Pro cdh-descent for cyclic homology and K-theory

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Matthew Morrow

Journal of the Institute of Mathematics of Jussieu, doi:10.1017/S1474748014000413 Abstract

In this paper we prove that cyclic homology, topological cyclic homology, and algebraic K-theory satisfy a pro Mayer–Vietoris property with respect to abstract blow-up squares of varieties, in both zero and finite characteristic. This may be interpreted as the well-definedness ofK-theory with compact support.

Key words: K-theory, cyclic homology, cdh-topology.

MSC: 19D55 (primary), 13D03 (secondary).

0 Introduction

The primary goal of this paper is to prove that algebraicK-theory satisfies a pro Mayer–

Vietoris property with respect to abstract blow-up squares of varieties, in both zero and finite characteristic, at least assuming strong resolution of singularities. This may also be interpreted as the well-definedness of a theory of K-groups with compact support.

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To state the main result, we first recall that an abstract blow-up square of schemes is a pull-back diagram

Y⁰ ^//

X⁰

Y ^//X

where X⁰ → X is proper, Y → X is a closed embedding, and the induced map on the open subschemes X⁰\Y⁰ → X\Y is an isomorphism. We denote by rY the r^th infinitesimal thickening of Y inside X, i.e., rY := SpecO_X/I^r where I ⊆ O_X is the sheaf of ideals definingY, and similarly for other closed embeddings.

Our pro descent theorem is the following:

Theorem 0.1(Pro cdh-descent forK-theory; see Thm.3.7). Letkbe an infinite, perfect field which has strong resolution of singularities. For any abstract blow-up square, as above, consisting of finite typek-schemes, the resulting square of pro spectra

K(X) ^//

K(X⁰)

{K(rY)}_r ^//{K(rY⁰)}_r

is homotopy cartesian. In other words, the canonical map of pro abelian relative K- groups

{K_n(X, rY)}_r−→ {K_n(X⁰, rY⁰)}_r is an isomorphism for all n∈Z.

The main applications of Theorem 0.1 are to zero cycles; since the Theorem may be accepted as a black box for these applications, they are presented separately in an accompanying paper [25]. There we solve cases of an outstanding conjecture of V. Srini- vas and S. Bloch [30, pg. 6] concerning the Levine–Weibel Chow group of zero cycles on singular varieties, and relate Chow groups with modulus, which play a prominent role in M. Kerz and S. Saito’s higher dimensional class field theory [19], to both Levine–Weibel Chow groups andK-theory via cycle class maps.

Theorem0.1may be interpreted as a definition ofK-theory with compact supportas follows. Given a separated k-variety X, we may choose a proper compactificationX, setY :=X\X, and then defineK^c(X) := holim_rK(X, rY). Theorem 0.1implies that this definition does not depend on the chosen compactificationX ofX, and we prove in Proposition4.4that there is a a pushforward localisation sequence relating K^c(U) and K^c(X) for any open subvariety U ⊆X, thereby justifying the nomenclature ofK^cas a theory with compact support.

In characteristic zero we actually establish Theorem0.1 in much greater generality, namely for any abstract blow-up square of Noetherian, quasi-excellent, Q-schemes of finite Krull dimension. This is obtained by first extending Haesemeyer’s argument [15,

§5–6], for checking whether a pre-sheaf of spectra satisfies cdh-descent, to this generality;

see Proposition 3.6. This also allows us to prove the vanishing part of Weibel’s K- dimension conjecture for such schemes; see Theorem4.7.

To contextualise Theorem 0.1 we should make a few comments about cdh-descent.

AlgebraicK-theory (as well as Andr´e–Quillen, Hochschild, topological Hochschild, cyclic, and topological cyclic homologies) does not satisfy descent in V. Voevodky’s cdh-topology

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[33], in contrast to, e.g., Weibel’s homotopy invariant K-theory [15], or periodic cyclic homology [4]. Since it is known that K-theory does satisfy Nisnevich descent, this precisely means that taking theK-theory of an abstract blow-up square does not necessarily yield a homotopy cartesian square of spectra. In fact, the failure of K-theory to satisfy cdh-descent in characteristic zero (resp. p >0) is precisely equal to the analogous obstruction in cyclic homology [3, 4, 15] (resp. topological cyclic homology [12]); this has led to new calculations of the K-theory of singular algebraic varieties, especially by G. Corti˜nas, C. Haesemeyer, M. Schlichting, M. Walker, and C. Weibel [4,5,7,8,6] in characteristic zero. The moral of Theorem 0.1is that the failure of K-theory to satisfy cdh-descent can be remedied by taking all infinitesimal thickenings of the exceptional subschemesY and Y⁰ into account.

We now briefly discuss how Theorem 0.1 is proved and simultaneously state our other main results. Using the aforementioned cdh comparison between K-theory and cyclic homology, Theorem0.1in characteristic zero may be reduced to an analogous pro Mayer–Vietoris assertion about cyclic homology, or even Hochschild or Andr´e–Quillen homology. We establish this pro Mayer–Vietoris property in considerable generality (see Theorem 2.10for the Andr´e–Quillen case):

Theorem 0.2 (Pro descent forHH andHCwrt. abstract blow-ups; see Thm. 3.3). Let k be a Noetherian ring, and let

Y⁰

//X⁰

Y ^//X

be an abstract blow-up square of Noetherian, finite Krull dimensional k-schemes. Then the canonical maps

{HH_n^k(X, rY)}_r−→ {HH_n^k(X⁰, rY⁰)}_r, {HC_n^k(X, rY)}_r −→ {HC_n^k(X⁰, rY⁰)}_r are isomorphisms of pro abelian groups for all n∈Z.

We should mention at this point that our Hochschild and cyclic homology groups are always computed in the derived sense, that is after replacing rings by free simplicial resolutions (see Section 3.1for details).

The key input to proving Theorem0.2 is new formal function theorems for Andr´e–

Quillen, Hochschild, and cyclic homology, in the style of Grothendieck’s formal functions theorem for coherent cohomology. In particular, in the case of Hochschild homology we prove the following (see Corollary 2.5for the Andr´e–Quillen case):

Theorem 0.3(Formal functions theorem forHH; see Thm.3.2(v)). LetAbe a Noethe- rian ring, I ⊆ A an ideal, and X a proper scheme over A of finite Krull dimension.

