aa r X i v : . [ m a t h . K T ] M a y EXCISION IN ALGEBRAIC K -THEORY REVISITED GEORG TAMME
Abstract.
By a theorem of Suslin, a Tor-unital (not necessarily unital)ring satisfies excision in algebraic K -theory. We give a new and directproof of Suslin’s result based on an exact sequence of categories of perfectmodules. In fact, we prove a more general descent result for a pullbacksquare of ring spectra and any localizing invariant. Besides Suslin’sresult, this also contains Nisnevich descent of algebraic K -theory foraffine schemes as a special case. Moreover, the role of the Tor-unitalitycondition becomes very transparent. Introduction
One of the main achievements in the algebraic K -theory of rings is thesolution of the excision problem, first rationally by Suslin–Wodzicki [SW92]and later integrally by Suslin [Sus95]: For a two-sided ideal I in a unitalring A one defines the relative K -theory spectrum K ( A, I ) as the homotopyfibre of the map of K -theory spectra K ( A ) → K ( A/I ), so that its homotopygroups K ∗ ( A, I ) fit in a long exact sequence · · · → K i ( A, I ) → K i ( A ) → K i ( A/I ) → K i − ( A, I ) → . . . If I is a not necessarily unital ring, one defines K ∗ ( I ) := K ∗ ( Z ⋉ I, I )where Z ⋉ I is the unitalization of I . For every unital ring A containing I as a two-sided ideal there is a canonical map Z ⋉ I → A . It induces amap K ∗ ( I ) → K ∗ ( A, I ) and one says that I satisfies excision in algebraic K -theory if this map is an isomorphism for all such A .Equivalently, I satisfies excision in algebraic K -theory if, for every ring A containing I as a two-sided ideal and any ring homomorphism A → B sending I isomorphically onto an ideal of B , the pullback square of rings A / / (cid:15) (cid:15) A ′ (cid:15) (cid:15) B / / B ′ (1)where A ′ = A/I , B ′ = B/I induces a homotopy cartesian square of non-connective K -theory spectra K ( A ) / / (cid:15) (cid:15) K ( A ′ ) (cid:15) (cid:15) K ( B ) / / K ( B ′ ) . (2) The author is supported by the CRC 1085
Higher Invariants (Universit¨at Regensburg)funded by the DFG.
A ring I is called Tor-unital if Tor Z ⋉ Ii ( Z , Z ) = 0 for all i >
0. Everyunital ring is Tor-unital, since if I is unital, then Z ⋉ I ∼ = Z × I and theprojection to Z is flat. Theorem 1 (Suslin) . If I is Tor-unital, then I satisfies excision in algebraic K -theory. In fact, both statements are equivalent [Sus95, Thm. A]. For Q -algebras,this was proven before by Suslin and Wodzicki [SW92, Thm. A]. Wodzicki[Wod89] gives many examples of Tor-unital Q -algebras, for instance all C ∗ -algebras. These results are the main ingredients in the proof of Karoubi’sconjecture about algebraic and topological K -theory of stable C ∗ -algebrasin [SW92, Thm. 10.9]. On the other hand, by work of Morrow [Mor14] ideals I in commutative noetherian rings are pro-Tor-unital in the sense that thepro-groups { Tor Z ⋉ I n i ( Z , Z ) } n vanish for all i > K -theoryin terms of Quillen’s plus-construction and relies on a careful study of thehomology of affine groups. By completely different methods we prove thefollowing generalization of Theorem 1: Theorem 2.
