Continuity of homology and K-theory

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Continuity of homology and K-theory

Bjørn Ian Dundas

NMF100. September 14, 2018, Bergen

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The goal of today’s talk/Disclaimer

The aim of today’s talk is to give enough background to appreciate:

Theorem [Clausen, Mathew, Morrow, 2018]Let p be a prime, A a noetherian (commutative) ring and I⊆A an ideal.

If A/p is “F-finite”¹then

K(lim

n A/Iⁿ)→lim

n K(A/Iⁿ) is a p-adic equivalence.

A great theorem is like an arch or a bridge spanning a chasm. The arch depends on its keystone, but also the voussoirs: if one is missing the arch will fall.

You’ll pardon me for pointing to my voussoirs

1my friends in algebraic geometry tell me that this is an innocent assumption:

it means thatA/pis finitely generated over itspth powers.

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K-theory

I one of the crowning achievements of the last century.

I an invariant encapsulating an enormous amount of information and was both a key structural framework and a spawning ground for a wide array of conjectures by many of the most prominent mathematicians (Grothendieck, Tate, Deligne, Borel, Quillen, Bott, Atiyah, Beilinson, Lichtenbaum, Milnor, Waldhausen, Friedlander, Serre, Suslin . . . )

I Early on, algebraic geometry and number theory were the central playgrounds, but K-theory quickly spread to manifold geometry, algebraic topology, functional analysis and mathematical physics.

I Unfortunately, not until around the turn of the millennium were we able to actually calculatemuch.²

I New tools: the trace methods of B¨okstedt, Hsiang and Madsen and the motivic cohomology of Suslin and Voevodsky.

I Since then a wealth of calculations has flowed from the hands of among others B¨okstedt, Geisser, Hesselholt, Madsen, Østvær, Rognes, Weibel . . .

2

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K-theory

2

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K-theory

2

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K-theory

2Some isolated fantastic results existed, e.g., Quillen, Borel, Suslin

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K-theory

2

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Continuity

For a functionf:

n→∞lim f(x_n) =f( lim

n→∞x_n)

“f commutes with taking the limit over a (convergent) sequence of numbers”

Questions

1. What should “continuity” mean for invariants?

2. What should “(convergent) sequence” mean?

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A concrete example of a “sequence” converging to the power series

Consider the ringA[[t]] of power series

a0+a1t+a2t²+. . .

with coefficients in a ringA(you may takeAto be the integersZas an example).

A power series is uniquely given by its sequence oftruncatedpolynomials a₀, a₀+a₁t, a₀+a₁t+a₂t², . . . ,

so, if we let A[[t]]/(tⁿ) be the ring of truncated polynomials,³we see that we have ring homomorphisms

· · · →A[[t]]/(t³)→A[[t]]/(t²)→A[[t]]/(t) =A (chop off the highest degree) andA[[t]] = limn(A[[t]]/(tⁿ)) is thelimit.

3careful: if you multiply two truncated polynomials, all resulting powerst^mform≥nare set to zero

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A concrete example of a “sequence” converging to the power series

a0+a1t+a2t²+. . .

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A concrete example of a “sequence” converging to the power series

a0+a1t+a2t²+. . .

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Question: What about invertible elements?

(an example of an “invariant”)

ExampleInA[[t]]/(t²)

(a₀+a₁t)(b₀+b₁t) =a₀b₀+ (a₀b₁+a₁b₀)t, so ifa₀is invertible, then (a₀+a₁t)(_a¹

0 −^a_a¹

0t) = 1.

In general, a truncated polynomial is invertible iff the constant term is invertible.

The same argument goes for power series: a0+a1t+. . . is invertible iff a0is invertible, or in other words, a unit⁴in A[[t]] is uniquely given by a sequence{a0+a1t+. . .antⁿ}n of invertible truncated polynomials

the units in A[[t]]is the limit of the units in the truncated polynomial rings:

units(A[[t]]) =units(lim

n A[[t]]/(tⁿ)) = lim

n units(A[[t]]/(tⁿ)).

units(A) is aninvariant⁵ in the ringA. The above is an instance of the fact that units (A) is a continuousinvariant.

4“the units”= “the (group of) invertible elements”

5if I replaceAby something isomorphic, units(A) is replaced by something isomorphic. . .

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Question: What about invertible elements?

0 −^a_a¹

0t) = 1.

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Question: What about invertible elements?

0 −^a_a¹

0t) = 1.

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Question: What about invertible elements?

0 −^a_a¹

0t) = 1.

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Question: What about invertible elements?

0 −^a_a¹

0t) = 1.

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Question: What about invertible elements?

0 −^a_a¹

0t) = 1.

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Hensel and Newton

(something relating to first year calculus)

Example. Let R=A[[t]] and I = (t) (all power series with constant term zero).

