Bjørn Ian Dundas

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Sⁿ⊗A Bjørn Ian Dundas

Happy Birthday Goal Setup Chromatic importance Higher THH of HFp

Higher THH of the integers Higher THH of rings of integers Thanks

Higher topological Hochschild homology

j/w Lindenstrauss and Richter

Bjørn Ian Dundas

To celebrate Tom Goodwillie.

Dubrovnik, June 23, 2014

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First listing in Math. Sci. Net (29 years ago):

Goodwillie, Thomas G. On the general linear group

and Hochschild homology. Ann. of Math. (2) 121

(1985), no. 2, 383–407

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Today’s goal

Calculate the “higher topological Hochschild homology” of some rings of integers, using ideas of Kriz; Basterra and Mandell and the calculations of B¨ okstedt, Breen

(conjectured by Goodwillie); Lindenstrauss and Madsen.

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Setup

k – commutative ring spectrum A – commutative k-algebra X – space

X ⊗ A – commutative k-algebra

¹

. S

¹

⊗ A = HH

^k

(A) (e.g., if k = S, S

¹

⊗ A = THH(A) = {[q] 7→ A

^∧^(q+1)

}).

T

ⁿ

⊗ A = HH

^k

(. . . HH

^k

(A) . . . ) (T

ⁿ

= S

¹

× · · · × S

¹

).

From the cellular structure of T

ⁿ

one can build T

ⁿ

⊗ A by means of S

^d

⊗ A’s.

1McClure, Schw¨anzl, Vogt. Fancy equivariant versions irrelevant for this talk, but things may be derived without notice.

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Some calculations

Theorem

A – discrete k = Q-algebra.

π

n

(S

¹

⊗ A) = L

n

j=0

H

n^(j)

(A) (Hodge decomp.

^a

) π

n

(S

^d

⊗ A) = L

i+dj=n

H

_i^(j)_+j

(A) (d odd

^b

)

a=splitting of Goodwillie tower (Kantorovitz-McCarthy (02))

bPirashvili (00)

Gen. k: just a filtration. Some other calculations, e.g., for Thom spectra (Blumberg, Cohen, Schlichtkrull)

Theorem (Schlichtkrull) k = S

S

ⁿ

⊗ MU ≃ MU∧B

ⁿ⁺¹

U

₊

.

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Chromatic importance

When k = S, the higher dimensional analog of Goodwillie-Jones’ “negative cyclic homology”

(T

ⁿ

⊗ A)

^hTⁿ

detects chromatic behavior of

K

⁽ⁿ⁾

(A) = K (. . . K (A) . . . ).

Theorem (n ≤ 3, Rognes; 3 < n ≤ p, Veen)

p prime

v

_n−1

6= 0 in

k(n - 1)

∗

K

⁽ⁿ⁾

(F

p

)

^a

and in k(n - 1)

∗

(T

ⁿ

⊗ HF

_p

)

^hTⁿ

.

^b

ak(n - 1)∗= Fp[vn−1], |vn−1| = 2pⁿ⁻¹− 2, connective Morava K -theory

bActually detected by a class in H^∗(GLn(Z) ⋉ Tⁿ; π∗(Tⁿ⊗ HFp)).

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Higher THH of HF

p

B¨ okstedt/Breen

(conjectured by Goodwillie) p prime

π

∗

(S

¹

⊗ HF

_p

) = F

_p

[x] = P(x), |x| = 2

Warning

(Commutative connective HZ-algebras) 6∼

(Commutative simplicial Z-algebras).

Also, note difference to the alg. case

π

∗

(H(S

¹

⊗ F

p

)) = Γ(x).

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Higher THH of HF

p

Basterra/Mandell (∞)

Partial results by Veen (n ≤ 2p arXiv13), Bobkova/Lindenstrauss/Poirier/Richter/Zakharevich (n ≤ 2p + 2 arXiv13)

π

∗

(S

ⁿ

⊗ HF

_p

) ∼ = B

ⁿ

(x),

|x| = 2, n > 0, where

B

¹

= P(x), B

ⁿ

(x) = Tor

^Bⁿ⁻¹^(x)

(F

_p

, F

_p

).

Bⁿ can be given concretely in terms of generators and relations,

c.f. Cartan’s calculation of H_∗(K (π, n); Fp)

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π

_∗

(S

²

⊗ HF

p

) ∼ = B

²

(x) = E (σx), |σx| = 3

Sketch: π

∗

(S

¹

⊗ HF

_p

) = B

¹

(x) (B¨ okstedt/Breen).