Then the canonical map

{HH_n^A(X)⊗_AA/I^r}_r−→ {HH_n^A/I^r(X×_AA/I^r)}_r is an isomorphism of pro A-modules for all n∈Z.

If A, as in the previous theorem, is moreover I-adically complete, then it follows that the canonical map HH_n^A(X) −→ lim←−rHHn^A/I^r(X×_AA/I^r) is an isomorphism for all n ∈ Z, which is the statement closest to the usual formulation of Grothendieck’s formal functions theorem for coherent cohomology.

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The proof of Theorem0.1in finite characteristic is similar in outline, except that the results concerning Andr´e–Quillen, Hochschild, and cyclic homology in Sections 2 and 3.1must be replaced by similar results for topological Hochschild and cyclic homology.

These results are not contained in the current paper, but may rather be found in recent joint work with B. Dundas [10], which includes in particular the THH analogue of Theorem 0.3. The few details missing from [10], where for example the analogue of Theorem0.2forTHH and TC was not explicitly stated, are given in Section 3.2.

Notation, etc.

All rings are commutative, associative, and unital. If A• is a simplicial abelian group, then we often abuse notation by identifyingA• with its associated complexN A•; in particular, we speak of the homology groupsHn(A•) ofA•, rather than more correctly, but equivalently, the simplicial homotopy groupsπn(A•) or the homology groupsHn(N A•).

Acknowledgments

I thank B. Dundas for the collaboration [10], without which Theorem0.1 would be re- stricted to characteristic zero. The majority of this work was done while I was supported by the Simons foundation as a postdoctoral fellow at the University of Chicago, and I am grateful to the foundation for their support.

1 Review of pro modules and Artin–Rees properties

1.1 Pro abelian groups and pro modules

Everything we need about categories of pro objects may be found in one of the standard references, such as the appendix to [2], or [17]. We will often use Pro(A-mod), the category of pro A-modules for some ring A, and ProAb, the category of pro abelian groups.

Let C be a category. In this paper, an object of ProC is simply an inverse system · · · → A₂ → A₁ of objects and morphisms in C, which is denoted {A_r}_r or very occasionallyA∞. Morphisms in ProC are given by the rule

Hom_ProC({A_r}_r,{B_s}_s) := lim←−

s

lim−→

r

HomC(A_r, B_s),

where the right side is a genuine pro-ind limit in the category of sets, and composition is defined in the obvious way. For example, a pro object {A_r}_r is isomorphic to zero (assuming that a zero object exists inC, hence also in ProC) if and only if for eachr≥1 there existss≥r such that the transition map A_s→A_r is zero.

There is a fully faithful embedding C → ProC. Assuming C has countable inverse limits, this has a right adjoint ProC → C, {A_r}_r 7→ lim←−rAr, which is left exact but not right exact. Moreover, ifC is an abelian category then so is ProC: given a inverse system of exact sequences

· · · −→An−1(r)−→An(r)−→An+1(r)−→ · · · , the “limit asr → ∞”, namely

· · · −→ {An−1(r)}_r−→ {A_n(r)}_r −→ {A_n+1(r)}_r−→ · · ·,

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is an exact sequence in ProC.

Pro spectral sequences play an important role in the paper, which we will discuss for concreteness only in the case of abelian groups. Suppose that

E_pq¹ (r) =⇒H_p+q(r),

forr≥1, are spectral sequences of abelian groups, which are functorial in that we have morphisms of spectral sequences · · · →E_pq^• (2) → E_pq^• (1). To avoid convergence issues, suppose that each spectral sequence is bounded, by a bound independent of r; e.g., each spectral sequence might be zero outside the first quadrant. Then we will often “let r → ∞” to obtain a spectral sequence of pro abelian groups

E_pq¹ (∞) :={E_pq¹ (r)}_r=⇒ {H_p+q(r)}_r. For further discussion and a dummy example, see [24, App. A].

1.2 Artin–Rees properties

For the sake of reference, we now formally state a fundamental Artin–Rees result which will be used several times; this result appears to have been first noticed and exploited by M. Andr´e [1, Prop. 10 & Lem. 11] and D. Quillen [27, Lem. 9.9]:

Theorem 1.1 (Andr´e, Quillen). Let A be a Noetherian ring, and I ⊆A any ideal.

(i) If M is a finitely generated A-module, then the pro A-module {Tor^A_n(A/I^r, M)}_r vanishes for all n >0.

(ii) The “completion” functor

− ⊗_AA/I^∞ :A-mod−→Pro(A-mod) M 7→ {M⊗_AA/I^r}_r is exact on the subcategory of finitely generated A-modules.

Sketch of proof. By picking a resolution P• of M by finitely generated projective A- modules and applying the classical Artin–Rees property to the pair d(P_n) ⊆Pn−1, one sees that for each r≥1 there exists s≥r such that the map

Tor^A_n(A/I^s, M)→Tor^A_n(A/I^r, M) is zero. This proves (i). (ii) is just a restatement of (i).

Next we quote a similar Artin–Rees result for Andr´e–Quillen homology from [24], a companion to this paper which treats pro excision problems; the basics of Andr´e–Quillen homology will be recalled in Section 2.1:

Theorem 1.2. Let k → A be a homomorphism of Noetherian rings, I ⊆ A an ideal, and M anA-module. Then:

(i) {Dⁱ_n(A/I^r|A, M/I^rM)}_r = 0 for alln≥0, i≥1.

(ii) The canonical map {Dⁱ_n(A|k, M/I^rM)}_r −→ {D_nⁱ(A/I^r|k, M/I^rM)}_r is an isomorphism for all n≥0, i≥0.

Proof. This is [24, Thm. 4.4], where the pro H-unitality hypothesis of the cited theorem is satisfied thanks to [24, Thm. 0.3].

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2 Formal function properties and pro descent for Andr´ e–Quillen homology

The aim of this section is to prove that André–Quillen homology of proper schemes satisfies certain formal function properties. These will be established in Section2.2, after preliminary material on André–Quillen homology is presented in Section2.1. From these formal function properties we show in Section2.3that André–Quillen homology satisfies the desired pro Mayer–Vietoris property with respect to abstract blow-up squares; this is a key step in deducing the same forK-theory in characteristic zero.