Assume that (1) is a homotopy pullback square of ring spectrasuch that the multiplication map A ′ ⊗ A A ′ → A ′ is an equivalence. Then thesquare (2) of non-connective K -theory spectra is homotopy cartesian. Here the tensor denotes the derived tensor product, and K -theory is thenon-connective K -theory of perfect modules. Example . Assume that (1) is a diagram of discrete rings. When viewed asa diagram of ring spectra, this is a homotopy pullback square if and only ifthe induced sequence of abelian groups0 → A → A ′ ⊕ B → B ′ → A ′ ⊗ A A ′ → A ′ is an equivalence if andonly if Tor Ai ( A ′ , A ′ ) = 0 for all i > A ′ with itself over A is isomorphic to A ′ via the multiplication.There are two basic cases where both conditions are satisfied: The firstone is that A ′ = A/I for a Tor-unital two-sided ideal I in A (see Example 24).This gives Suslin’s result. The second one is that (1) is an elementary affineNisnevich square, i.e., all rings are commutative, A ′ = A [ f − ] is a localiza-tion of A , A → B is an ´etale map inducing an isomorphism A/ ( f ) ∼ = B/ ( f ),and B ′ = B [ f − ] (see Example 25). Note that by [AHW17, Prop. 2.3.2]the family of coverings of the form { Spec( A [ f − ]) → Spec( A ) , Spec( B ) → Spec( A ) } generate the Nisnevich topology on the category of affine schemes(of finite presentation over some base). Thus Theorem 2 also implies Nis-nevich descent for the algebraic K -theory of affine schemes.In general, the condition that A ′ ⊗ A A ′ → A ′ be an equivalence is equiv-alent to LMod( A ) → LMod( A ′ ) being a localization, where LMod denotesthe ∞ -category of left modules in spectra. In particular, under this con-dition LMod( A ′ ) is a Verdier quotient of LMod( A ). The usual methodthat is used, for example, to produce localization sequences in K -theory XCISION IN ALGEBRAIC K -THEORY REVISITED 3 (see [Sch11, §
3] for an overview, [NR04, Thm. 0.5] for the case of a non-commutative localization where a similar condition on Tor-groups appears),would be to apply Neeman’s generalization of Thomason’s localization the-orem [Nee92, Thm. 2.1] in order to deduce that also the induced functor onthe subcategories of compact objects, which are precisely the perfect mod-ules, Perf( A ) → Perf( A ′ ) is a Verdier quotient. However, Neeman’s theoremdoes not apply here, since the kernel of LMod( A ) → LMod( A ′ ) need not becompactly generated. Indeed, there is an example by Keller [Kel94, §
2] of aring map A → A ′ satisfying the hypotheses of Theorem 2, where this kernelhas no non-zero compact objects at all and Perf( A ′ ) is not a Verdier quotientof Perf( A ).Instead, under the conditions of Theorem 2 we prove a derived version ofMilnor patching (Theorem 26) saying that (1) induces a pullback diagramof ∞ -categories of left modules, i.e.,LMod( A ) ≃ LMod( A ′ ) × LMod( B ′ ) LMod( B ) . Its proof is inspired by a similar patching result for connective modulesover connective ring spectra due to Lurie [Lur17b, Thm. 16.2.0.2]. We usethis to show that LMod( A ) can be embedded as a full subcategory in thelax pullback LMod( A ′ ) × → LMod( B ′ ) LMod( B ) (see Section 1) and to identifythe Verdier quotient with LMod( B ′ ). Now the Thomason–Neeman theoremapplies and gives an exact sequence of small stable ∞ -categories(3) Perf( A ) i −→ Perf( A ′ ) × → Perf( B ′ ) Perf( B ) π −→ Perf( B ′ ) , i.e., the composite π ◦ i is zero and the induced functor from the Verdierquotient of the middle term by Perf( A ) to Perf( B ′ ) is an equivalence upto idempotent completion. This implies the assertion of Theorem 2 notonly for algebraic K -theory, but for any invariant which can be defined forsmall stable ∞ -categories and which sends exact sequences of such to fibresequences. In fact, in Section 1 we prove the existence of the analog ofthe exact sequence (3) for any so-called excisive square of small stable ∞ -categories (Theorem 15). In Section 2 we then prove that any square of ringspectra satisfying the hypotheses of Theorem 2 yields an excisive square of ∞ -categories of perfect modules (see Theorem 28). These are the two mainresults of the paper. Remark . The failure of excision in K -theory is measured in (topologi-cal) cyclic homology: Corti˜nas [Cor06] proved that the fibre of the rationalGoodwillie–Jones Chern character from rational algebraic K -theory to neg-ative cyclic homology satisfies excision, i.e., sends the pullback square ofrings (1) with B → B ′ surjective to a homotopy pullback square of spec-tra without any further condition. Geisser and Hesselholt [GH06] provedthe analogous result with finite coefficients, replacing the Goodwillie–JonesChern character by the cyclotomic trace map from K -theory to topologicalcyclic homology. Both use pro versions of the results of Suslin and Wodz-icki. Building on these results, Dundas and Kittang [DK08, DK13] provethat the fibre of the cyclotomic trace satisfies excision also for connectivering spectra, and with integral coefficients (under the technical assumptionthat both, π ( B ) → π ( B ′ ) and π ( A ′ ) → π ( B ′ ) are surjective). GEORG TAMME
In this general situation, i.e., without assuming any Tor-unitality con-dition, one still has the sequence (3), but the induced functor f from theVerdier quotient to Perf( B ′ ) need not be an equivalence up to idempotentcompletion. It would therefore be interesting to find conditions on an in-variant E that guarantee that E ( f ) is still an equivalence. From the resultsmentioned above we know that E ( f ) is an equivalence for E the fibre of thecyclotomic trace.We use ∞ -categorical language. More concretely, we use the model ofquasi-categories, which are the fibrant objects for the Joyal model structureon simplicial sets, as developed by Joyal [Joy08] and Lurie in his books[Lur09, Lur17a, Lur17b]. Acknowledgements.
