇁⊗ Σ2B ∆2 //ΩA ⇁⊗ ΣB ∆1 // A ⇁⊗ B. Now we want to show that any two choices for ∆ , and hence for any other ∆ , 1 i are isomorphic. In addition to the resolutions (3.1) and (3.2), let (3.5) 0−→B −→I0′ −→I1′ −→... be another injective resolution of B, and (3.6) ...−→P ′ −→P ′ −→A−→0 1 0 another projective resolution of A. Lifting the identity map on A, extending the identity map on B, and taking the tensor product results in a commutative 3D version of diagram (3.3). By the naturality of the connecting homomorphism in 7 the snake lemma, we have a commutative diagram with exact rows ... //Tor (A,Σ′B) // A⊗B //A⊗I0′ // ... 1 ▲▲▲▲▲▲▲▲▲▲&&&& ✈;; ✈✈✈✈✈✈✈;; ⇁′ A⊗ B α ∼= (cid:15)(cid:15) (cid:15)(cid:15) ... //Tor (A,ΣB) // A⊗B // A⊗I0 //... 1 ▼▼▼▼▼▼▼▼▼▼&&&& (cid:15)(cid:15) ✈:: ✈✈✈✈✈✈✈✈:: ⇁ A⊗B By[6,Lemma3.2)], αis anequality. Onthe otherhand,the right-handsideofthe 3D-versionof (3.3) yields a commutative diagram of solid arrows -- Tor (A,Σ′B) -- vv♠♠♠♠♠♠♠♠ 1 (cid:15)(cid:15) Tor (A,ΣB) 1 (cid:15)(cid:15) 0 // Ω′A ⇁⊗ Σ′B //Ω′A⊗(cid:15)(cid:15) Σ′B // Ω′A⊗I1′ ⇁ xx♣∼=♣♣♣♣♣♣β (cid:15)(cid:15) ww♥♥♥♥♥♥♥♥♥♥ yyssssssss 0 // ΩA ⊗ ΣB //ΩA⊗ΣB // ΩA⊗I1 (cid:15)(cid:15) (cid:15)(cid:15) P ′⊗Σ′B // // P ′⊗I1′ 0 0 (cid:15)(cid:15) vv♠♠♠♠♠♠♠♠♠ (cid:15)(cid:15) xxqqqqqqq P ⊗ΣB // //P ⊗I1 0 0 withexactrowsandcolumns. Thedottedarrowsalsocomefromdiagram(3.3)and makethe trianglescontainingthemcommute. By[6,Lemmas8.1and8.2],β isthe canonicalisomorphism. UsingthefactthatthemapTor (A,ΣB)−→ΩA⊗ΣBisa 1 monomorphism,wehavethatthecurvedsquarealsocommutes. Splicingitwiththe left-hand squarecontaining α fromthe preceding diagram,we have a commutative square Ω′A ⇁⊗ Σ′B ∆′1 //A ⇁⊗′ B ∼= β ∼= α (cid:15)(cid:15) (cid:15)(cid:15) ΩA ⇁⊗ ΣB ∆1 //A ⇁⊗ B with the vertical maps being canonical isomorphisms. This proves Proposition 3.1.2. Any two choices for ∆ , and hence for any ∆ , based on the 1 i diagram (3.3) are canonically isomorphic. (cid:3) Arguments very similar to the ones just used yield 8 ALEXMARTSINKOVSKYANDJEREMYRUSSELL ⇁ ⇁ Proposition 3.1.3. The homomorphism ∆ : ΩA ⊗ ΣB −→ A ⊗ B, and hence 1 any ∆ , is functorial in both A and B. (cid:3) i For any integer n ∈ Z (including negative values), the process of constructing the directed system (3.4) may be repeated with Ωk+nA in place of ΩkA, yielding directed systems ⇁ (3.7) M (A,B):=Ωk+nA ⊗ ΣkB, k,k+n≥0 n Definition 3.1.4. The asymptotic stabilization T (A, ) of the left tensor product n in degree n with coefficients in the right Λ-module A is T (A, )(B):=T (A,B) n n (3.8) := lim Ωk+nA ⇁⊗ ΣkB =limM (A,B) ←− ←− n k,k+n≥0 It is easy to see that the T (A, ) : Λ-Mod −→ Ab, n ∈ Z are covariant n additive functors from the