2.1 Andr´e–Quillen homology for schemes

In this preliminary section we recall the fundamentals of Andr´e–Quillen homology [1, 26, 28], though we expect that the reader already has some familiarity with it, before extending it from algebras to schemes in the standard way using hypercohomology, as was done for Hochschild and cyclic homology in [38,34].

Letk→A be a homomorphism of rings; let P• →A be a simplicial resolution ofA by freek-algebras, and set

LA|k:= Ω¹_P_•_|k⊗_P_•A.

ThusLA|kis a simplicialA-module which is free in each degree; it is called thecotangent complex of the k-algebra A. The cotangent complex up to homotopy depends only on A, since the free simplicial resolutionP• →A is unique up to homotopy.

Given simplicial A-modules M•, N•, the tensor product and alternating powers are new simplicial A-modules defined degreewise: (M• ⊗_A N•)n = Mn ⊗_ANn and (V_i

AM•)_n=V_i

AM_n.

In particular we set Lⁱ_A|k := Vi

ALA|k for each i≥1. The Andr´e–Quillen homology of thek-algebra A, with coefficients in any A-moduleM, is defined by

D_nⁱ(A|k, M) :=H_n(LⁱA|k⊗_AM), forn≥0,i≥1. When M =A the notation is simplified to

Dⁱ_n(A|k) :=Dⁱ_n(A|k, A) =H_n(Lⁱ_A|k).

Wheni= 1 the superscript is often omitted, writingDn(A|k, M) =Hn(LA|k⊗_AM) and Dn(A|k) =Hn(LA|k) instead. If k→A is essentially of finite type andkis Noetherian, thenDⁱ_n(A|k, M) is a finitely generatedA-module for alln, iand for all finitely generated A-modules M.

To avoid any ambiguity, we also remark that the notationDⁱ_n(A|k, M) is defined in the same way ifi≤0 or n <0. However,Dⁱ_n(A|k, M) = 0 if n <0 and

D_n⁰(A|k, M) =

(M n= 0, 0 else, sinceL⁰_A|k⊗_AM 'M.

Now we extend the definition of Andr´e–Quillen homology to schemes; since we restrict to Noetherian schemes of finite Krull dimension, we may follow a classical approach, as we explain in the next remark:

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Remark 2.1 (Hypercohomology of non-bounded below complexes). Our main results concerning Andr´e–Quillen, and later Hochschild and cyclic, homology of schemes concern Noetherian schemes or even varieties; therefore we will restrict throughout to Noetherian schemes of finite Krull dimension to simplify hypercohomology of complexes which are not necessarily bounded below.

The Zariski hypercohomology of a cochain complex of sheavesM^• on a Noetherian schemeX of finite Krull dimension is defined as follows [23, Appendix C]: there exists a quasi-isomorphism M^{• ∼}→ F^•, where F^• is a cochain complex of acyclic sheaves on X, and one shows that

H^∗(X, M^•) :=H^∗(F^•(X))

is independent ofF. We may use the Cartan–Eilenberg approach to construct F^•: first pick functorial acyclic resolutionsM^p→ F^∼ ^p• of length≤dimXfor eachp∈Z(e.g., the Godement resolution), and then set F^• = TotF^••; note that at most 1 + dimX terms appear in each direct sumFⁿ=L

p+q=nF^{p q}. The familiar spectral sequences

E₁^pq=H^q(X, M^p) =⇒ H^p+q(X, M^•)

0E₂^pq =H^p(X, H^q(M^•)) =⇒ H^p+q(X, M^•)

remain valid and bounded. We will also require two variations on Deligne’s spectral sequence for the hypercohomology of a bounded-below filtered complex [9, 1.4.5], which are not hard to prove given the boundedness assumptions we impose:

(i) Let M^•• be a bounded (i.e., M^{p q} is non-zero for only finitely many p, q in each bounded region of the plane), double cochain complex of sheaves. Then there is a bounded spectral sequence

E₂^pq =H^p(X, H_v^q(M^••)) =⇒ H^p+q(X,TotM^••), where H_v indicates taking cohomology in the second index.

(ii) Let M^• be a cochain complex of sheaves equipped with a descending filtration F M^• which satisfies F⁰M^• = M^• and F^pM^• = 0 for p 0. Then there is a bounded spectral sequence

E₁^pq =H^p+q(X,gr^qM^•) =⇒ H^p+q(X, M^•).

Now we are prepared to define Andr´e–Quillen homology of schemes. Letkbe a ring and fix i≥0. First notice that ifAis a k-algebra then is possible to choose a simplicial resolution P•(A)→A by freek-algebras in a way which is functorial in A; e.g., via the comonad associating toAthe freek-algebra generated by the setA, see [37, E.g. 8.6.16].

So, given a schemeXoverk, letPe•(X) denote the simplicial sheaf ofk-algebras obtained by degree-wise sheafifyingU 7→P•(O_X(U)); also putLⁱ_X_|k := Ωⁱ

Pe•(X)/k⊗

Pe•(X)O_X. Given a quasi-coherent O_X-module M, and assuming that X is Noetherian of finite Krull dimension, define the Andr´e–Quillen homology of X, relative to k, with coefficients in M to be

Dⁱ_n(X|k, M) :=H⁻ⁿ(X,Lⁱ_X|k,neg⊗_O_X M),

where the subscript neg indicates replacing the simplicial sheaf Lⁱ_X|k by its associated chain complex and then negating the indexing to get a cochain complex of sheaves.

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We continue the standard notational abuses of omitting i when it equals 1 and omittingM when it equalsO_X. Note, from the first two properties below, thatDⁱ_n(X|k) can be non-zero forn both positive and negative.

The following consequences are either automatic from the construction, or follow from the previous remark on hypercohomology and mimicking the arguments for Hochschild homology given in [38,34], bearing in mind that Andr´e–Quillen homology of rings behaves well under localisation:

i= 0

ii= 0= 0: SinceL⁰_X_|k,neg⊗O_X M 'M, we have D_n⁰(X|k, M)∼=H⁻ⁿ(X, M).