I would like to express my sincere gratitude to thereferee for the efforts taken to improve both the exposition and the resultsof this paper. The referee gave a hint which led to a simplification of theproof of the main result of the first version of this paper, and also suggestedto formulate the general categorical Theorem 18 in terms of excisive squaresand to deduce the excision result via Theorem 28. I would also like tothank Justin Noel and Daniel Sch¨appi for discussions about (lax) pullbacksof ∞ -categories.1. Pullbacks and exact sequences of stable ∞ -categories In this section, we discuss the pullback and the lax pullback of a diagram A → C ← B of ∞ -categories. In the stable case, we relate these by exactsequences. We further prove our first main result (Theorem 18) saying thatany excisive square of small stable ∞ -categories (see Definition 14) yields apullback square upon applying any localizing invariant.Let I = ∆[1] ∈ sSet be the standard simplicial 1-simplex. For any ∞ -category C , we denote by C I = Fun( I, C ) the arrow category of C . Theinclusion { , } ⊆ I induces the source and target maps s, t : C I → C .Consider a diagram of ∞ -categories B q (cid:15) (cid:15) A p / / C. (4) Definition 5.
The lax pullback A × → C B of (4) is defined via the pullbackdiagram A × → C B (pr , pr ) (cid:15) (cid:15) pr / / C I ( s,t ) (cid:15) (cid:15) A × B p × q / / C × C (5)in simplicial sets. We give a sufficient condition in a forthcoming article with Markus Land: it sufficesthat the natural map E ( C ) → E ( π ( C )) is an equivalence for any connective E -ringspectrum C . XCISION IN ALGEBRAIC K -THEORY REVISITED 5 By [Joy08, Ch. 5, Thm. A] the map C I ( s,t ) −−→ C × C is a categoricalfibration, i.e., a fibration in the Joyal model structure. Since the lower andupper right corners in (5) are ∞ -categories, this implies that A × → C B isindeed an ∞ -category, and that (5) is homotopy cartesian with respect tothe Joyal model structure. Remark . The objects of A × → C B are triples of the form ( a, b, g : p ( a ) → q ( b )) where a , b are objects of A , B respectively and g is a morphism p ( a ) → q ( b ) in C . If ( a, b, g ) and ( a ′ , b ′ , g ′ ) are two objects of A × → C B , the mappingspace between these sits in a homotopy cartesian diagram of spacesMap(( a, b, g ) , ( a ′ , b ′ , g ′ )) / / (cid:15) (cid:15) Map C I ( g, g ′ ) (cid:15) (cid:15) Map A ( a, a ′ ) × Map B ( b, b ′ ) / / Map C ( p ( a ) , p ( a ′ )) × Map C ( q ( b ) , q ( b ′ )) . Indeed, using Lurie’s Hom R -model for the mapping spaces [Lur09, § Remark . Denote by C ( I ) ⊆ C I the full subcategory spanned by the equiv-alences in C . It follows from [Joy08, Prop. 5.17] that the pullback of thediagram C ( I )( s,t ) (cid:15) (cid:15) A × B p × q / / C × C in simplicial sets models the homotopy pullback of ∞ -categories A × C B .In particular, we can identify A × C B with the full subcategory of A × → C B spanned by those objects ( a, b, g ) where g is an equivalence in C . Lemma 8. (i)
Let K be a simplicial set and δ : K → A × → C B a di-agram. If the compositions of δ with the projections to A and B admit colimits and these colimits are preserved by p and q respec-tively, then δ admits a colimit, which is preserved by the projectionsto A and B . The same statement holds for diagrams in A × C B . (ii) If A and B are idempotent complete, then A × → C B and A × C B areidempotent complete. (iii) If A , B , and C are presentable and p and q commute with colim-its, then both ∞ -categories A × → C B and A × C B are presentable.Moreover, a functor from a presentable ∞ -category D to A × C B or A × → C B preserves colimits if and only if the compositions with theprojections to A and B do. (iv) If A , B , and C are stable, and p and q are exact, then both ∞ -categories A × → C B and A × C B are stable. For the definition of a presentable ∞ -category see [Lur09, Def. 5.5.0.1],for that of an idempotent complete ∞ -category [Lur09, § ∞ -category [Lur17a, Def. 1.1.1.9]. GEORG TAMME
