Algebraic Cycles, Branes and De Rham-Witt

Jan Stienstra

Nagoya, 23 November 2010

talk at workshopWitt vectors, Foliations and absolute De Rham cohomology

Algebraic Cycles: web of intersecting subvarieties of a variety.

Branes: boundary conditions for open strings in a manifold.

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X smooth, projective algebraic variety over alg. closed field k.

Zd(X) := free abelian group on the set of irreducible

subvarieties of codimension d in X.

Bd(X) := subgroup of Zd(X) generated by cycles div(f),

the divisor of zeros and poles of a rational function f

on a codimension d− 1 subvariety of X.

codimension d algebraic cycle on X is an element of Zd(X).

rational equivalence:

α, β ∈ Zd(X) : α ∼ β ⇔ α− β ∈ Bd(X)

d-th Chow group of X

CHd(X) := Zd(X) /Bd(X)

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Chow ring of X

CH•(X) =⊕

d≥0

CHd(X)

Product on CH•(X) given by intersecting algebraic cycles.

Do not use set theoretic unions and intersections!

Multiplicities are subtle!

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K0(X) := Grothendieck group of the category of

locally free OX-modules of finite type.

K0(X) is a λ-ring, with λi-operation coming from the

ith exterior power operation on locally free sheaves.

Gr•K0(X) := graded ring associated with the

γ-filtration on K0(X).

K0(coh(X)) := Grothendieck group of the category of

coherent sheaves on X.

Gr•K0(coh(X)) := graded group associated with filtration on

K0(coh(X)) by codimension of support

of coherent sheaf on X.

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For smooth X the natural map is an isomorphism:

K0(X)≃−→ K0(coh(X))

Grothendieck-Riemann-Roch:

For X smooth projective over a field there are natural homomor-

phisms of graded rings, which are isomorphisms modulo torsion

Gr•K0(X)≃−→ Gr•K0(coh(X))

≃←− CH•(X)

Grothendieck’s Chern character gives an isomorphism forevery d

GrdK0(X)⊗Q≃←− ξ ∈ K0(X)⊗Q | ψn(ξ) = nd ξ , ∀n

here ψn is the n-th Adams operation.

Remark: Quillen has defined higher algebraic K-groups Ki(X)(i ≥ 0) of the category of locally free sheaves of finite type onX. These also carry an action by Adams operations. The cor-

responding eigenspaces are the motivic cohomology groups

of X. Bloch has defined the higher Chow groups of X.

Upon tensoring with Q the higher Chow groups coincide withthe motivic cohomology groups.

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Quillen’s higher algebraic K-theory of exact categories.

For commutative ring R with unit Kd(R) is the dth algebraic K-group of the category of finitely generated projectiveR-modules.

For scheme X let Kd,X denote the Zariski sheaf associated with

the pre-sheaf

affine open U = Spec(R) 7→ Kd(R)

Theorem: CHd(X) ≃ Hd(X,Kd,X)

For d = 1 this is in fact the classical formula

Pic(X) ≃ H1(X,O∗X)

For d = 2 this result is due to Bloch.For general d it was conjectured by Gersten and proven by

Quillen.

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Question (Spencer Bloch, 1970’s):

Determine, for smooth projective X, the structure of the functor

augmented Artinian local k-algebras −→ Abelian groups

A 7→ Hm(X,Kn,X×A/X)

Here Kn,X×A/X is the Zariski sheaf associated with the pre-sheaf

affine open U = Spec(R) 7→ ker(Kn(R⊗k A)→ Kn(R))

Classical theory tells that for m = n = 1 this functor is pro-

representable by the formal group of the Picard variety of X.

Artin and Mazur have considered the case n = 1 and arbi-trary m. They gave conditions for when this functor is pro-

representable by a formal group.

e.g. n = 1 , m = 2 gives the formal Brauer group of X.

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In 1976 Bloch proposed the case m = n = 2 to me as a researchproblem for my PhD thesis.

One PhD thesis and three post-doc jobs later I formulated theanswer as follows (published in Crelle 355 (1985)):

Theorem. Let X be a smooth projective variety over a perfect

field k of characteristic p > 0.Then there is a functorial homomorphism for every m and n:(Hm(X,WΩ•X)⊗R WΩn−1−•

A

)−→ Hm(X,Ks

n,X×A/X)

of which the kernel and the cokernel are both essentially 0.

Main ingredients in the proof of the theorem are

• detailed information about the De Rham-Witt complex givenin the works of Bloch, Deligne, Illusie and Raynaud

• an alternative construction of the De Rham-Witt complexbased on the K-theory of categories of finitely generated

projective modules equiped with an endomorphism.