Agreement on affines: If X = SpecA is an affine k-scheme, then Dⁱ_n(X|k, M) = D_nⁱ(A|k, M).

Mayer–Vietoris property: IfU, V are an open cover ofX, then there is a long exact, Mayer–Vietoris sequence

sequence

E^pq₂ =H^p(X,Dⁱ_−q(X|k, M)) =⇒Dⁱ_−p−q(X|k, M),

where Dⁱ_−q(X|k, M) denotes the Zariski sheafification of U 7→ Dⁱ_−q(U|k, M|_U).

This shows thatDⁱ_n(X|k, M) = 0 for n <−dimX.

Moreover,

Homology l.e.s.: If 0→M →N →P →0 is a short exact sequence of quasi-coherent O_X-modules, then there is a long exact sequence of Andr´e–Quillen homology

· · · −→D_nⁱ(X|k, M)−→Dⁱ_n(X|k, N)−→Dⁱ_n(X|k, P)−→ · · ·

(Proof: The long exact hypercohomology sequence associated to a short exact sequence of complexes of sheaves.)

Higher Jacobi–Zariski spectral sequence: Letk→Abe a homomorphism of rings, letX be a Noetherian scheme overA of finite Krull dimension, letM be a quasi- coherent O_X-module, and fix i ≥ 0. Then there is a natural, bounded spectral sequence ofA-modules E_pq¹ =⇒ Dⁱ_p+q(X|k, M) whose columns may be described as follows:

(i) Suppose p <−i orp >0. ThenE_pq¹ = 0.

(ii) Suppose −i ≤ p ≤ 0. Then the p^th column of the E¹-page is given by a bounded spectral sequence

E_αβ² =D_α^−p A|k, D^i+p_β (X|A, M)

=⇒E_p,α+β−p¹ . (Note thatE_αβ² = 0 if α orβ is <−dimX.)

(Proof: In the case thatX is affine, this is due to C. Kassel and A. Sletsjøe [18].

Examining the proof in the affine case reveals thatLⁱ_X|khas a filtration with graded pieces gr^pLⁱ_X|k 'L^p_A|k⊗_AL^i−p_X/A for p = 0, . . . , i. Tensoring by M, and applying the variations on Deligne’s spectral sequence for a filtered complex given in the Remark2.1, one easily obtains the desired spectral sequence.)

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Example 2.2. Suppose thatX is smooth overk, or more generally thatXhas an affine open cover by the spectra of geometrically regular k-algebras. Then Lⁱ_X|k'Ωⁱ_X|k, and soDⁱ_n(X|k)∼=H⁻ⁿ(X,Ωⁱ_X|k) for alln∈Z,i≥0.

This completes our discussion of Andr´e–Quillen homology for schemes.

2.2 Formal function properties for AQ homology of schemes

Now that the basic properties of Andr´e–Quillen homology for schemes have been established, we show that it satisfies formal function properties akin to Grothendieck’s formal functions theorem. We begin with a lemma in the affine case:

Lemma 2.3. Let A → B be an essentially finite type morphism of Noetherian rings, I ⊆B an ideal, andM a finitely generated B-module. Then the canonical map

{D_nⁱ(B|A, M)⊗_BB/I^r} −→ {Dⁱ_n(B|A, M/I^rM)}_r is an isomorphism for all n, i≥0.

Proof. Given anotherB-module N, there is a natural, first quadrant spectral sequence E_pq² = Tor^B_p(Dⁱ_q(B|A, M), N) =⇒Dⁱ_p+q(B|A, M ⊗_BN).

This is proved in the usual way by choosing a resolution of N by flat B-modules and applying the Kunneth spectral sequence for a tensor product of complexes.

Applying this withN =B/I^r gives spectral sequences

E_pq² (r) = Tor^B_p(Dⁱ_q(B|A, M), B/I^r) =⇒Dⁱ_p+q(B|A, M/I^rM),

and taking the limit over r yields a first quadrant spectral sequence of pro B-modules:

E_pq² (∞) ={Tor^B_p(D_qⁱ(B|A, M), B/I^r)}_r =⇒ {D_p+qⁱ (B|A, M/I^rM)}_r.

Our hypotheses on A, B and M ensure that the B-modules Dⁱ_q(B|A, M) are finitely generated, whence Theorem 1.1(i) implies that E_pq² (∞) = 0 for p > 0. We thus obtain edge map isomorphisms{Dⁱ_n(B|A, M)⊗_BB/I^r}→ {D^' _nⁱ(B|A, M/I^rM)}, as desired.

In the following results a sheaf of ideals I ⊆ O_X plays a prominent role; to keep notation clear, we write rY for ther^th infinitesimal thickening of the closed subscheme defined byI, i.e.,

rY := SpecO_X/I^r.

Theorem 2.4. Let A be a Noetherian ring and π :X → SpecA a Noetherian scheme over A of finite Krull dimension; let I ⊆ O_X be an ideal sheaf and M a coherent O_X-module. Then:

(i) The canonical map {Dⁱ_n(X|A, M/I^rM)}_r −→ {Dⁱ_n(rY|A, M/I^rM)}_r is an isomorphism for all n∈Z, i≥0.

(ii) Assuming that π is essentially of finite type, the canonical map of pro sheaves {D_nⁱ(X|A, M)⊗_O_X O_X/I^r}_r −→ {D_nⁱ(X|A, M/I^rM)}_r is an isomorphism for all n∈Z, i≥0.

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Now suppose further that π is proper and thatI =IO_X for some ideal I ⊆A. Then:

(iii) D_nⁱ(X|A, M) is a finitely generatedA-module for all n∈Z, i≥0.

(iv) The canonical map {D_nⁱ(X|A, M)⊗_AA/I^r}_r −→ {D_nⁱ(X|A, M/I^rM)}_r is an isomorphism for alln∈Z, i≥0.

Proof. (i): By the Mayer–Vietoris long exact descent sequence, and the usual two inductions starting with an affine open cover ofX (note thatX is quasi-separated, since it is Noetherian), this claim reduces to the case thatX is affine, in which case it is Theorem 1.2(ii).

