Degree-wise continuity and continuity of HH, THH, TR r , etc

Combining the restriction spectral sequence of Proposition 4.1 with our earlier finite generation results and the Artin–Rees theorem, we may now establish our “degree-wise continuity” results, starting with the following lemma:

Lemma 4.2. Let A be a Noetherian ring, I ⊆ A an ideal, M a finitely generated A-module, andn≥0. Consider the canonical maps:

{HH_n(A, M)⊗_AA/I^s}_s−−→ {HH⁽ⁱⁱ⁾ _n(A, M/I^sM)}_s−→ {HH⁽ⁱ⁾ _n(A/I^s, M/I^sM)}_s {THH_n(A, M)⊗_AA/I^s}_s−−→ {THH⁽ⁱⁱ⁾ _n(A, M/I^sM)}_s−→ {THH⁽ⁱ⁾ _n(A/I^s, M/I^sM)}_s Then maps (i) are isomorphisms. If A is further assumed to be m-AQ-finite for some m≥0, and M is annihilated by m, then maps (ii) are also isomorphisms.

Proof. (i): By Proposition 4.1, for eachs≥1 there is a first quadrant spectral sequence ofA-modules

E_ij²(s) =HHi(A/I^s,Tor^A_j(A/I^s, M/I^sM)) =⇒ HHi+j(A, M/I^sM).

These assemble to a spectral sequence of proA-modules

E_ij²(∞) ={HH_i(A/I^s,Tor^A_j (A/I^s, M/I^sM))}_s =⇒ {HH_i+j(A, M/I^sM)}_s. But{Tor^A_j(A/I^s, M/I^sM)}_s= 0 for allj≥1, by Corollary 1.2, so this spectral sequence collapses to edge map isomorphisms {HH_n(A, M/I^sM)}_s → {HH^' _n(A/I^s, M/I^sM)}_r of proA-modules for alln≥0; this proves isomorphism (i) forHH.

Isomorphism (i) forTHH can be proved in exactly the same way asHH, using Brun’s THH version of Proposition 4.1 in [3, Thm. 6.2.10]. However, we will show that it can

also be deduced from the HH case using the Pirashvili–Waldhausen spectral sequence, which was described in Section 1.3.1.

The P.–W. spectral sequences for A and its modules M/I^sM assemble to a spectral sequence of proA-modules

F_ij²(∞) ={HH_i(A, THH_j(Z, M/I^sM))}_s =⇒ {THH_i+j(A, M/I^sM)}_s.

Similarly, the P.–W. spectral sequences for the ringsA/I^s and modulesM/I^sM assemble to a spectral sequence of proA-modules

0F_ij²(∞) ={HH_i(A/I^s, THHj(Z, M/I^sM))}_s =⇒ {THH_i+j(A/I^sM, M/I^sM)}_s. By naturality there is a morphism of spectral sequences⁰F(∞) →F(∞), and the proof of (i) will be complete if we show it is an isomorphism on the terms on the second pages, i.e., that

{HH_i(A, THHj(Z, M/I^sM))}_s → {HH^' _i(A/I^s, THHj(Z, M/I^sM))}_s (†) for all i, j ≥ 0. However, B¨okstedt’s calculation of THH(Z,−) as torsion or cortorsion, and Corollary 1.4, imply that

{THH_j(Z, M/I^sM)}_s∼={THH_j(Z, M)⊗_AA/I^s}_s. So the map (†) is an isomorphism if and only if the map

{HH_i(A, THHj(Z, M)⊗_AA/I^s)}_s −→ {HH_i(A/I^s, THHj(Z, M)⊗_AA/I^s)}_s is an isomorphism; since THH_j(Z, M) is a finitely generated A-module by B¨okstedt’s caclcuation, this is indeed an isomorphism by the already establishedHH case of (i).