category of left Λ-modules to the category of abelian groups. It is plain that the T (A, ) are injectively stable. The next result shows n that 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- ≥0 ≥0 phisms of functors Tn(A,Σk )∼=Tn−k(A, ) and Tn(ΩjA, )∼=Tn+j(A, ) Proof. The directed systems (including the structure maps) for the components of the former (respectively, latter) pair of functors at any right Λ-module can be obviously chosen to be shifts of each other. (cid:3) Now we wantto discuss the vanishing of the functors T (A, ). The first result • is an an immediate consequence of the definitions. Proposition 3.1.6. If the right global dimension of Λ is finite then T (A, )=0 n for all integers n. (cid:3) Proposition 3.1.7. If the flat dimension of A is finite, then T (A, )=0 for all n integers n. ⇁ Proof. Asthediagram(3.3)shows,wehaveaninjectionΩA ⊗ ΣB −→Tor (A,ΣB). 1 ⇁ In particular, Ωn+kA ⊗ ΣkB, n+k,k ≥ 1 embeds in Tor (Ωn+k−1A,ΣkB). But 1 the latter vanishes for n+k−1≥fl.dim A. (cid:3) It is known that the vanishing of stable cohomology in one degree implies its vanishinginalldegrees. WedonotknowifasimilarstatementistrueforT (A, ). • A partial answer is provided by Proposition 3.1.8. If T (A, ) = 0 for some integer n, then T (A, ) = 0 for n m all m<n. If, in addition, Λ is quasi-Frobenius, then T (A, )=0 for all m∈Z. m Proof. The first assertion is an immediate consequence of the first isomorphism of Lemma 3.1.5. Suppose now that Λ is quasi-Frobenius. Since projective modules 9 are injective, for any positive integer k, any right Λ-module B is a kth cosyzygy module in an injective resolution of ΩkB, i.e., B ≃ΣkΩkB. Therefore, Tn+k(A,B)∼=Tn+k(A,ΣkΩkB)∼=Tn(A,ΩkB)=0. (cid:3) 3.2. The second construction. NextwewanttoshowthatProposition3.1.7fol- lowsfromamoregeneralresult,namely,thattheasymptoticstabilizationT (A,B) • can be computed via the Tor functors. Our goal is to construct a commutative di- agram ... //Tor (ΩA,Σ2B) // Tor (A,ΣB) // A⊗B ③③③③③1③③<< ❑❑❑❑❑❑❑❑%%%% ::✉✉✉✉✉✉✉✉1:: ❊❊❊❊❊❊❊"""" ✁@@✁✁✁✁✁✁@@ ..<<③.③③ // ΩA⇁⊗ΣB //A⇁⊗B wherethebottomsequenceisgivenby(3.4),andthearrowsinthetopsequenceare connecting homomorphisms. Clearly, once such a diagram has been constructed, the limits of the horizontal rows will be isomorphic, showing that the asymptotic stabilization can indeed be constructed using the Tor functors. Moreover,we shall alsoshowthatallnortheastarrowsaremonicandallsoutheastarrowsareepic. This immediately implies that all stages in the original construction of the asymptotic stabilization can be recoveredvia the epi-mono factorizations of the top arrows. The construction requires explicit choices, so for a right Λ-module A we choose a projective resolution ...