(ii): Assumeπ is essentially of finite type. This claim also reduces to the affine case, which is Lemma2.3, but since this is the only occurrence of pro sheaves, we provide a few more details. Fixingn∈Zand i≥0, letK_r andC_r respectively denote the kernel and cokernel of the canonical map

Dⁱ_n(X|A, M)⊗O_X O_X/I^r−→ Dⁱ_n(X|A, M/I^rM).

Let {U_i} be a finite cover of X by affine opens. Then, given r ≥ 1 and applying the previous lemma, there exists s ≥ r such that the maps Ks(Ui) → Kr(Ui) and Cs(Ui) → Cr(Ui) are zero; since these are quasi-coherent sheaves, it follows that the mapsK_s →K_r andC_s→C_r are also zero, whence{K_r}_r and{C_r}_r vanish, completing the proof.

For the rest of the proof we assume that π is proper and that I = IO_X for some idealI ⊆A.

(iii): EachO_X-moduleD_nⁱ(X|A, M) is coherent sinceXis of finite type overA. Since X is proper overA, this implies that each of the cohomology groups of D_nⁱ(X|A, M) is a finitely generatedA-module; so the claim follows from the hypercohomology spectral sequence.

(iv): As explained in the proof of (iii), the hypercohomology spectral sequence E₂^pq =H^p(X,D_−qⁱ (X|A, M)) =⇒Dⁱ_−p−q(X|A, M)

consists of finitely generatedA-modules. Since{− ⊗_AA/I^r}_r is an exact functor on the category of finitely generated A-modules, by Theorem 1.1(ii), we may apply it to the spectral sequence to obtain a spectral sequence of pro A-modules

E₂^pq(∞) ={H^p(X,Dⁱ_−q(X|A, M))⊗_AA/I^r}_r =⇒ {D_−p−qⁱ (X|A, M)⊗_AA/I^r}_r. This spectral sequence maps to the limit of the hypercohomology spectral sequences for theO_X-modules M/I^rM, namely

0E₂^pq(∞) ={H^p(X,Dⁱ_−q(X|A, M/I^rM))}_r =⇒ {D_−p−qⁱ (X|A, M/I^rM)}_r, and so to complete the proof it is enough to show that the canonical maps E₂^pq(∞)→

0E₂^pq(∞) are all isomorphisms.

Firstly, sinceDⁱ_−q(X|A, M) is a coherent sheaf andπis proper, Grothendieck’s formal functions theorem¹[13, Cor. 4.1.7] states that the canonical map

E₂^pq(∞) ={H^p(X,D_−qⁱ (X|A, M))⊗_AA/I^r}_r −→ {H^p(X,D_−qⁱ (X|A, M)⊗_O_XO_X/I^r)}_r

1This theorem is normally stated as the isomorphism lim←−^r H^p(X, N)⊗AA/I^r→^' lim←−^rH^p(X, N/I^rN) for any coherentOX-moduleN. But a quick examination of the proof in EGA shows that the stronger isomorphism of proA-modules{H^p(X, N)⊗AA/I^r}r

→ {H' ^p(X, N/I^rN)}r also holds.

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is an isomorphism. Secondly, part (ii) implies that the canonical map

{H^p(X,D_−qⁱ (X|A, M)⊗_O_X O_X/I^r)}_r −→ {H^p(X,D_−qⁱ (X|A, M/I^rM))}_r=⁰E₂^pq(∞) is also an isomorphism, completing the proof.

It is part (iv) of the previous theorem which will be so essential in Section 2.3; a secondary application of it is the following corollaries, which will not be needed but which most closely mimic Grothendieck’s formal functions theorem for coherent cohomology.

Corollary 2.5. Let A be a Noetherian ring, X a proper scheme over A of finite Krull dimension, I ⊆A an ideal, and M a coherent O_X-module. Then the canonical map

{D_nⁱ(X|A, M)⊗_AA/I^r}_r −→ {D_nⁱ(rY|A/I^r, M/I^rM)}_r is an isomorphism for all n∈Z, i≥0, where rY =X×_AA/I^r.

Proof. Using parts (iv) and (i) of Theorem 2.4, it remains only to prove that the canonical map {D_nⁱ(rY|A, M/I^rM)}_r → {Dⁱ_n(rY|A/I^r, M/I^rM)}_r is an isomorphism.

From the higher Jacobi–Zariski spectral sequence, this follows from the vanishing of {D_nⁱ(A/I^r|A)}_r; the details are as follows:

Applying the higher Jacobi–Zariski spectral sequences to the homomorphism A → A/I^r and A/I^r-scheme rY, we obtain a bounded spectral sequence of pro A-modules E_pq¹ (∞) =⇒ {Dⁱ_p+q(rY|A, M/I^rM)}_rwhich is supported in the range−i≤p≤0, where it is described by bounded, right-half-plane spectral sequences

E_αβ² (∞) ={D_α^−p(A/I^r|A, D_β^i+p(rY|A/I^r, M/I^rM))}_r=⇒E_p,α+β−p¹ (∞).

For any r, α > 0, Theorem 1.2(i) may be applied to the ideal I^r and module D^i+p_β (rY|A/I^r, M/I^rM) to deduce that there exists s > 1 such that the second of the following maps, hence the composition, is zero:

D^−p_α (A/I^sr|A, D^i+p_β (srY|A/I^sr, M/I^srM))−→D^−p_α (A/I^s|A, D^i+p_β (rY|A/I^r, M/I^rM))

−→D^−p_α (A/I^r|A, D^i+p_β (rY|A/I^r, M/I^rM)) That is,E_αβ² (∞) = 0 forα6= 0. Also,E_0β² (∞) = 0 forp6= 0. So theE-spectral sequences degenerate to edge map isomorphisms

E_p,n−p¹ (∞)∼=

(0 p6= 0

Dⁱ_n(rY|A/I^r, M/I^rM) p= 0

Thus the E-spectral sequence degenerates to the desired edge map isomorphisms.

Corollary 2.6 (Formal functions theorem for Andr´e–Quillen homology). Under the hypotheses of the previous corollary, the canonical map

D_nⁱ(X|A, M)^b−→lim←−

r

D_nⁱ(rY|A/I^r, M/I^rM)

is an isomorphism for all n∈Z, i≥0, where Dⁱ_n(X|A, M)^b denotes the I-adic completion of D_nⁱ(X|A, M).