The projectionR→R/I =Asends a power seriesp=p(t) to its constant term ¯p=p(0),

“pis invertible iff ¯pis”.

In equations: Letf(x) =p·x−1 (NB: poly. w/coeff. inR)

“f has a root iff ¯f(x) = ¯p·x−1 has a root inR/I”

For arbitraryR andI we say that R→R/I is Henselif “Hensel’s criterion” is satisfied:

any polynomial f(x)w/coeff. in R has a root provided 1. f¯(x)(poly. w/coeff in R/I ) has a rootx¯0∈R/I and 2. f⁰(¯x0)is invertible in R/I . ⁶

Hensel’s criterion guarantees that Newton’s method will give you a sequence of partial solutions.

We’re mostly concerned with the special case whereR is completewrt. I, that is to say; an elementx∈Ris given uniquely by giving the sequence {x mod Iⁿ}n:

R= lim

n R/Iⁿ.

6In the example, point 2. is redundant sincef⁰(¯x0) = ¯p= ¯x₀⁻¹by 1.

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Hensel and Newton

R= lim

n R/Iⁿ.

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Hensel and Newton

R= lim

n R/Iⁿ.

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Hensel and Newton

R= lim

n R/Iⁿ.

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Hensel and Newton

R= lim

n R/Iⁿ.

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Invariants – another old example: FLT

Fermat’s last theorem was seemingly proved in 1847 by Gabriel Lam´e:

For each prime⁷ p, factor

z^p=x^p+y^p= (x+y)(x+ζ_py)· · ·(x+ζ_p^p−1y)

(whereζ_p=e^2πi^p ∈C); use unique factorization into prime factors, and . . . !

Liouville: Unfortunately this is wrong

7for instance,p= 23. We will only consider odd primes since even primes are odd

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Invariants – another old example: FLT

Fermat’s last theorem was seemingly proved in 1847 by Gabriel Lam´e:

For each prime⁷ p, factor

z^p=x^p+y^p= (x+y)(x+ζ_py)· · ·(x+ζ_p^p−1y)

(whereζ_p=e^2πi^p ∈C); use unique factorization into prime factors, and . . . !

Liouville: Unfortunately this is wrong

7for instance,p= 23. We will only consider odd primes since even primes are odd

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Kummer: unique factorization (UF) depends on the prime!

UF inZ[ζ_p]⊆Cis true forsomeprimes p, but false for others.⁹

Whether we have UF depends on an abelian group “ ˜K0(Z[ζp])”¹⁰ 8

being trivial.

8Z[ζp] consists of all integral linear combinations of thepth roots of unity in the complex fieldC.

Z[ζp] is an example of a “number ring”:Z[ζp] is the “integers” in a finite extension of the rationals

9for instance forp= 23.

10K˜0(Z[ζp]) is an invariant in the ringZ[ζp] called the “ideal class group”, the “Picard group”, the “reduced Grothendieck group” or the “0th reduced K-group”

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Kummer: unique factorization (UF) depends on the prime!

UF inZ[ζ_p]⊆Cis true forsomeprimes p, but false for others.⁹

Whether we have UF depends on an abelian group “ ˜K0(Z[ζp])”¹⁰ 8

being trivial.

8Z[ζp] consists of all integral linear combinations of thepth roots of unity in the complex fieldC.

Z[ζp] is an example of a “number ring”:Z[ζp] is the “integers” in a finite extension of the rationals

9for instance forp= 23.

10K˜0(Z[ζp]) is an invariant in the ringZ[ζp] called the “ideal class group”, the “Picard group”, the “reduced Grothendieck group” or the “0th reduced K-group”

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Kummer, ˜ K

₀

(Z[ζ

_p

]) and Fermat

More information can be gleaned from ˜K₀(Z[ζ_p]):

The prime p is said to be regularif the only g ∈K˜0Z[ζp] with g^p= 1is g = 1.

In fact,Kummer proved FLT for the regular primes.

Regular primes seem quite common: of the primes less than 100 only 37, 59 and 67 are irregular, but to this day one does not know if there are infinitely many.

Another way of saying thatp>3 is regular is to say that it does not divide the numerators of ζ(−1), ζ(−3), . . . , ζ(4−p)

whereζis the Riemann zeta-function, i.e., the continuation ofζ(s) =P∞

n=1n^−s.Thesevalues ofζ(1−n) are easy to calculate, (known forn<10⁸).

Example: ζ(−11) =₃₂₇₆₀⁶⁹¹ , so 691 is irregular. ζ(−31) = 37·208360028141

16320 , so 37 is irregular.

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Kummer, ˜ K

₀

(Z[ζ

_p

]) and Fermat

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Kummer, ˜ K

₀

(Z[ζ

_p

]) and Fermat

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Kummer, ˜ K

₀

(Z[ζ

_p

]) and Fermat

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The invariant K

₀

A

Introduced much later by Grothendieck to express his Riemann-Roch theorem

The Grothendieck groupK0Ais an invariant in the ringA¹¹ It reveals some of the structure ofAand

if I replaceAby something isomorphicK0Ais replaced by something isomorphic.