(ii): Now assume further thatA ism-AQ-finite and that M is killed by m. Universal coefficient spectral sequences forHH assemble to a spectral sequence of pro A-modules

0E_ij²(∞) ={Tor^A_i (A/I^s, HH_j(A, M))}_s =⇒ {HH_i+j(A, M/I^sM)}_s.

But the the A-modules HHj(A, M) are finitely generated for all q ≥ 0 by assumption and Lemma 3.1, so {Tor^A_i (A/I^s, HHj(A, M))}_s = 0 for i ≥ 1 by the Artin–Rees The-orem 1.1(i). Thus we again obtain edge map isomorphisms {HH_n(A, M)⊗_AA/I^r}_r →^' {HH_n(A, M/I^rM)}_r, completing the proof of isomorphism (ii) for HH. The proof for THH is verbatim equivalent.

We now reach our main degree-wise continuity results, which establishes Theorem 0.4 of the Introduction:

Theorem 4.3. Let A be a Noetherian, F-finite Z(p)-algebra, I ⊆A an ideal, and v ≥1.

Then the canonical maps

(i) {HH_n(A;Z/p^v)⊗_AA/I^s}_s−→ {HH_n(A/I^s;Z/p^v)}_s

(ii) {TR^r_n(A;Z/p^v)⊗_Wr(A)W_r(A/I^s)}_s−→ {TR^r_n(A/I^s;Z/p^v)}_s are isomorphisms for alln≥0, r≥1.

Proof. (i): This follows from the HH part of Lemma 4.2 in a relatively straightforward way: to keep the proof clear we will use∞ notation for all the proA-modules. As in the proof of Theorem 3.7(i), there is a long exact sequence ofA-modules

· · · −→HHn−1(A, A[p^v])−→HH_n(A;Z/p^v)−→HH_n(A, A/p^vA)−→ · · · ,

all of which are finitely generated. Hence we may base change byA/I^∞, as in the Artin–

Rees Theorem 1.1(ii), to obtain a long exact sequence of proA-modules

· · · →HHn−1(A, A[p^v])⊗_AA/I^∞→HH_n(A;Z/p^v)⊗_AA/I^∞→HH_n(A, A/p^vA)⊗_AA/I^∞→ · · · . (1) Replacing A by A/I^∞, and using the isomorphisms of Corollary 1.4 with M = A, there is also a long exact sequence of proA-modules

· · · →HHn−1(A/I^∞, A[p^v]⊗_AA/I^∞)→HHn(A/I^∞;Z/p^v)→HHn(A/I^∞, A/p^vA⊗_AA/I^∞)→ · · ·. (2) The obvious map from (1) to (2) induces an isomorphism on all the terms with coefficients inA[p^v] andA/p^vA, by Theorem 3.6 and Lemma 4.2, hence also induces an isomorphism on the Z/p^vZ terms, as desired.

(ii): If r= 1 then the claim is that the canonical map{THH_n(A;Z/p^v)⊗_AA/I^s}_s→ {THH_n(A/I^s;Z/p^v)}_sis an isomorphism; the proof of this is verbatim equivalent to part (i). So now assumer >1 and proceed by induction.

We may compare the fundamental long exact sequences with finite coefficients for bothA and A/I^s as follows:

· · · ^//π_n(THH(A;Z/p^v)_hCpr) ^//

TR_n^r+1(A;Z/p^v) ^//

TR^r_n(A;Z/p^v) ^//

· · ·

· · · ^//π_n(THH(A/I^s;Z/p^v)_hCpr) ^//TR^r+1_n (A/I^s;Z/p^v) ^//TR^r_n(A/I^s;Z/p^v) ^//· · · As we saw in the proof of Theorem 3.7(ii), the top row consists of finitely generated Wr+1(A)-modules; so by Theorem 2.7(i), it remains exact after base changing by the proW_r+1(A)-algebraW_r+1(A/I^∞). Simultaneously assembling the bottom row into pro W_r+1(A)-modules therefore yields a map of long exact sequences of proW_r+1(A)-modules (note that we have rotated the diagram for the sake of space):

... ...