Proof. Take lim←−r of the pro isomorphism of the previous corollary.

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Remark 2.7. Let A be a Noetherian ring, X a proper scheme over A of finite Krull dimension, and I ⊆ A an ideal with respect to which A is assumed to be adically complete. Applying Corollary 2.6 to M = O_X and noting that D_nⁱ(X|A) is a finitely generatedA-module, hence already I-adically complete, we obtain

Dⁱ_n(X|A)−→^' lim←−

r

Dⁱ_n(rY|A/I^r).

2.3 Pro descent for AQ homology with respect to blow-up squares With the formal function theorems for Andr´e–Quillen homology established, we may now proved that Andr´e–Quillen homology satisfies the desired pro Mayer–Vietoris property with respect to abstract blow-up squares. We begin by developing further the results of Theorem2.4in the case thatπ is a an isomorphism away fromV(I):

Lemma 2.8. Let k→A be a homomorphism of Noetherian rings,I ⊆A an ideal, and X a proper scheme over A of finite Krull dimension such that the induced morphism X\V(IO_X) → SpecA\V(I) is an isomorphism. Then the following canonical maps are isomorphisms for alln∈Z, i≥0:

(i) {D_nⁱ(X|A, I^rO_X)}_r−→ {I^rD_nⁱ(X|A)}_r

(∗)∼= 0 (vanishing(∗) not necessarily valid if i=n= 0).

(ii) {D_nⁱ(A|k, I^r)} −→ {D_nⁱ(X|k, I^rO_X)}_r.

Proof. (i): The short exact sequences 0 → I^rO_X → O_X → O_X/I^rO_X → 0 induce a long exact sequence of proA-modules

· · · −→ {D_nⁱ(X|A, I^rO_X)}_r−→Dⁱ_n(X|A)−→ {Dⁱ_n(X|A,O_X/I^rO_X)}_r−→ · · · According to Theorem 2.4(iv), {D_nⁱ(X|A,O_X/IO_X)}_r ∼= {Dⁱ_n(X|A)⊗_AA/I^r}_r, from which it follows that the long exact sequences breaks into short sequences and induce the desired isomorphisms {D_nⁱ(X|A, I^rO_X)}_r→ {I^' ^rD_nⁱ(X|A)}_r.

Regarding the vanishing claim, suppose first thati= 0; thenD⁰_n(X|A) =H⁻ⁿ(X,O_X), which is a finitely generated A-module (since X is proper over A) which is supported on V(I) when n 6= 0 (since X \V(IO_X) ∼= SpecA\V(I)), and hence is killed by a power ofI. Next suppose i > 0; in this case the vanishing claim merely requires that X be essentially of finite type over A; induction on the size of an affine open cover of X reduces us to proving that if B is an essentially finite type A-algebra then D_nⁱ(B|A) is killed by a power ofI. ButDⁱ_n(B|A) is a finitely generatedB-module supported on V(IB), since Andr´e–Quillen homology behaves well under localisation; thus it is killed by a power ofI, completing the proof of all the vanishing claims.

(ii): The higher Jacobi–Zariski spectral sequence states that for eachr ≥1 there is a natural, third quadrant, bounded, spectral sequence

E_pq¹ (r) =⇒D_p+qⁱ (X|k, I^rO_X)

which vanishes outside −i ≤ p ≤ 0, and whose columns in the range −i≤ p ≤ 0 are described by natural, first quadrant spectral sequences

E_αβ² (r) =D^−p_α (A|k, D_β^i+p(X|A, I^rO_X)) =⇒E_p,α+β−p¹ (r).

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According to part (i), {E_αβ² (r)}_r= 0 unless β = 0 and p=−i, whence the limit of the spectral sequences collapse to edge map isomorphisms

{D_nⁱ(A|k, D⁰₀(X|A, I^rO_X))}_r→ {D^' _nⁱ(X|k, I^rO_X)}_r.

Since {D₀⁰(X|A, I^rO_X)}_r ∼={I^rD₀⁰(X|A)}_r = {I^rH⁰(X,O_X)}_r by part (i) and the usual description of Andr´e–Quillen homology when i = 0, we will have completed the proof as soon as we show that the canonical map

{I^r}_r−→ {I^rH⁰(X,O_X)}_r

is an isomorphism of proA-modules. WriteB:=H⁰(X,O_X), which is a finiteA-algebra with the property that A_p →^' B ⊗_AA_p for each prime ideal p ⊆ A not containing I; hence the kernel K and cokernel C of the structure map A→ B are killed by a power of I. Now consider the following commutative diagram of proA-modules:

0 ^//K ^//

(1)

A ^//

(2)

B ^//

(3)

C ^//

(4)

0

0 ^//{K⊗_AA/I^r}_r ^//{A/I^r}_r ^//{B⊗_AA/I^r}_r ^//{C⊗_AA/I^r}_r ^//0 The top row is an exact sequence of finitely generated A-modules, whence the bottom row is an exact sequence of proA-modules by Theorem1.1(ii). SinceK andCare killed by a power of I, maps (1) and (4) are isomorphisms, and so the central square in the diagram is bicartesian. It follows that maps (2) and (3) have isomorphic kernels; i.e., {I^r}_r→ {I^' ^rB}_r, as required.

Corollary 2.9. Let k be a Noetherian ring, let X⁰→X be a proper morphism between Noetherian k-schemes of finite Krull dimension, and suppose that I ⊆ O_X is an ideal sheaf such that the induced morphism X⁰ \V(IO_X⁰) → X\V(I) is an isomorphism.

Then the canonical map

{Dⁱ_n(X|k,I^r)}_r−→ {Dⁱ_n(X⁰|k,I^rO_X⁰)}_r is an isomorphism for all n∈Z, i≥0.

Proof. By the usual two inductions starting with an affine open cover ofX (since X is Noetherian, hence quasi-separated) we may assume thatX = SpecA is affine, in which case the statement is precisely Lemma 2.8(ii) (with X rechristened X⁰).