S

²

⊗ HF

_p

= (D

²

a

S¹

D

²

) ⊗ HF

_p

∼ = (D

²

⊗ HF

_p

) ∧

_S1⊗HFp

(D

²

⊗ HF

_p

)

≃ HF

_p

∧

^L_S1⊗HFp

HF

_p

, Tor SS:

Tor

^π∗^∗^(S¹^⊗^HF^p⁾

(F

p

, F

p

) = Tor

^P(x)∗

(F

p

, F

p

) ⇒ π

∗

(S

²

⊗ HF

p

), no room for differentials or extensions:

π

∗

(S

²

⊗ HF

_p

) ∼ = B

²

(x) ∼ = E (σx).

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π

_∗

(S

ⁿ

⊗ HF

p

) ∼ = B

ⁿ

(x)

B¹= P(x), Bⁿ(x) = Tor^Bⁿ⁻¹^(x)(Fp, Fp)

π

∗

(S

²

⊗ HF

p

) ∼ = B

²

(x) ∼ = E (σx).

Unique ht. type of commutative HF

p

-algebras,

²

so S

²

⊗ HF

_p

≃ HF

_p

∨ Σ

³

HF

_p

≃ H(F

_p

[S

³

]) (square zero) as commutative HF

p

-algebras(!). . .

³

2Kriz? C.f. Lazarev’s (04) writeup of Basterra/Mandell’s identification of TAQ(HFp)

3Veen and BLPRZ do it algebraically. I have not seen Basterra/Mandell’s proof, but will return to some of the problems that have to be addressed

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Higher THH of integers

D, Lindenstrauss, Richter (∞)

π

∗

(S

ⁿ

⊗ HZ) is torsion in positive dimensions, and V (0)

∗

(S

ⁿ

⊗ HZ) ∼ = B

ⁿ

(x) ⊗ B

ⁿ⁺¹

(y ),

^a

n > 0, |x| = 2p, |y | = 2p − 2.

aV(0) = S/p is the mod p Moore spectrum.

B¹= P(x), Bⁿ(x) = Tor^Bⁿ⁻¹^(x)(Fp, Fp)

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Higher THH of integers

Relative constructions: A → B, ∗ → X ; X ⊗ (A, B) = (X ⊗ A)∧

A

B

⁴

commutative ring spectrum.

Examples:

S

¹

⊗ (A, B) = THH(A, B),

S

ⁿ

⊗ (HZ, HF

_p

) ≃ V (0)∧(S

ⁿ

⊗ HZ) S

ⁿ

⊗ ( ℓ

p

, HF

p

) ≃ V (1)∧(S

ⁿ

⊗ ℓ

p

)

⁵

4∗ ⊗ A = A

5ℓpAdams summand, V (1) = S/(p, v1)

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Higher THH of integers

Linearization: If A is a discrete ring, then the map of commutative ring spectra

(S

¹

⊗ HA)

_p

→ H(S

¹

⊗ A)

_p

is 2p − 1-connected

⁶

Hence, in a range, the Postnikov sections of (S

¹

⊗ HA)

p

are H of simplicial commutative rings.

6B¨okstedt’s calc. of THH(Z)

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Higher THH of integers

V (0)_∗(Sⁿ⊗ HZ) ∼= π∗(Sⁿ⊗ (HZ, HFp)) ∼= Bⁿ(x) ⊗ Bⁿ⁺¹(y )

sketch proof: B¨ o: ok for n = 1:

V (0)

∗

(S

¹

⊗ HZ) ∼ = B

¹

(x) ⊗ B

²

(y ) ∼ = P(x) ⊗ E (σy ),

|x| = 2p, |y | = 2p − 2, |σy | = 2p − 1.

Mapping to the first Postnikov section we get S

¹

⊗ (HZ, HF

p

)

^//

Postnikov

^//

HF

p

HF

p // '!&"%#$

..

//

S

²

⊗ (HZ, HF

p

)

78

7

8warning: pushouts are in commutative HFp-algebras.

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Higher THH of integers

sketch proof: B¨ o: ok for n = 1:

V (0)

∗

(S

¹

⊗ HZ) ∼ = B

¹

(x) ⊗ B

²

(y ) ∼ = P(x) ⊗ E (σy ),

|x| = 2p, |y | = 2p − 2, |σy | = 2p − 1.

Mapping to the first Postnikov section we get S

¹

⊗ (HZ, HF

p

)

^//

HF

p

[S

^2p−1

]

^//

HF

p

HF

p // '!&"%#$

..