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Theorem. Let X be a smooth projective variety over a perfectfield k of characteristic p > 0.Then there is a functorial homomorphism for every m and n:(Hm(X,WΩ•X)⊗R WΩn−1−•

A

)−→ Hm(X,Ks

n,X×A/X)

of which the kernel and the cokernel are both essentially 0.

• Ksn,X×A/X is the symbol part of Kn,X×A/X, i.e.

the image under multiplication in K-theory of

(1 +OX ⊗k mA)⊗Z O∗X×A ⊗Z . . .⊗Z O

∗X×A ;

mA is the maximal ideal of A.

• R = W (k)[F, V, d] is the Raynaud ring;W (k) the p-typical Witt vectors of k;F is Frobenius, V Verschiebung,

d the derivation in the De Rham-Witt complex of X:

WOXd→ WΩ1

Xd→ . . .

d→WΩi

Xd→ . . .

• Hm(X,WΩ•X) :=

Hm(X,WOX)d→ Hm(X,WΩ1

X)d→ . . .

d→ Hm(X,WΩi

X)d→ . . .

viewed as an R-module.

• WΩ•A is the formal De Rham-Witt complex of A.

• We call a functor G : augm.Art.loc.k-algs → Ab.grps

essentially zero if for every A there is a surjection A′ ։ Ainducing the zero map G(A′)→ G(A).

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The alternative construction of the De Rham-Witt complex works

for any commutative unital ring R, without assumptions

about the characteristic or about regularity.

End(R) denotes the exact category whose

• objects: pairs (M,α) consisting of a finitely generated pro-jective R-module M and an R-linear endomorphism α ofM

• morphisms: (M,α)→ (M ′, α′) is an R-linear mapf : M →M ′ s.t. fα = α′f .

• short exact sequences: underlying sequence of R-modulesis exact

Ki(End(R)), for i = 0, 1, 2, . . ., denote Quillen’s K-groups of the

exact category End(R).

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K∗(End(R)) =⊕

i≥0

Ki(End(R))

is a graded commutative ring with product induced by tensorproduct

(M,α)⊗ (M ′, α′) = (M ⊗M ′, α⊗ α′)

There is a Frobenius operator Fn for every n ≥ 1 induced

by(M,α) 7→ (M,αn)

There is a Verschiebung operator Vn for every n ≥ 1 induced

by

(M,α) 7→ (M⊕n,

0 0 . . . 0 α

1 0 . . . 0 00 1 . . . 0 0... 0 . . . 0

...0 0 . . . 1 0

)

There is also a map

d : Ki(End(R)) → Ki+1(End(R)) ,

constructed as follows.

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Consider the polynomial ring in one variable Z[u] and the bi-exact functor

End(R) × End(Z[u]) −→ End(R)

(M,α) , (N, β) 7→ (M ⊗Z[u] N , 1⊗ β)

here M is considered as a Z[u]-module: um = αm, ∀m ∈M .

This induces maps

Ki(End(R)) ⊗Z Kj(End(Z[u])) −→ Ki+j(End(R)) .

The map

d : Ki(End(R)) → Ki+1(End(R)) ,

comes from a particular element in K1(End(Z[u])).

Remark. The above construction of the map d is a special caseof a general construction of functorial operations onK∗(End(R)).

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Relations:

V1 = F1 = 1

FnFm = Fnm , VnVm = Vnm (∀n,m)

FnVn = n1 (∀n)

VpFp = p1 if p prime and pR = 0

FnVm = VmFn if (n,m) = 1

Vn d = n d Vn , d Fn = nFn d (∀n)

Fn d Vn = d if n odd , F2 d V2 = d if 2R = 0

2 d2 = 0; d2 = 0 if 2R = 0

Fn(a b) = (Fn a) (Fn b) (∀n)

Vn(aFnb) = (Vna) b (∀n)

d(a b) = (da) b + (−1)ia (db)

for all a ∈ Ki(End(R)) and b ∈ Kj(End(R))

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Exact functors

End(R) −→ P (R) , (M,α) 7→ M

P (R) −→ End(R) , M 7→ (M, 0)

where P (R) = category of fin. gen. projective R-modules.

These induce direct sum decomposition:

Ki(End(R)) = Ki(R) ⊕ Ki(End(R))

with

Ki(End(R)) := ker(Ki(End(R))→ Ki(R))

Theorem(Almkvist, Grayson)

Let W(R) denote the ring of big Witt vectors of R.Then the map

K0(End(R)) −→ (1 + tR[[t]])× = W(R)

[M,α]− [M, 0] 7→ det(1− tα)−1

is an injective homomorphism of rings, which commutes withthe Frobenius and Verschiebung operators.