−→P −→P −→A−→0 1 0 and use the definition of Tor (A, ) via the exact sequence 1 (3.9) 0−→Tor (A, )−→ΩA⊗ −→P ⊗ −→A⊗ −→0. 1 0 ForaprojectiveresolutionofΩAwechoosetheprojectiveresolutionofAtruncated in degree 1. This allows us to claim that Tor (A, ) = Tor (ΩA, ), where we i+1 i do mean an equality rather than an abstract isomorphism. For a short exact sequence (3.10) 0−→C −→D −→E →0 of left A-modules, recall the construction the connecting homomorphisms Tor (A,E)−→Tor (A,C) i+1 i in the corresponding long exact sequence of the Tor functors. The case i = 0 consists of evaluating the sequence (3.9) on the short exact sequence above and then using the snake lemma. For positive values of i, we describe the construction when i = 1 and then use the dimension shift. To this end, we replace A with ΩA and build a snake diagram as in the case i=0. The new diagram and the original one have a common row, ΩA⊗C −→ΩA⊗D −→ΩA⊗E −→0, 10 ALEXMARTSINKOVSKYANDJEREMYRUSSELL which allows to glue the two diagrams together: Tor1(ΩA,C) //Tor1(ΩA,D) //Tor1(ΩA,E) ✞ ✞ ✠ ✞ ✞ ✠ ✞ ✞ ✠ ✞ ✞ ✠ ✞ ✞ ✠ ✞ ✞ ✠ ✞ ✞ ✠ ✞ ✞ ✠ ✞ ✞ ✠ ✞ ✞ ✠ ✞ ✞ ✠ ✞ ✞ ✠ (cid:3)(cid:3)✞✞ (cid:3)(cid:3)✞✞ (cid:4)(cid:4)✠✠ Ω2A⊗C //Ω2A⊗D //Ω2A⊗E ☞ ✞ ✠ ☞ ✞ ✠ ☞ ✞ ✠ ☞☞☞☞☞☞☞ ✞✞✞✞✞✞✞✞ ✠✠✠✠✠✠✠✠ ☞ ✞ ✠ (cid:5)(cid:5)☞☞ (cid:3)(cid:3)✞✞ (cid:4)(cid:4)✠✠ P1⊗C //P1⊗D //P1⊗E ✍ ✡ ✡ ✍ ✡ ✡ ✍ ✡ ✡ Tor1((cid:15)(cid:15)A,C✍✍✍)✍✍✍✍✍ //Tor1(A(cid:15)(cid:15) ,D✡✡)✡✡✡✡✡✡✡ //Tor1((cid:15)(cid:15)A,E✡✡)✡✡✡✡✡✡✡ ✍ ✡ ✡ ✍ ✡ ✡ ΩA⊗(cid:15)(cid:15) (cid:7)(cid:7)✍✍C γ //ΩA⊗(cid:15)(cid:15) (cid:4)(cid:4)✡✡D ////ΩA⊗(cid:15)(cid:15) (cid:4)(cid:4)✡✡E α T δ (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) P0⊗C // //P0⊗D ////P0⊗E β (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) A⊗C //A⊗D ////A⊗E Notice that the connecting homomorphism ǫ Tor (A,E)=Tor (ΩA,E)−→ΩA⊗C 2 1 in the horizontal part of the diagram factors through Kerα = Tor (A,C). In- 1 deed, the commutativity of the square T shows that βαǫ = δγǫ = 0. Since β is monic, αǫ=0 andǫ factorsthroughTor (A,C). As a result, we have a connecting 1 homomorphism Tor (A,E)−→Tor (A,C) and the desired long exact sequence. 2 1 Returning to the left Λ-module B, we specialize the short exact sequence (3.10) to the cosyzygy sequence (3.11) 0−→ΣB −→I1 −→Σ2B −→0. The foregoing argument then yields a commutative square Tor (A,Σ2B) // Tor (A,ΣB) 2 ◗◗◗◗◗◗◗◗◗◗◗◗◗(( 1 (cid:15)(cid:15)(cid:15)(cid:15) Tor (ΩA,Σ2B) //ΩA⊗ΣB 1 where the diagonalmap is the connecting homomorphismin the horizontalpart of thediagramonpage10. Thecompositionofthismapwithγ :ΩA⊗ΣB →ΩA⊗I1 ⇁ is zero, hence it factors through the kernel of γ, which is by definition ΩA ⊗ ΣB.