Proof. (i) The assumptions and [Lur09, Prop. 5.1.2.2] (applied to the pro-jection C × I → I ) imply that the composition of δ with the projection to C I also admits a colimit. Now the claim follows from [Lur09, Lemmas 5.4.5.4,5.4.5.2].(ii) Let Idem be the nerve of the 1-category with a single object X andHom( X, X ) = { id X , e } , where e ◦ e = e . An ∞ -category D is idempotentcomplete if and only if any diagram Idem → D admits a colimit. It followsfrom [Lur09, Prop. 4.4.5.12, Lemma 4.3.2.13] that every functor between ∞ -categories D → D ′ preserves colimits of diagrams indexed by Idem. Hencethe claim follows from part (i).By construction of the lax pullback, it suffices to check the remainingassertions for pullbacks and functor categories.(iii) For the functor category see [Lur09, Prop. 5.5.3.6, Cor. 5.1.2.3] andfor the pullback [Lur09, Prop. 5.5.3.12].(iv) See [Lur17a, Prop. 1.1.3.1] for the functor category, [Lur17a, Prop.1.1.4.2] for the pullback. (cid:3) From now on, we will mainly be concerned with stable ∞ -categories.Recall that by [Lur17a, Thm. 1.1.2.14] the homotopy category Ho( A ) of astable ∞ -category A is a triangulated category. Recollection . We recall the ∞ -categorical version of Verdier quotients. Fora detailed discussion see [BGT13, § Lst denote the ∞ -category of pre-sentable stable ∞ -categories and left adjoint (equivalently, colimit preserv-ing) functors, and let Cat ex ∞ be the ∞ -category of small stable ∞ -categoriesand exact functors. Both admit small colimits. Given a fully faithful functor A → B in either of these, B/A denotes its cofibre. By [BGT13, Prop. 5.9,5.14] the functor B → B/A induces an equivalence of the Verdier quotientHo( B ) / Ho( A ) with Ho( B/A ).A sequence A → B → C in Pr Lst or Cat ex ∞ is called exact if the compositeis zero, A → B is fully faithful, and the induced map B/A → C is an equiva-lence after idempotent completion. It follows from [BGT13, Prop. 5.10] andthe above that A → B → C is exact if and only Ho( A ) → Ho( B ) → Ho( C ) isexact (up to factors) in the sense of triangulated categories (see e.g. [Sch11,Def. 3.1.10]).If C is a localization of B , i.e., the functor B → C has a fully faithfulright adjoint, and A → B induces an equivalence of A with the kernel of B → C , i.e., the full subcategory of objects of B that map to a zero objectin C , then A → B → C is exact.For the remainder of this section, we assume that (4) is a diagram ofstable ∞ -categories and exact functors.The pair of functors B → A × B , b (0 , b ), and B → C I , b (0 → q ( b )),induces a functor r : B → A × → C B . Similarly, the functors A → A × B , a ( a, A → C I , a ( p ( a ) → s : A → A × → C B . This is Cor. 4.4.5.15 in the 2017 version of HTT, available at the author’s homepage.
XCISION IN ALGEBRAIC K -THEORY REVISITED 7 Proposition 10.
Assume that (4) is a diagram of stable ∞ -categories andexact functors. We have a split exact sequence B r / / A × → C B pr / / pr { { A, s y y i.e., the sequence is exact, pr and s are right adjoints of r , pr , respectively,and id B ≃ pr ◦ r , pr ◦ s ≃ id A via unit and counit, respectively.Proof. By construction, we have id B = pr ◦ r and we claim that this is aunit transformation for the desired adjunction (see [Lur09, Prop. 5.2.2.8]).That is, we have to show that for any object b in B and ( a ′ , b ′ , g ′ ) in A × → C B the map Map( r ( b ) , ( a ′ , b ′ , g ′ )) → Map( b, b ′ ) induced by pr is an equivalence.This map is the second component of the left vertical map in the diagramMap( r ( b ) , ( a ′ , b ′ , g ′ )) / / (cid:15) (cid:15) Map((0 → q ( b )) , g ′ ) (cid:15) (cid:15) Map(0 , a ′ ) × Map( b, b ′ ) / / Map(0 , p ( a ′ )) × Map( q ( b ) , q ( b ′ )) . By Remark 6 this diagram is homotopy cartesian. Since the functor C → C I , c (0 → c ), is a left adjoint of t : C I → C , the right vertical map isan equivalence. Hence the left vertical map is an equivalence (use thatMap(0 , a ′ ) and Map(0 , p ( a ′ )) are contractible).Similarly, one shows that s is a right adjoint of pr . Since the counitpr ◦ s → id A is an equivalence, s is fully faithful. Since moreover r inducesan equivalence of B with the kernel of pr , the sequence in the statement ofthe lemma is exact by Recollection 9. (cid:3) We let π be the composition of functors A × → C B pr −−→ C I Cone −−−→ C , whereCone : C I → C sends a morphism in C to its cofibre. Proposition 11.