Moreover the Teichmuller lifting is given by

R → K0(End(R)) → W(R)

x 7→ [R, x]− [R, 0] 7→ (1− xt)−1

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The inverse map from the image of K0(End(R)) in W(R) backto K0(End(R)) is given by

(1−

n∑

j=1

rjtj

)−1

7→

R⊕n,

0 0 . . . 0 rn1 0 . . . 0 rn−1

0 1 . . . 0 rn−2... 0 . . . 0

...0 0 . . . 1 r1

− [R⊕n, 0]

We will describe a decreasing filtration by homogeneous ideals

FilnK∗(End(R)) , n = 1, 2, 3, . . .

such that the Frobenius maps Fm, Verschiebung maps Vm andthe derivation d extend to the completion

K∗(End(R))∧ := lim←n

K∗(End(R))/FilnK∗(End(R))

and such thatK0(End(R))∧ = W(R) .

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For a commutative unital ring A let Nil(A) denote the full exactsubcategory of End(A) with objects those (M,α) for which α is

nilpotent.

The fundamental theorem of K-theory gives an isomor-phism for every i ≥ 0:

NKi+1(A) ≃ Ki(Nil(A))

where

NKi+1(A) := coker(Ki+1(A)→ Ki+1(A[u]))

Ki(Nil(A)) := ker(Ki(Nil(A))→ Ki(A))

for the maps induced by the inclusion A → A[u] and the forget-ful functor Nil(A)→ P (A), respectively.

Let s0 , s1 : Ki+1(A[u]) → Ki+1(A) denote the homorphisms

induced by the substitutions u 7→ 0 and u 7→ 1, respectively.Then s1 − s0 gives a well defined homorphism

s1 − s0 : NKi+1(A) −→ Ki+1(A) .

By composing the latter homomorphism with the isomorphism

from the fundamental theorem we obtain, for every i ≥ 0, ahomomorphism

∫ 1

0

: Ki(Nil(A)) −→ Ki+1(A) .

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The bi-exact functor

End(R) × Nil(A) −→ Nil(R⊗Z A)

(M,α) , (N, β) 7→ (M ⊗Z N , α⊗ β)

induces

Ki(End(R)) ⊗Z Kj(Nil(A)) −→ Ki+j(Nil(R⊗Z A))

a , b 7→ ab

and by composition with∫ 1

0 :

Ki(End(R)) ⊗Z Kj(Nil(A)) −→ Ki+j+1(R⊗Z A)

a , b 7→

∫ 1

0

ab

Remark: The above pairing (a, b) 7→∫ 1

0 ab underlies the map

(Hm(X,WΩ•X)⊗R WΩn−1−•

A

)−→ Hm(X,Ks

n,X×A/X)

in my theorem.

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Nil(A) carries an increasing filtration by full exact subcategories:

FilnNil(A) := (M,α) | αn = 0 .

This yields an increasing filtration on the K-groups:

FilnKj(Nil(A)) := image(Kj(FilnNil(A))→ Kj(Nil(A)))

Now define

FilnKi(End(R)) :=

a ∈ Ki(End(R))

∣∣∣∣∣∀A , ∀j , ∀b ∈ Kj(FilnNil(A))∫ 1

0 ab = 0 in Ki+j+1(R⊗Z A)

Ki(End(R))∧ := lim←n

Ki(End(R))/FilnKi(End(R))

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We define the generalized De Rham-Witt complex of R asthe completion of the smallest subring of K∗(End(R))∧ which

contains W(R) and is closed under the operations d, Fn, Vn.

The generalized formal De Rham-Witt complex of a com-mutative unital ring A is similarly defined from K∗(Nil(A)) and

the multiplicative group

W(A) := (1 + uAnil[u])× ⊂ NK1(A) = K0(Nil(A))

of polynomials with constant term 1 and all other coefficientsnilpotent in A. No need for completion.

If R and A are rings of prime characteristic p one can split off theDe Rham-Witt complex WΩ•R of R and the formal De Rham-

Witt complex WΩ•A of A from the above generalized forms bymeans of the idempotent operator (µ is the Mobius function):

∑

n, p∤n

µ(n)

nVnFn =

∏

ℓ prime 6=p

(1−

1

ℓVℓFℓ

)

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In physics branes are boundary conditions for open strings ina target manifoldM.

One speaks of D-branes (D for Dirichlet) if the boundary con-

ditions specify the position of the endpoints of the string.

One speaks of D-branes of B-type if the boundary condi-tions require the string to end on a holomorphic submanifold inthe target manifoldM.