{TR^r_n(A;Z/p^v)⊗_Wr+1(A)Wr+1(A/I^s)}_s

OO //{TR^r_n(A/I^s;Z/p^v)}_s

OO

{TR^r+1_n (A;Z/p^v)⊗_Wr+1(A)W_r+1(A/I^s)}_s

OO //{TR^r+1_n (A/I^s;Z/p^v)}_s

OO

{π_n(THH(A;Z/p^v)_hCpr)⊗_Wr+1(A)W_r+1(A/I^s)}_s

OO //{π_n(THH(A/I^s;Z/p^v)_hCpr)}_s

OO

...

OO

...

OO

By the five lemma and inductive hypothesis, it is enough to prove that the bottom horizontal arrow in the diagram, namely

πn(THH(A;Z/p^v)_hCpr)⊗_Wr+1(A)Wr+1(A/I^∞)−→πn(THH(A/I^∞;Z/p^v)_hCpr), is an isomorphism for alln≥0. Both sides are the abutment of natural group homology spectral sequences, so it is enough to check that the map of spectral sequence is an isomorphism on the second page, namely that the canonical map

Hi(Cp^r, THHj(A;Z/p^v))⊗_Wr+1(A)Wr+1(A/I^∞)−→Hi(Cp^r, THHj(A/I^∞;Z/p^v)) (†) is an isomorphism for all i, j ≥ 0. Since Hi(Cp^r, THHj(A;Z/p^v)) is a finitely generated A-module, the left side of (†) is precisely H_i(C_p^r, THH_j(A;Z/p^v))⊗_AA/I^∞ by Theorem 2.7(ii); meanwhile, the right side isH_i(C_p^r, THH_j(A;Z/p^v)⊗_AA/I^∞) by part (ii).

Therefore it is now enough to prove that the map

Hi(Cp^r, THHj(A;Z/p^v))⊗_AA/I^∞−→Hi(Cp^r, THHj(A;Z/p^v)⊗_AA/I^∞)

is an isomorphism; this follows from the finite generation ofTHHj(A;Z/p^v) and Corollary 1.3.

From the previous degree-wise continuity theorem we obtain a description of the re-lationship between HH, THH, and TR^r of A and its I-adic completion Ab = lim←−sA/I^s, proving Corollary 0.5 of the Introduction:

Corollary 4.4. Let A be a Noetherian, F-finite Z(p)-algebra, and I ⊆A an ideal; let Ab denote the I-adic completion of A. Then all of the following maps (not just the composi-tions) are isomorphisms for alln≥0 and v, r≥1: Proof. We claim that each of the following canonical maps is an isomorphism:

HHn(A;Z/p^v)⊗_AAb−→lim←−

s

HHn(A;Z/p^v)⊗_AA/I^s−→lim←−

s

HHn(A/I^s;Z/p^v).

Firstly,HH_n(A;Z/p^v) is a finitely generatedA-module by Theorem 3.7, andAis Noethe-rian, so standard commutative algebra, e.g. [24, Thm. 8.7], implies that the first map is an isomorphism. Secondly, Theorem 4.3(i) implies that the second map is an isomorphism.

However, Lemma 2.8 implies thatAbis also a Noetherian, F-finite Z(p)-algebra, so ap-plying the same argument toAbwith respect to the idealIAbwe obtain another composition of isomorphisms

The proofs of the isomorphisms forTR^r are exactly the same as for HH, except that forTR^r one must also note thatWr(A) is Noetherian by Langer–Zink (Theorem 2.6) and use Lemma 2.3.