Assume that (4) is a diagram of stable ∞ -categories andexact functors. Assume furthermore that q : B → C admits a fully faithfulright adjoint v : C → B . Then the composite ρ : C v −→ B r −→ A × → C B is a fully faithful right adjoint of π .Proof. Since v is fully faithful by assumption, and r is fully faithful byProposition 10, the functor ρ is fully faithful. The functor Cone : C I → C has the right adjoint β mapping c to (0 → c ) [Lur17a, Rem. 1.1.1.8]. Bythe construction of r we have a canonical equivalence pr ◦ r ≃ β ◦ q . Hencethe counit of the adjoint pair (Cone , β ) induces a natural transformation π ◦ r = Cone ◦ pr ◦ r ≃ Cone ◦ β ◦ q → q and hence π ◦ r ◦ v → q ◦ v . Composingwith the counit of the adjoint pair ( q, v ) we get a natural transformation η : π ◦ ρ = π ◦ r ◦ v → id C . We claim that η is a counit transformation forthe desired adjunction. This will imply the claim by [Lur09, Prop. 5.2.2.8].We thus have to show that the composition(6) Map(( a, b, g ) , ρ ( c )) π −→ Map( π (( a, b, g )) , π ( ρ ( c ))) η −→ Map( π (( a, b, g )) , c )is an equivalence for every object ( a, b, g ) in A × → C B and any object c in C . GEORG TAMME
From Remark 6 we have a homotopy pullback square of spacesMap(( a, b, g ) , ρ ( c )) pr / / (cid:15) (cid:15) Map( g, (0 → q ( v ( c )))) (cid:15) (cid:15) Map( a, × Map( b, v ( c )) / / Map( p ( a ) , × Map( q ( b ) , q ( v ( c ))) . (7)Since v is fully faithful, q ( v ( c )) ≃ c and the lower horizontal map is an equiv-alence by adjunction. Hence the upper horizontal map pr is an equivalence,too. The (Cone , β )-adjunction yields an equivalence(8) Map( g, (0 → q ( v ( c )))) ≃ −→ Map(Cone( g ) , q ( v ( c ))) . By construction, (6) is the composition of the equivalences pr in (7) and(8) and the map induced by the counit q ( v ( c )) → c , which is an equivalenceby fully faithfulness of v . Hence (6) is an equivalence, as desired. (cid:3) Corollary 12.
Assume that (4) is a diagram in Pr Lst . If the right adjointof B → C is fully faithful, then the sequence A × C B → A × → C B π −→ C is exact.Proof. An object ( a, b, g ) of A × → C B belongs to A × C B if and only if g is anequivalence, if and only if Cone( g ) ≃
0. This shows that the composite istrivial and that A × C B is precisely the kernel of π . The claim now follows,since π admits a fully faithful right adjoint by Proposition 11. (cid:3) Let A ′ be a small stable ∞ -category. Then the ∞ -category Ind( A ′ ) of Ind-objects of A ′ [Lur09, Def. 5.3.5.1] is presentable [Lur09, Thm. 5.5.1.1] andstable [Lur17a, Prop. 1.1.3.6]. A stable ∞ -category A is called compactlygenerated if there exists a small stable ∞ -category A ′ and an equivalenceInd( A ′ ) ≃ A (see [Lur09, Def. 5.5.7.1] and the text following it). If this is thecase, then A ′ → A induces an equivalence of the idempotent completion of A ′ [Lur09, § A ω of the compact objectsin A [Lur09, Lemma 5.4.2.4]. In particular, if A is compactly generated, A ω is (essentially) small and Ind( A ω ) ≃ A . Whether a stable ∞ -category isidempotent complete or compactly generated only depends on its homotopycategory [Lur17a, Lemma 1.2.4.6, Rem. 1.4.4.3]. Proposition 13.
Assume that (4) is a diagram in Pr Lst in which A and B are compactly generated and the functors p : A → C and q : B → C mapcompact objects to compact objects. Then A × → C B is compactly generated aswell and ( A × → C B ) ω ≃ A ω × → C ω B ω .Proof. By Lemma 8(iii) the ∞ -category A × → C B is presentable and henceadmits all small colimits. Let D ′ := A ω × → C ω B ω . This is an (essentially)small full stable subcategory of A × → C B . It follows from [Lur09, Lemmas5.4.5.7, 5.3.4.9] that D ′ consists of compact objects in A × → C B . Hence theinduced functor Ind( D ′ ) → A × → C B is fully faithful. Since the functors r : B → A × → C B and s : A → A × → C B preserve colimits by Lemma 8(iii) andsince A and B are compactly generated, it follows that the essential imageof Ind( D ′ ) in A × → C B contains A and B . Proposition 10 implies that every XCISION IN ALGEBRAIC K -THEORY REVISITED 9 object X of A × → C B sits in a fibre sequence X ′ → X → X ′′ with X ′ ∈ B and X ′′ ∈ A . Hence the essential image of Ind( D ′ ) must be all of A × → C B , andhence the latter is compactly generated. Since A ω and B ω are idempotentcomplete, so is D ′ by Lemma 8(ii). Hence D ′ ≃ ( A × → C B ) ω . (cid:3) Definition 14. An excisive square of small stable ∞ -categories is a com-mutative square D / / (cid:15) (cid:15) B q (cid:15) (cid:15) A p / / C (9)in Cat ex ∞ such that the induced squareInd( D ) / / (cid:15) (cid:15) Ind( B ) (cid:15) (cid:15) Ind( A ) / / Ind( C )(10)in Pr Lst is a pullback square and Ind( B ) → Ind( C ) is a localization, i.e., itsright adjoint is fully faithful.The following is the categorical version of our first main result. Theorem 15.