Chien-Hao Liu and Shing-Tung Yau are working on a projectto find structures in algebraic geometry to model D-branes of B-

type.

see e.g. their paperD-branes and Azumaya noncommutative geometry: From Polchin-ski to Grothendieck

Two key ideas:

• Polchinski: stacks of branes (≈ branes with multiplicities)

should be described by matrix valued functions onM.

• Grothendieck: instead of substructures ofM use structuresthat map intoM.

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Liu and Yau implement these ideas in what they call anAzumaya scheme with fundamental module.

The visible part of such a structure consists of a scheme X and

a locally free OX-module of finite type E .

The (as yet) invisible part of such a structure can only be probedthrough the set of morphisms from it to other schemes Y(acting as target spaces likeM).

Liu and Yau define these sets of morphisms as

Mor((X, E), Y ) :=

coherent OX×Y -modules E on

X × Y with p1∗E = E

They remark that it may suffice to “probe” the structure of theAzumaya scheme with fundamental module by taking Y from a

fixed (well chosen) collection of “basic spaces”.

They also say (parenthetically)

“Furthermore, from .......... , one expects that one finally has toconsider everything in the derived(-category) sense.”

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Comments and Questions.

• The assignment

Y 7→ Mor((X, E), Y )

is a covariant functor from the category of schemes to thecategory of sets.

One should compare this with standard constructions inmoduli problems where one uses a contravariant functor:

F : schemes→ sets

and shows that there is a scheme Z such that

F (Y ) = Mor(Y, Z) for all schemes Y .

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• What does it mean:has to consider everything in the derived(-category) sense?

Are X and Y fixed and can only E vary?

For fixed X and Y the E ’s form an exact category:

Brane(X; Y ) :=

coherent OX×Y -modules E on X × Y s.t.

p1∗E is locally free of finite type on X

Should one then take the corresponding derived category?

Or can one take its K-theory K∗(Brane(X; Y )) instead?

Should one then vary Y and consider for fixed X the co-variant functor

Y 7→ K∗(Brane(X; Y ))

on the category of schemes?

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Liu-Yau in affine context:

X = Spec(R)

Y = Spec(A)

E = finitely generated projective R-module M

ThenMor((X, E), Y ) = Homrings(A,EndR(M)) .

For fixed (R,M) the question becomes:

Study the contravariant functor:

commutative unital rings −→ sets

A 7→ Homrings(A,EndR(M))

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For fixed R and A we have the exact category

End(R;A) :=

pairs (M,ϕ)

∣∣∣∣M fin. gen. proj. R-mod.ϕ : A→ EndR(M) ring hom.

Example:

End(R; Z[u]) = End(R)

So there is a link with De Rham-Witt when Azumaya schemes

are “probed” by morphisms to the affine line.

A ring homomorphism f : A′ → A induces an exact functor

End(R;A) −→ End(R;A′) , (M,ϕ) 7→ (M,ϕf)

This makes the construction contravariantly functorial in A.

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Recall the question Spencer Bloch asked in 1970’s:

Determine, for smooth projective X, the structure of the functor

augmented Artinian local k-algebras −→ Abelian groups

A 7→ Hm(X,Kn,X×A/X)

and notice behind the facade of K-theory, sheaves and coho-

mology on the affine level the exact category

fin. gen. proj. R ⊗k A-modules

Being an augmented Artinian local algebra over the field k thering A is a finite dimensional k-vector space.

Therefore every fin. gen. proj. R ⊗k A-module M is also

a fin. gen. proj. R-module with A-module structure given bya ring homomorphism ϕ : A→ EndR(M)

A ring homomorphism f : A′ → A induces an exact functor

f.g.pr. R ⊗k A′-mod.

−⊗R⊗kA′(R⊗kA)

−→ f.g.pr. R⊗k A-mod.

This makes the construction covariantly functorial in A.

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As Liu and Yau remarked, it may suffice to “probe” the struc-ture of the Azumaya scheme with fundamental module by taking

Y from a fixed (well chosen) collection of “basic spaces”.

Do toric varieties constitute a good collection of “basic spaces”for that purpose?

Since toric varieties are locally modeled on commutative semi-groups this would mean in the affine context :

For a commutative unital ring R and and a commutative semi-

group S look at the exact category of representations of S infinitely generated projective R-modules:

Rep(R;S) :=

pairs (M,ϕ)

∣∣∣∣M fin. gen. proj. R-mod.ϕ : S → EndR(M) s.-grp hom.

Restricting to representations by nilpotent endomorphisms shouldprobably be the same as working with Artinian local algebras.

Question:

Working with Artinian local algebras is a kind of analysis.Are there other sensible ways to bring in (non-Archimedian)

analysis?

c.f. Payne’s work on analytification and tropicalization usinghomomorphisms from commutative semi-groups into the semi-

ring (R≥0 , max , ·).

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