Whereas the previous two continuity results have concerned individual groups, we now prove the spectral continuity of THH, TR^r, etc. under our usual hypotheses; this establishes Theorem 0.6 of the Introduction:

Theorem 4.5. LetA be a Noetherian, F-finite Z(p)-algebra, andI ⊆A an ideal; assume thatA is I-adically complete. Then, for all 1≤r ≤ ∞, the following canonical maps of spectra are weak equivalences after p-completion:

TR^r(A;p)−→holim

s TR^r(A/I^s;p) TC^r(A;p)−→holim

s TC^r(A/I^s;p)

Proof. We must prove that each of the given maps of spectra is a weak equivalence with Z/p^vZ-coefficients for all v ≥ 1. Firstly, fixing 1 ≤ r < ∞, the homotopy groups of holimsTR^r(A/I^s;Z/p^v) fit into short exact sequences

0→lim←−

s

1TR^r_n+1(A/I^s;Z/p^v)→πn(holim

s TR^r(A/I^s;Z/p^v))→lim←−

s

TR^r_n(A/I^s;Z/p^v)→0.

Theorem 4.3(i) implies that the left-most term is lim←−¹sTR^r_n+1(A;Z/p^v)⊗_Wr(A)Wr(A/I^s), which vanishes because of the surjectivity of the transition maps in the pro abelian group {TR^r_n+1(A;Z/p^v)⊗_Wr(A)W_r(A/I^s)}_s. In conclusion, the natural map

πn(holim

s TR^r(A/I^s;Z/p^v))−→lim←−

s

TR^r_n(A/I^s;Z/p^v)

is an isomorphism for all n ≥ 0. But since A is already I-adically complete, Corollary 4.4 states thatTR^r_n(A;Z/p^v)→lim←−sTR^r_n(A/I^s;Z/p^v) is also an isomorphism for all n≥ 0. So the map TR^r(A;Z/p^v) → holim_sTR^r(A/I^s;Z/p^v) induces an isomorphism on all homotopy groups, as required to prove the first weak equivalence.

The claims for TC^r, TR^∞ = TR, and TC^∞ = TC then follow since homotopy limits commute.

In the remainder of this section we consider straightforward consequences of the pre-vious three results in special situations. We begin with the case in whichp is nilpotent inA:

Corollary 4.6. Let Abe a Noetherian, F-finiteZ(p)-algebra, andI ⊆Aan ideal; assume that p is nilpotent in A, and let Ab denote the I-adic completion of A. Then all of the following maps (not just the compositions) are isomorphisms for all n≥0, r≥1:

{HH_n(A)⊗_AA/I^s}_s−→ {HH_n(A/I^s)}_s

{TR^r_n(A;p)⊗_Wr(A)W_r(A/I^s)}_s−→ {TR^r_n(A/I^s;p)}_s HH_n(A)⊗_AAb−→HH_n(A)b −→lim←−sHH_n(A/I^s)

TR^r_n(A;p)⊗_Wr(A)Wr(A)b −→TR^r_n(A;b p)−→lim←−sTR^r_n(A/I^s;p)

Moreover, the maps of spectra in the statement of Theorem 4.5 are weak equivalences without p-completing.

Proof. Note that p is also nilpotent in W_r(A), by Lemma 2.9(ii). So, fixing r ≥ 1, we may pickv0 such that the groups

HHn(A), HHn(A/I^s), THHn(A), THH(A/I^s), TR^r_n(A;p), TR^r_n(A/I^s;p)

are annihilated by p^v for all n≥0, s≥1. Hence the spectra appearing in Theorem 4.5 are allp-complete, and the isomorphisms follow from Theorem 4.3 and Corollary 4.4 by examining the short exact sequences for homotopy groups with finite coefficients, in the usual way.

Remark 4.7. Some of the statements of Corollary 4.6 hold for rings other than Noethe-rian, F-finite Z(p)-algebras in which p is nilpotent. In particular, if A is a Noetherian ring which is AQ-finite overZ(e.g., an essentially finite type Z-algebra) and I ⊆A is an ideal, then the proHH andTHH =TR¹ isomorphisms of Corollary 4.6 hold; indeed, this follows immediately from Lemma 4.2 withm= 0 and M =A.