//

S

²

⊗ (HZ, HF

p

)

78

7unique ht. type

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Higher THH of integers

sketch proof: B¨ o: ok for n = 1:

V (0)

∗

(S

¹

⊗ HZ) ∼ = B

¹

(x) ⊗ B

²

(y ) ∼ = P(x) ⊗ E (σy ),

|x| = 2p, |y | = 2p − 2, |σy | = 2p − 1.

Mapping to the first Postnikov section we get S

¹

⊗ (HZ, HF

p

)

^//

HF

p

[S

^2p−1

]

^//

HF

p

HF

p //

HF

p

[S

^2p+1

]

^//

S

²

⊗ (HZ, HF

p

)

78

7unique ht. type

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Higher THH of integers

sketch proof: B¨ o: ok for n = 1:

V (0)

∗

(S

¹

⊗ HZ) ∼ = B

¹

(x) ⊗ B

²

(y ) ∼ = P(x) ⊗ E (σy ),

|x| = 2p, |y | = 2p − 2, |σy | = 2p − 1.

Mapping to the first Postnikov section we get S

¹

⊗ (HZ, HF

p

)

^//

HF

p

[S

^2p−1

]

^//

HF

p

HF

p

//

HF

p

∧

_HF_p_[S2p−1]

HF

p

HF

p //

HF

p

[S

^2p+1

]

^//

S

²

⊗ (HZ, HF

p

)

78

7unique ht. type

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Higher THH of integers

S

¹

⊗ (HZ, HF

p

)

^//

HF

p

[S

^2p−1

]

^//

HF

p

HF

p

//

HF

p

∧

_HF_p_[S2p−1]

HF

p

HF

p //

HF

p

[S

^2p+1

]

^//

S

²

⊗ (HZ, HF

p

) So,

S

²

⊗ (HZ, HF

p

) ≃ HF

p

[S

^2p+1

]∧

_HF_p

HF

p

∧

_HF_p_[S^2p−1_]

HF

p

≃ H

F

p

[S

^2p+1

] ⊗

_F_p

F

p

⊗

_F_p_[S2p−1]

F

p

.

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Higher THH of integers

S

²

⊗ (HZ, HF

p

) ≃ H

F

p

[S

^2p+1

] ⊗

_F_p

F

p

⊗

_F_p_[S^2p−1_]

F

p

.

Tor SS now gives the result for n = 2:

V (0)

∗

(S

²

⊗ HZ) ∼ = π

∗

(S

²

⊗ (HZ, HF

_p

))

∼ = B

²

(x) ⊗ B

³

(y ) ∼ = E (σx) ⊗ Γ(σ

²

y ).

|σx| = 2p + 1, |σ

²

y | = 2p.

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Higher THH of integers

In higher dimension: do all directions simultaneously, remembering that

S

²

⊗ (HZ, HF

p

) ≃ H

F

p

[S

^2p+1

] ⊗

_F_p

F

p

⊗

_F_p_[S2p−1]

F

p

,

and that

S

ⁿ

⊗ (HZ, HF

p

) −−−−→ HF

p

  y

HF

_p

−−−−→ S

ⁿ⁺¹

⊗ (HZ, HF

_p

)

is a homotopy pushout of commutative HF

p

-algebras; and

using a multisimplicial Bar-resolution.

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Higher THH of rings of integers

D, Lindenstrauss, Richter (∞)

A is a number ring n > 0

(S

ⁿ

⊗ A)

p

≃ Y

p∈q∈Spec(A)

(S

ⁿ

⊗ Ab

^q

)

p

1

Unramified: F finite field;

π

∗

(S

ⁿ

⊗ (W (F), F)) ∼ = B

_Fⁿ

(x) ⊗ B

_Fⁿ⁺¹

(y ).

^a

|x| = 2p, |y | = 2p − 2

2

Wildly ramified: if Ab

^q

is wildly ramified, F = A/q:

π

∗

(S

ⁿ

⊗ (Ab

q

, F)) ∼ = B

_Fⁿ

(x) ⊗ B

_Fⁿ⁺¹

(y )

|x| = 2, |y | = 0.

aB_F¹(x) = F[x] and B_Fⁿ⁺¹(x) = Tor^B^Fⁿ(F, F)

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Higher THH of rings of integers

Unramified case follows from π

∗

(S

ⁿ

⊗ (HZ, HF

p

)) and the wildly ramified case from Lindenstrauss and Madsen’s calculation when n = 1.

(Weirdly enough: it is the tamely ramified case we don’t yet

know how to handle)

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Quote of the day

I would like to use the opportunity to express my admiration for Tom Goodwillie.

⁹

Thank you.

9Dundas, 1997