Assume that (9) is an excisive square of small stable ∞ -categories. Then there is an exact sequence (11) D i −→ A × → C B π −→ C. Proof.
If we apply Corollary 12 to the pullback diagram (10), we get theexact sequence Ind( D ) → Ind( A ) × → Ind( C ) Ind( B ) → Ind( C )in Pr Lst . Clearly, the first and the third term in this sequence are compactlygenerated. Proposition 13 implies that also the middle term is compactlygenerated, and that the functors preserve compact objects. Recall fromRecollection 9 that we can test exactness on the level of homotopy categories.Thus we may apply the Thomason–Neeman localization theorem [Nee92,Thm. 2.1] to conclude that the induced sequence of compact objects is exact.But up to idempotent completion this is exactly (11). (cid:3)
We now apply this to localizing invariants.
Definition 16. A weakly localizing invariant is a functor E : Cat ex ∞ → T from Cat ex ∞ to some stable ∞ -category T which sends exact sequences inCat ex ∞ to fibre sequences in T . Example . Any localizing invariant in the sense of [BGT13] is weakly local-izing. Concrete examples are non-connective algebraic K -theory `a la Bass-Thomason [BGT13, § T HH [BGT13, § p -typical topological cyclic homology T C for some prime p [BGT13, § T is the ∞ -category of spectra. Theorem 18.
Assume that (9) is an excisive square of small stable ∞ -categories, and let E : Cat ex ∞ → T be a weakly localizing invariant. Then theinduced square in T E ( D ) / / (cid:15) (cid:15) E ( B ) (cid:15) (cid:15) E ( A ) / / E ( C )(12) is cartesian.Proof. Applying E to the exact sequence (11) provided by Theorem 15 yieldsthe fibre sequence(13) E ( D ) E ( i ) −−→ E ( A × → C B ) E ( π ) −−−→ E ( C )in T . On the other hand, applying E to the split exact sequence of Propo-sition 10 gives an equivalence(14) E ( s ) ⊕ E ( r ) : E ( A ) ⊕ E ( B ) ≃ −→ E ( A × → C B )with inverse induced by the projections pr , pr . Combining (13) and (14),we get a fibre sequence(15) E ( D ) → E ( A ) ⊕ E ( B ) → E ( C )where the first map is induced by the given functors D → A and D → B .The map E ( A ) → E ( C ) is induced by the functor a Cone( p ( a ) → ≃ Σ p ( a ). Since the endofunctor Σ : C → C induces − id on E ( C ), the map E ( A ) → E ( C ) in (15) is the negative of the map induced by the functor p : A → C . Finally, the map E ( B ) → E ( C ) in (15) is induced by thefunctor b Cone(0 → q ( b )) ≃ q ( b ). Thus (15) being a fibre sequence in T implies that (12) is cartesian. (cid:3) Remark . This theorem can also be used to prove the Mayer-Vietorisproperty of algebraic K -theory for the Zariski topology [TT90, Thm. 8.1]for quasi-compact quasi-separated schemes without using Thomason’s local-ization theorem [TT90, Thm. 7.4]. Together with Example 3 one may thendeduce Nisnevich descent for noetherian schemes in general.2. Application to ring spectra
In this section, we apply the constructions of Section 1 to the ∞ -categoriesof (perfect) modules over an E -ring spectrum, discuss Tor-unitality, and weprove our second main result (Theorem 28) saying that a pullback square ofring spectra where one map is Tor-unital (Definition 21) yields an excisivesquare upon applying Perf( − ). From this we finally deduce Theorems 1and 2 of the Introduction.The ∞ -categories of E -ring spectra and their modules are discussed in[Lur17a, Ch. 7]. For an E -ring spectrum A , we write LMod( A ) for thestable ∞ -category of left A -module spectra, which we will simply call left A -modules henceforth. A left A -module is called perfect if it belongs to thesmallest stable subcategory Perf( A ) of LMod( A ) which contains A and isclosed under retracts. By [Lur17a, Prop. 7.2.4.2], LMod( A ) is compactlygenerated and the compact objects are precisely the perfect A -modules. XCISION IN ALGEBRAIC K -THEORY REVISITED 11 Example . Any discrete ring A can be considered as an E -ring spectrum.Then Ho(LMod( A )) is equivalent to the unbounded derived category of A in the classical sense [Lur17a, Rem. 7.1.1.16]. Definition 21.
A map f : A → A ′ of E -ring spectra is called Tor-unital ifthe following equivalent conditions are satisfied:(i) The map A ′ ⊗ A A ′ → A ′ given by multiplication is an equivalence.(ii) The map A ′ → A ′ ⊗ A A ′ induced from A → A ′ by A ′ ⊗ A ( − ) is anequivalence.(iii) If I is the fibre of A → A ′ in LMod( A ), we have A ′ ⊗ A I ≃ Lemma 22.