Suppose now, in addition to be being Noetherian and AQ-finite, thatA is an F-finite Z(p)-algebra (e.g., an essentially finite typeZ(p)-algebra). Then the proTR^risomorphisms of Corollary 4.6 hold: this is proved by verbatim repeating the proof of Theorem 4.3(ii) integrally instead of with finite coefficients.

Now, in stark contrast with the case in which p is nilpotent, we consider the case whereI =pA; here our methods yield a new proof, albeit under different hypotheses, of a result of Geisser and Hesselholt, as we will discuss in Remark 4.9:

Corollary 4.8. LetAbe a Noetherian, F-finiteZ(p)-algebra. Then the following canonical maps are isomorphisms for all n≥0 and v, r≥1:

HH_n(A;Z/p^v)−→ {HH_n(A/p^sA;Z/p^v)}_s TR^r_n(A;Z/p^v)−→ {TR^r_n(A/p^sA;Z/p^v)}_s TC_n^r(A;Z/p^v)−→ {TC_n^r(A/p^sA;Z/p^v)}_s

Moreover, for1≤r ≤ ∞, all of the following maps (not just the compositions) of spectra are weak equivalences afterp-completion:

TR^r(A;p)−→TR^r(A^b_p;p)−→holim_sTR^r(A/p^sA;p), TC^r(A;p)−→TC^r(A^b_p;p)−→holimsTC^r(A/p^sA;p).

Proof. The groups HHn(A;Z/p^v) and TR^r_n(A;Z/p^v) are annihilated by p^v. Since we proved in Lemma 2.4 thatp^vW_r(A) containsW_r(p^sA) for s0, we deduce that

{HH_n(A;Z/p^v)⊗_AA/p^sA}_s=HH_n(A;Z/p^v),

{TR^r_n(A;Z/p^v)⊗_Wr(A)Wr(A/p^sA)}_s=TR^r_n(A;Z/p^v).

Hence the desired proHH and TR^r isomorphisms follow from Theorem 4.3. The proTC isomorphism then follows in the usual way by applying the five lemma to the long exact sequence relatingTC^r,TR^r, and TR^r−1.

SinceA^b_pis also a Noetherian, F-finiteZ(p)-algebra by Lemma 2.8, and sinceA^b_p/p^sA^b_p ∼= A/p^sA, applying isomorphism (ii) to both Aand A^b_p yields

TR^r_n(A;Z/p^v)∼={TR^r_n(A/p^s;Z/p^v)}_s∼=TR^r_n(A^b_p;Z/p^v)

for any integer r ≥ 1. This proves that TR^r(A;p)^b_p ' TR^r(A^b_p;p)^b_p; the same follows for TC^r,TR andTC by taking homotopy limits.

The second column of maps between spectra are weak equivalences afterp-completing by Theorem 4.5.

Remark 4.9. The isomorphisms of Corollary 4.8 were proved by Geisser and Hesselholt [10, §3] for any (possibly non-commutative) ringA in which p is a non-zero divisor. The key assertion is the proTHH isomorphism, namely

THHn(A;Z/p^v)→ {THH^' _n(A/p^sA;Z/p^v)}_s,

as they have a short argument to deduce the corresponding TR^r isomorphism from this by induction.

This THH isomorphism can also be proved using our methods as follows: mimick-ing the proof of Lemma 4.2 via Proposition 4.1 and the Artin–Rees vanishmimick-ing result {Tor^A_n(A/I^s, A/I^s)}_s = 0, it is enough to show that

{Tor^A_n(A/p^sA, A/p^sA)}_s= 0

for alln >0 wheneverp is a non-zero divisor of a possibly non-commutative ringA. But in such a situation we may calculate Tor using the projective resolution 0→A ^×p

s

−−→A→ A/p^sA→0 ofA/p^sA, and from this calculation it easily follows that the map

Tor^A₁(A/p^2sA, A/p^2sA)−→Tor^A₁(A/p^sA, A/p^sA) is zero, as required.