We are now equipped to prove our main theorem about Andr´e–Quillen homology of schemes, namely that it satisfies the pro Mayer–Vietoris property with respect to abstract blow-up squares. Recall that an abstract blow-up square of schemes is a pull- back diagram

Y⁰ ^//

X⁰

Y ^//X

where X⁰ →X is proper, Y → X is a closed embedding, and the induced map on the open subschemes X⁰\Y⁰→X\Y is an isomorphism.

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To state our descent result we need the following piece of notation which we did not introduce in Remark2.1when discussing hypercohomology: ifM^• is a cochain complex of sheaves, andM^{• ∼}→ F^•is a quasi-isomorphism to a cochain complex of acyclic sheaves, then set

H(X, M^•) :=F^•(X),

which is well-defined up to quasi-isomorphism and satisfiesH^∗(H(X, M^•)) =H^∗(X, M^•) by definition.

The following theorem is true more generally with a coherent moduleM in place of O_X, but we omit it to simplify the notation:

Theorem 2.10 (Pro descent for AQ homology wrt. abstract blow-ups). Let k be a Noetherian ring, and let

Y⁰

//X⁰

Y ^//X

be an abstract blow-up square of Noetherian, finite Krull dimensionalk-schemes. Then the following square of cochain complexes of pro k-modules is homotopy cartesian

H(X,Lⁱ_X_|k,neg) ^//

H(X⁰,Lⁱ_X⁰_|k,neg)

{H(rY,Lⁱ_rY_|k,neg)}_r ^//{H(rY⁰,Lⁱ_rY⁰_|k,neg)}_r resulting in a long exact, Mayer–Vietoris sequence of pro k-modules

· · · −→Dⁱ_n(X|k)−→ {Dⁱ_n(rY|k)}_r⊕ {Dⁱ_n(X⁰|k)}_r−→Dⁱ_n{(rY⁰|k)}_r−→ · · · Proof. According to Theorem 2.4(i), the cohomologies of the complexes in the bottom row of the diagram are unchanged if we replace them by

{H(X,Lⁱ_X|k,neg⊗_O_X O_X/I^r)}_r−→ {H(X⁰,Lⁱ_X⁰_|k,neg⊗_O

X0 O_X⁰/I^rO_X⁰)}_r, whereI is the sheaf of ideals definingY. Having made this replacement, we must show that the homotopy fibres of the two resulting vertical arrows

H(X,Lⁱ_X|k,neg)

H(X⁰,Lⁱ_X⁰_|k,neg)

{H(X,Lⁱ_X_|k,neg⊗_O_X O_X/I^r)}_r {H(X⁰,Lⁱ_X⁰_|k,neg⊗_O

X0 O_X⁰/I^rO_X⁰}_r) are quasi-isomorphic. These homotopy fibres are respectively

{H(X,Lⁱ_X|k,neg⊗_O_X I^r)}_r and {H(X⁰,Lⁱ_X⁰|k,neg⊗_O

X0 I^rO_X⁰)}_r,

between which the canonical map is indeed a quasi-isomorphism since{Dⁱ_n(X|k,I^r)}_r→^' {D_nⁱ(X⁰|k,I^rO_X⁰)}_r for all n∈Z,i≥0, by Corollary2.9.

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3 Formal function properties and pro descent for HH , HC , THH , TC , and K -theory

Now we extend the main results of Section 2 to Hochschild and cyclic homology of schemes, and their topological counterparts, and then deduce thatK-theory also satisfies the pro Mayer–Vietoris property with respect to abstract blow-up squares of varieties.

3.1 Hochschild and cyclic homology

Here we discuss the theories of derived Hochschild and cyclic homology for schemes which will concern us.

Let k → A be a homomorphism of commutative rings. Given an A-module M, we let H∗^naive,k(A, M) denote the “usual” Hochschild homology of A as a k-algebra with coefficients in M; it is the homology of the Hochschild complex C_•^k(A, M). In particular, HH∗^naive,k(A) = H∗^naive,k(A, A) denotes the usual Hochschild homology of A as ak-algebra. However, we will work throughout with the derived version of Hochschild homology, for which we use the notation H_∗^k(A, M). That is, letting P• → A be a simplicial resolution of A by freek-algebras, H_∗^k(A, M) is defined to be the homology (of the diagonal) of the bisimplicial A-module

C_q^k(P_p, M) =M⊗_kP_p^⊗^k^q. (p, q≥0) In the special case A=M we writeHH_∗^k(A) =H_∗^k(A, A).

Next we discuss cyclic homology. Firstly, HC∗^naive,k(A) denotes the usual cyclic homology of the k-algebra A, which is the homology of the cyclic bicomplex CC_••^k (A).

Just as for Hochschild homology we prefer to denote by HC_∗^k(A) the derived version, defined as the homology of (the diagonal of) the simplicial tricomplexCC_••^k(P•), where P• → A is a simplicial resolution of A by free k-algebras. The usual SBI sequence remains valid in the derived setting:

· · · −→HH_n^k(A)−→^I HC_n^k(A)−→^S HC_n−2^k (A)−→ · · ·^B

Next consider a Noetherian schemeX of finite Krull dimension overk. As in Section 2.1, let Pe_•^k(X) denote the simplicial sheaf ofk-algebras obtained by degree-wise sheafi- fying U 7→P•(O_X(U)), whereP•(O_X(U))→ O_X(U) is a functorially chosen simplicial resolution by free k-algebras. Given a quasi-coherent O_X-moduleM, set

C_•^k(X, M) :=M⊗_kPe_•^k(X)^⊗^k^•

and letC_neg^k (X, M) be the cochain complex of quasi-coherentO_X-modules, supported in negative degrees, obtained by negating the numbering of the chain complex ofC_•^k(X, M).

Define the derived Hochschild homology of X, relative to k, with coefficients in M to be its hypercohomology

H_∗^k(X, M) :=H^−∗(X, C_neg^k (X, M)).

In the special caseM =O_X we writeC_•^k(X) =C_•^k(X,O_X) andHH_∗^k(X) =H_∗^k(X,O_X).

The obvious analogues of the “Agreement on affines”, “Mayer–Vietoris property”, “Hy- percohomology spectral sequence”, and “Homology l.e.s.” stated for Andr´e–Quillen homology in Section 2.1are true for this derived Hochschild homology of schemes.