A morphism A → A ′ of E -ring spectra is Tor-unital if andonly if the forgetful functor LMod( A ′ ) → LMod( A ) is fully faithful.Proof. By [Lur17a, Prop. 4.6.2.17] the forgetful functor v is right adjoint to A ′ ⊗ A − : LMod( A ) → LMod( A ′ ). It is fully faithful if and only if the counit A ′ ⊗ A M → M is an equivalence for every A ′ -module M . Taking M = A ′ ,we see that fully faithfulness of v implies Tor-unitality of A → A ′ . Theconverse follows, since LMod( A ′ ) is generated by A ′ under small colimitsand finite limits, and the tensor product preserves both. (cid:3) Now consider any pullback square of E -ring spectra A / / (cid:15) (cid:15) A ′ (cid:15) (cid:15) B / / B ′ . (16) Lemma 23.
Assume that (16) is a pullback square of E -ring spectra inwhich A → A ′ is Tor-unital. Then also B → B ′ is Tor-unital. Moreover, thecanonical map A ′ ⊗ A B → A ′ ⊗ A B ′ induced from B → B ′ is an equivalence. See Remark 27 for a partial converse.
Proof.
Write I for the fibre of A → A ′ . Since A → A ′ is Tor-unital, A ′ ⊗ A I ≃
0. As by assumption (16) is a pullback square, the fibre of B → B ′ is equivalent (as left A -module) to I , hence A ′ ⊗ A B → A ′ ⊗ A B ′ is anequivalence, too. By Lemma 22 the counit A ′ ⊗ A M → M is an equivalencefor every A ′ -module M . In particular, A ′ ⊗ A B ′ → B ′ is an equivalence.Summing up, the canonical map A ′ ⊗ A B → B ′ is an equivalence. Thus B ′ ⊗ B B ′ ≃ ( A ′ ⊗ A B ) ⊗ B B ′ ≃ A ′ ⊗ A B ′ ≃ B ′ and B → B ′ is Tor-unital. (cid:3) Example . Let A → B be a morphism of discrete unital rings sending atwo-sided ideal I of A isomorphically onto an ideal of B . Then the Milnorsquare A / / (cid:15) (cid:15) A/I (cid:15) (cid:15) B / / B/I is a pullback diagram in rings. Since B → B/I is surjective, this diagram isalso a pullback when considered as a diagram of E -ring spectra. The map A → A/I is Tor-unital if and only if Tor Ai ( A/I, A/I ) = 0 for all i > I is Tor-unitalin the classical sense that Tor Z ⋉ Ii ( Z , Z ) = 0 for all i >
0, then Lemma 23applied to the Milnor square Z ⋉ I / / (cid:15) (cid:15) Z (cid:15) (cid:15) A / / A/I implies that A → A/I is Tor-unital for any ring A containing I as a two-sidedideal. Example . Assume that A is a commutative, unital discrete ring, and let f ∈ A . Then A → A [ f − ] is Tor-unital. Assume further that A → B is an´etale ring map which induces an isomorphism A/ ( f ) ∼ −→ B/ ( f ). Then thediagram A / / (cid:15) (cid:15) A [ f − ] (cid:15) (cid:15) B / / B [ f − ] , viewed as a diagram of E -ring spectra, is a pullback square. Indeed, this isequivalent to the exactness of the sequence0 → A → A [ f − ] ⊕ B → B [ f − ] → , which may be checked directly. Alternatively, one may use the Mayer–Vietoris exact sequence of ´etale cohomology groups0 → A → A [ f − ] ⊕ B → B [ f − ] → H (Spec( A ) , O Spec( A ) ) , which may be deduced from [Mil80, Prop. III.1.27], together with the van-ishing of the higher ´etale cohomology of quasi-coherent sheaves on affineschemes.The following is a derived version of Milnor patching: Theorem 26.
Assume that (16) is a pullback square of E -ring spectrawhere the morphism A → A ′ is Tor-unital. Then extension of scalars inducesan equivalence LMod( A ) ≃ LMod( A ′ ) × LMod( B ′ ) LMod( B ) . Proof.