We’ll enjoy knowing one way to constructK0A - it is through a process called “group completion”:

the process that takes the positive integers{1,2, . . .} and considers fractions “_Q^P” subject to the equality

P·X Q·X = P

Q

giving the positive rationals (as an abelian group under multiplication).

11Kummer was interested in the caseA=Z[ζp]⊆C

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The invariant K

₀

A

P·X Q·X = P

Q

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The invariant K

₀

A

P·X Q·X = P

Q

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The invariant K

₀

A

Group completion: the process that takes the positive integers{1,2, . . .}and considers fractions “_Q^P” subject to the equality ^P·X_Q·X = ^P_Q to get the nonnegative rationals (an abelian group under multiplication).

First, consider the case whenAis a field (whichZ[ζ_p] is not).

Instead of the positive integers, consider the finite dimensional vector spaces Aⁿ. If we consider “fractions” ^A_A^mn”¹² with the equality

A^m×A^x Aⁿ×A^x =A^m

Aⁿ

we have done nothing but added the “negative dimensional vector spaces A⁻ⁿ” and demanded that A^m×Aⁿ=A^m+n for all n,m∈Z.

Boring! K0(field)∼=Z.

12at this stage I don’t want to distinguish between different vector spaces of the same dimension, so I consider “isomorphism classes”

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The invariant K

₀

A

Group completion: the process that takes the positive integers{1,2, . . .}and considers fractions “_Q^P” subject to the equality ^P·X_Q·X = ^P_Q to get the nonnegative rationals (an abelian group under multiplication).

First, consider the case whenAis a field (whichZ[ζ_p] is not).

Instead of the positive integers, consider the finite dimensional vector spaces Aⁿ. If we consider “fractions” ^A_A^mn”¹² with the equality

A^m×A^x Aⁿ×A^x =A^m

Aⁿ

we have done nothing but added the “negative dimensional vector spaces A⁻ⁿ” and demanded that A^m×Aⁿ=A^m+n for all n,m∈Z.

Boring! K0(field)∼=Z.

12at this stage I don’t want to distinguish between different vector spaces of the same dimension, so I consider “isomorphism classes”

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The invariant K

₀

A

Group completion: the process that takes the positive integers and considers fractions “^P_Q” subject to the equality ^P·X_Q·X = ^P_Q to get the nonnegative rationals.

Boring! K0(field) ={Aⁿ|n∈Z} ∼=Z.

IfAis not a field, there are more things to consider than just theAⁿs: there are “A-modules”¹³ P,Q with

P×Q=A^m without eitherP orQ necessarily being of the formAⁿ. ¹⁴

K₀A is the abelian group you get by considering such fractions _Q^P andK˜₀A is what you get if you kill the subgroup Z∼={Aⁿ|n∈Z} ⊆K₁A.

One of the main results of classical algebraic number theory:

K˜0OF is finite for any number ring OF. ¹⁵

13“A-modules are just like vector spaces except that the “scalars” are elements inA.

14These are called(finitely generated) projective A-modules.

For instance, ifA=Z×Zwith componentwise addition/multiplication, thenZ× {0}is a projectiveA-module not of the formAⁿ. Similar things happen,e.g.,forA=Z[ζ23]

15“number ring”: the ring of integers in a finite extension ofQ;e.g.,Z[ζp]

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The invariant K

₀

A

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The invariant K

₀

A

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The next invariants: K

₁

A, K

₂

A, . . .

LetF be a finite extension ofQandOF its ring of integers. Dedekind’s zeta function ζF(s) = X

06=I⊂O_F

|OF/I|^−s

(Riemann’s ifF=Q) contains a wealth of number theoretic information, but it is somewhat densely packed and you need to know some pieces in order to extract others.

For instance, ˜K0OF is packed together with information about the group units (OF) of invertible elements inOF, also crucial for Kummer’s proof. ¹⁶

1. More generally,K1A=GL(A)/E(A), measuring to what extent Gaussian elimination is possible inA: E(A) consists of the “elementary matrices” sitting inside the group GL(A) = units(M(A)) of invertible matrices. ¹⁷ Note: K1(OF) = units (OF).

2. K₂Ameasures howuniqueGaussian elimination is. . .

16Exact sequence 0→units(O_F)→units(F)→L

m∈Max(O_F)Z→K˜0O_F →0

17a similar argument as the one above (as applied to matrices) gives thatK1commutes with limits: ı.e..,K1

is continuous

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The next invariants: K

₁

A, K

₂

A, . . .

06=I⊂O_F

|OF/I|^−s

is continuous

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The next invariants: K

₁

A, K

₂

A, . . .