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We omit the analogous construction of derived cyclic homology for schemes since the procedure should now be clear. As in the affine case, the usual SBI sequence remains valid

· · · −→HH_n^k(X)−→^I HC_n^k(X)−→^S HC_n−2^k (X)−→ · · ·^B

The relationships between Andr´e–Quillen homology, derived Hochschild/cyclic homology, and usual Hochschild/cyclic homology are summarised by the next lemma:

Lemma 3.1. Let kbe a ring, X a Noetherian scheme of finite Krull dimension overk, andM a quasi-coherentO_X-module. Then:

(i) If X is flat over k then the canonical maps

H_n^k(X, M)→H_n^naive,k(X, M) and HC_n^k(X)→HC_n^naive,k(X)

are isomorphisms for alln ∈Z, where H^naive,k and HC^naive,k denote the “usual”

Hochschild and cyclic homology of a scheme, as constructed in, e.g., [34].

(ii) There is a bounded spectral sequence of k-modules

E_pq² =D^q_p(X|k, M) =⇒H_p+q^k (X, M).

Proof. We briefly explain the proofs of these standard results in the affine case X = SpecA, after which the case of a scheme should be clear. (i): IfAis flat overkthen the mapM⊗_kP•^⊗^k^q →M⊗_kA^⊗^k^qis a quasi-isomorphism for allq ≥0, and so the diagonal of the bisimplicialA-moduleC_•^k(P•, M) is quasi-isomorphic toC_•^k(A, M). (ii): This is the spectral sequence associated to the bisimplicial A-module C_•^k(P•, M), since C_•^k(P_p, M) has homology Ω^∗_P

p/k⊗_P_pM for each p≥0.

The following formal function results for Hochschild homology are the analogues of those given for Andr´e–Quillen homology in Section2:

Theorem 3.2. Letk→Abe a homomorphism of Noetherian rings, andπ :X→SpecA a Noetherian scheme overA of finite Krull dimension; letI ⊆ O_X be an ideal sheaf and M a coherent O_X-module. Then:

(i) The canonical map {H_n^A(X, M/I^rM)}_r −→ {H_n^A(rY, M/I^rM)}_r is an isomorphism for all n∈Z.

(ii) Assuming that π is essentially of finite type, the canonical map of pro sheaves {H^A_n(X, M)⊗_O_X O_X/I^r}_r−→ {H^A_n(X, M/I^rM)}_r is an isomorphism for alln∈ Z.

Now suppose further that π is proper and I=IO_X for some idealI ⊆A. Then (iii) H_n^A(X, M) is a finitely generated A-module for all n∈Z,

and the following canonical maps are isomorphisms for alln∈Z: (iv) {H_n^A(X, M)⊗_AA/I^r}_r−→ {H_n^A(X, M/I^rM)}_r.

(v) {H_n^A(X, M)⊗_AA/I^r}_r−→ {H_n^A/I^r(rY, M/I^rM)}_r.

Now suppose still further that the induced morphism X\V(IO_X) → SpecA\V(I) is an isomorphism. Then the following canonical maps are isomorphisms for alln∈Z:

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(vi) {H_n^A(X, I^rO_X)}_r −→ {I^rHH_n^A(X)}_r

(∗)∼= 0 (vanishing (∗) not necessarily valid ifn= 0).

(vii) {H_n^k(A, I^r)}_r−→ {H_n^k(X, I^rO_X))}_r.

Proof. Thanks to the Andr´e–Quillen to Hochschild homology spectral sequence for schemes, i.e., Lemma 3.1(ii), these claims immediately reduce to the analogous asser- tions for Andr´e–Quillen homology, which we have already proved. For the sake of reference: (i)–(iv) correspond to Theorem 2.4; (v) to Corollary 2.5; and (vi)–(vii) to Lemma 2.8.

Given a closed embeddingY →X of Noetherian schemes of finite Krull dimension, we will write HH_∗^k(X, Y), HC_∗^k(X, Y) for the relative derived Hochschild and cyclic homology groups, fitting into long exact sequences

· · · −→HH_n^k(X, Y)−→HH_n^k(X)−→HH_n^k(Y)−→ · · ·

· · · −→HC_n^k(X, Y)−→HC_n^k(X)−→HC_n^k(Y)−→ · · ·

Now we prove the analogue of Theorem 2.10 for HH and HC, namely that they also satisfy the pro Mayer–Vietoris property for abstract blow-up squares:

Theorem 3.3(Pro descent forHHandHCwrt. abstract blow-ups). Letkbe a Noethe- rian ring, and let

Y⁰

//X⁰

Y ^//X

be an abstract blow-up square of Noetherian, finite Krull dimensional k-schemes. Then the canonical maps

{HH_n^k(X, rY)}_r−→ {HH_n^k(X⁰, rY⁰)}_r, {HC_n^k(X, rY)}_r −→ {HC_n^k(X⁰, rY⁰)}_r are isomorphisms of pro abelian groups for all n∈Z.

Proof. The claim for HH is equivalent to the statement that the following square of cochain complexes of pro k-modules is homotopy cartesian, where His the hypercohomology replacement functor introduced immediately proceeding Theorem 2.10:

H(X, C_neg^k (X)) ^//

H(X⁰, C_neg^k (X⁰))

{H(rY, C_neg^k (rY))}_r ^//{H(rY⁰, C_neg^k (rY⁰))}_r

This is proved exactly as it was for Andr´e–Quillen homology in Theorem2.10, using the appropriate parts of Theorem 3.2. More precisely, by Theorem 3.2(i) we may replace the bottom of the diagram by

{H(X, C_neg^k (X,O_X/I^r))}_r −→ {H(X, C_neg^k (X⁰,O_X⁰/I^rO_X⁰))}_r

without changing the cohomologies of the complexes, where I is the sheaf of ideals defining Y. Then the map between the homotopy fibres of the vertical arrows becomes

{H(X, C_neg^k (X,I^r))}_r−→ {H(X, C_neg^k (X⁰,I^rO_X⁰))}_r,

Pro cdh-descent for cyclic homology and K-theory

Contents

0 Introduction

1 Review of pro modules and Artin–Rees properties

2 Formal function properties and pro descent for Andr´ e–Quillen homology

3 Formal function properties and pro descent for HH , HC , THH , TC , and K -theory