Let F be the functor LMod( A ) → LMod( A ′ ) × LMod( B ′ ) LMod( B )induced by extension of scalars. Since both ∞ -categories are presentableand F preserves colimits by Lemma 8(iii), F admits a right adjoint G .Explicitly, if ( M, N, g ) is an object of LMod( A ′ ) × LMod( B ′ ) LMod( B ), then G ( M, N, g ) is the pullback in left A -modules G ( M, N, g ) ≃ M × B ′ ⊗ B N N where the map M → B ′ ⊗ B N is the composition M → B ′ ⊗ A ′ M g −→ B ′ ⊗ B N .We claim that the unit P → ( A ′ ⊗ A P ) × B ′ ⊗ B ( B ⊗ A P ) ( B ⊗ A P ) XCISION IN ALGEBRAIC K -THEORY REVISITED 13 of the adjunction is an equivalence for any A -module P . Since also G com-mutes with colimits, it suffices to check this for P = A . In that case theclaim follows from the assumption that (16) is a pullback square. Hence F is fully faithful.It now suffices to show that the right adjoint G of F is conservative. Forthis it is enough to show that G detects zero objects. So let ( M, N, g ) bean object of the pullback and assume that G ( M, N, g ) ≃
0. There is a fibresequence of left A -modules G ( M, N, g ) → M ⊕ N → B ′ ⊗ B N and hence the map(17) M ⊕ N ≃ −→ B ′ ⊗ B N is an equivalence. Extending scalars from A to A ′ we get an equivalence(18) A ′ ⊗ A M ⊕ A ′ ⊗ A N ≃ −→ A ′ ⊗ A B ′ ⊗ B N. From Lemma 23 we know that A ′ ⊗ A B → A ′ ⊗ A B ′ is an equivalence. SinceLMod( B ) is generated by B under colimits and finite limits, we concludethat A ′ ⊗ A P → A ′ ⊗ A B ′ ⊗ B P is an equivalence for every left B -module P .Applying this with P = N , we see that the restriction of (18) to the secondsummand is an equivalence. Hence A ′ ⊗ A M ≃
0. Since M is an A ′ -module,Lemma 22 implies that the counit is an equivalence A ′ ⊗ A M ≃ M , i.e., M ≃
0. But then also B ′ ⊗ B N ≃ B ′ ⊗ A ′ M ≃
0, and hence N ≃ (cid:3) Remark . Without the Tor-unitality assumption Theorem 26 does nothold, see [Lur17b, Warning 16.2.0.3] for a counter example.However, if one assumes instead that (16) is a pullback square of con-nective ring spectra with π ( B ) → π ( B ′ ) surjective, then [Lur17b, Prop.16.2.2.1] implies that restricting the functors F and G from the proof ofTheorem 26 to the subcategories of connective modules gives inverse equiv-alences LMod( A ) ≥ ⇆ LMod( A ′ ) ≥ × LMod( B ′ ) ≥ LMod( B ) ≥ . One can use this to show that in this situation, Tor-unitality of B → B ′ implies Tor-unitality of A → A ′ : Let I be the fibre of B → B ′ . Since π ( B ) → π ( B ′ ) is surjective, I is connective. Since B → B ′ is Tor-unital, B ′ ⊗ B I ≃
0. Hence we may view (0 , I,
0) as an object of the pullbackLMod( A ′ ) ≥ × LMod( B ′ ) ≥ LMod( B ) ≥ . The functor G sends (0 , I,
0) to the A -module 0 × I ≃ I . By the above the counit F ( I ) ≃ F ( G (0 , I, → (0 , I, A ′ ⊗ A I → A → A ′ is Tor-unital. Theorem 28.
Assume that (16) is a pullback square of E -ring spectrawhere the morphism A → A ′ is Tor-unital. Then the square Perf( A ) / / (cid:15) (cid:15) Perf( B ) (cid:15) (cid:15) Perf( A ′ ) / / Perf( B ′ )(19) is excisive. In particular, if E : Cat ex ∞ → T is a weakly localizing invariant,then the induced square E (Perf( A )) / / (cid:15) (cid:15) E (Perf( B )) (cid:15) (cid:15) E (Perf( A ′ )) / / E (Perf( B ′ )) in T is cartesian.Proof. Applying Ind to diagram (19) yields the diagramLMod( A ) / / (cid:15) (cid:15) LMod( B ) (cid:15) (cid:15) LMod( A ′ ) / / LMod( B ′ )This is a pullback diagram by Theorem 26. As A → A ′ is Tor-unital,so is B → B ′ by Lemma 23. Hence the right adjoint of LMod( B ) → LMod( B ′ ), which is the forgetful functor, is fully faithful by Lemma 22.So the square (19) is excisive. Now the second assertion follows by applyingTheorem 18. (cid:3) Proof of Theorems 1 and 2.
If we apply Theorem 28 with E = K , we im-mediately get Theorem 2.Now let I be a ring which is Tor-unital in the classical sense, and let A beany unital ring containing I as a two-sided ideal. Then the Milnor square Z ⋉ I / / (cid:15) (cid:15) Z (cid:15) (cid:15) A / / A/I, viewed as square of E -ring spectra, is a pullback square (see Example 24).By assumption, the top horizontal map is Tor-unital in our sense. Hencewe may apply Theorem 2 to deduce that the map on relative K -groups K ∗ ( I ) = K ∗ ( Z ⋉ I, I ) → K ∗ ( A, I ) is an isomorphism. (cid:3)
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