06=I⊂O_F

|OF/I|^−s

is continuous

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The next invariants: K

₁

A, K

₂

A, . . .

06=I⊂O_F

|OF/I|^−s

is continuous

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Higher K-theory is defined as K

₀

is, but respecting symmetries. . .

K0Ais the set consisting of

0. fractions _P^P⁺− of iso classes of finitely generated projectiveA-modules, 1. subject to the relation _P^P−⁺^×X×X = _P^P−⁺

K(A) is the space consisting of 0. a point for each pair (P⁺,P⁻),

1. a path from (P⁺,P⁻) to (P₁⁺,P₁⁻) for each pair of isomorphismP^±×X ∼=P₁^±, 2. a surface between two such paths for each compatible isomorphismX ∼=X⁰ (i.e., an

invertible matrix ifX =X⁰ =Aⁿ) . . .

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Higher K-theory is defined as K

₀

is, but respecting symmetries. . .

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Higher K-theory is defined as K

₀

is, but respecting symmetries. . .

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Higher K-theory is defined as K

₀

is, but respecting symmetries. . .

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Higher K-theory

invertible matrix ifX =X⁰ =Aⁿ) . . . Then

0. K0Ais the set of path components ofK(A) 1. K₁Ais the fundamental group ofK(A), . . .

n. if Ais a number ring, all the homotopy groups KnAare involved in the zeta functions, giving an vastly more powerful invariant

Picture K-theory as a highly structured zeta-function with periodicity phenomena¹⁸ and localizations mirroring the functional equation and meromorphic continuation.

18like the one of the Lichtenbaum-Quillen conjecture, but also higher periodicity coming from stable homotopy theory

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Higher K-theory

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Higher K-theory

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Higher K-theory

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Higher K-theory

K-theory as a “highly structured zeta-function with periodicity phenomena and localizations mirroring the functional equation and meromorphic continuation”.

However, K-theory is not restricted to algebra and number theory. It is an important invariant in geometry (algebraic and topological), physics, . . . ¹⁹

For instance,starting with finite sets instead of projective modules, K-theory gives you the

“sphere spectrum” – a base ring more fundamental than the integers²⁰ – encoding homotopy groups of spheres.

19. . . diffeomorphisms of manifolds, vector bundles, functional analysis, algebraic geometry, . . . For some examples beyond my core competence, you may for instance open the Wikipedia page on K-theory (physics) and read stuff like “In condensed matter physics K-theory has also found important applications,. . . ”

20The sphere spectrum is the ground ring for many of the homology theories we will be talking about

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Higher K-theory

K-theory as a “highly structured zeta-function with periodicity phenomena and localizations mirroring the functional equation and meromorphic continuation”.

However, K-theory is not restricted to algebra and number theory. It is an important invariant in geometry (algebraic and topological), physics, . . . ¹⁹

For instance,starting with finite sets instead of projective modules, K-theory gives you the

“sphere spectrum” – a base ring more fundamental than the integers²⁰ – encoding homotopy groups of spheres.

19. . . diffeomorphisms of manifolds, vector bundles, functional analysis, algebraic geometry, . . . For some examples beyond my core competence, you may for instance open the Wikipedia page on K-theory (physics) and read stuff like “In condensed matter physics K-theory has also found important applications,. . . ”

20The sphere spectrum is the ground ring for many of the homology theories we will be talking about

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An inconvenient truth

K-theory is notcontinuous: for instance, the difference between K₂(Z/p[[t]])and the corresponding limit of truncated polynomial algebras is an uncountable rational vector space!

However, this should be seen as a feature of the difference between algebra and topology and not as a bug,²¹ and there is a remedy peeling away these unwanted rational vector spaces, called(profinite) completion.

– The passage fromK(A) to its completionKb(A) kills this unwanted noise.

– For finite torsion and free summands in the two will always correspond.

– Completion itself is continuous and does not contribute any complexity.

21similar to the fact that when you look at it discretely the set of invertible realsR− {0}={±1} ×R⁺is enormous, but topologically it has just two components, each of which is contractible – and rational vector spaces

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An inconvenient truth

K-theory is notcontinuous: for instance, the difference between K₂(Z/p[[t]])and the corresponding limit of truncated polynomial algebras is an uncountable rational vector space!

However, this should be seen as a feature of the difference between algebra and topology and not as a bug,²¹ and there is a remedy peeling away these unwanted rational vector spaces, called(profinite) completion.

– The passage fromK(A) to its completionKb(A) kills this unwanted noise.

– For finite torsion and free summands in the two will always correspond.

– Completion itself is continuous and does not contribute any complexity.

21similar to the fact that when you look at it discretely the set of invertible realsR− {0}={±1} ×R⁺is enormous, but topologically it has just two components, each of which is contractible – and rational vector spaces