Algebraic geometry, categories and trace formula

Bertrand To¨en (CNRS, Toulouse)

Clay Research Conference, Oxford, September 2017

Algebraic geometry, categories and trace formula 1 / 32 Homogeneous polynomials F1,..., Fp ∈ C[X0,..., Xn]

n X := {(x0,..., xn)/Fi (x) = 0} ⊂ PC.

Problem: read the topology of X in terms of the Fi ’s.

Topology of algebraic varieties

Algebraic geometry, categories and trace formula 2 / 32 Topology of algebraic varieties

Homogeneous polynomials F1,..., Fp ∈ C[X0,..., Xn]

n X := {(x0,..., xn)/Fi (x) = 0} ⊂ PC.

Problem: read the topology of X in terms of the Fi ’s.

Algebraic geometry, categories and trace formula 2 / 32 Topology of algebraic varieties

Typical answers in low dimension

(n = 1, p = 1): X finite set of cardinality deg(F1) counted with multiplicities.

(n = 2 and p = 1): X is a compact Riemann surface and

(d − 1)(d − 2) g(X ) = d = deg(F ) 2 1 (g(X ) is the arithmetic genus if X not smooth).

Algebraic geometry, categories and trace formula 3 / 32 Theorem

X p+q p q χ(X ) = (−1) dim H (X , ΩX ). p,q

q Here: ΩX is the sheaf of holomorphic differential q-forms on X . The right hand side can be determined purely in terms of the Fi (”GAGA” theorem).

Euler characteristic of algebraic varieties Simple topological invariant: Euler characteristic χ(X )

X i i χ(X ) := (−1) dim H (X , Q). i

Algebraic geometry, categories and trace formula 4 / 32 q Here: ΩX is the sheaf of holomorphic differential q-forms on X . The right hand side can be determined purely in terms of the Fi (”GAGA” theorem).

Euler characteristic of algebraic varieties Simple topological invariant: Euler characteristic χ(X )

X i i χ(X ) := (−1) dim H (X , Q). i

Theorem

X p+q p q χ(X ) = (−1) dim H (X , ΩX ). p,q

Algebraic geometry, categories and trace formula 4 / 32 Euler characteristic of algebraic varieties Simple topological invariant: Euler characteristic χ(X )

X i i χ(X ) := (−1) dim H (X , Q). i

Theorem

X p+q p q χ(X ) = (−1) dim H (X , ΩX ). p,q

q Here: ΩX is the sheaf of holomorphic differential q-forms on X . The right hand side can be determined purely in terms of the Fi (”GAGA” theorem).

Algebraic geometry, categories and trace formula 4 / 32 Euler characteristic of algebraic varieties

X p+q p q χ(X )= (−1) dim H (X , ΩX ). p,q

topological invariant= algebraic invariant

Algebraic geometry, categories and trace formula 5 / 32 But, it has also an independent proof: Z (GB) χ(X ) = Ctop(X ) X Z X p+q p q (HRR) Ctop(X ) = (−1) dim H (X , ΩX ). X p,q

Euler characteristic of algebraic varieties The theorem follows from the existence of the Hodge decomposition

i M p q H (X , Q) ⊗ C ' H (X , ΩX ). p+q=i

Algebraic geometry, categories and trace formula 6 / 32 Euler characteristic of algebraic varieties The theorem follows from the existence of the Hodge decomposition

i M p q H (X , Q) ⊗ C ' H (X , ΩX ). p+q=i

But, it has also an independent proof: Z (GB) χ(X ) = Ctop(X ) X Z X p+q p q (HRR) Ctop(X ) = (−1) dim H (X , ΩX ). X p,q

Algebraic geometry, categories and trace formula 6 / 32 Homogeneous polynomials F1,..., Fp ∈ k[X0,..., Xn](k an n algebraically closed field) X := {(x0,..., xn)/Fi (x) = 0} ⊂ Pk .

X i i X p+q p q χ(X ) := (−1) dim Het (X , Q`)= (−1) dim H (X , ΩX ) i p,q

i Het (X , Q`) are the `-adic cohomology groups introduced by p q Grothendieck. H (X , ΩX ) sheaf cohomology for the Zariski topology. i p q Warning: In general Het (X , Q`) and ⊕p+q=i H (X , ΩX ) dont have the same dimension !

Euler characteristic of algebraic varieties Gauss-Bonnet and Hirzebruch-Riemann-Roch are both true over arbitrary fields.

Algebraic geometry, categories and trace formula 7 / 32 i p q Warning: In general Het (X , Q`) and ⊕p+q=i H (X , ΩX ) dont have the same dimension !

Euler characteristic of algebraic varieties Gauss-Bonnet and Hirzebruch-Riemann-Roch are both true over arbitrary fields. Homogeneous polynomials F1,..., Fp ∈ k[X0,..., Xn](k an n algebraically closed field) X := {(x0,..., xn)/Fi (x) = 0} ⊂ Pk .

X i i X p+q p q χ(X ) := (−1) dim Het (X , Q`)= (−1) dim H (X , ΩX ) i p,q

i Het (X , Q`) are the `-adic cohomology groups introduced by p q Grothendieck. H (X , ΩX ) sheaf cohomology for the Zariski topology.

Algebraic geometry, categories and trace formula 7 / 32 Euler characteristic of algebraic varieties Gauss-Bonnet and Hirzebruch-Riemann-Roch are both true over arbitrary fields. Homogeneous polynomials F1,..., Fp ∈ k[X0,..., Xn](k an n algebraically closed field) X := {(x0,..., xn)/Fi (x) = 0} ⊂ Pk .

X i i X p+q p q χ(X ) := (−1) dim Het (X , Q`)= (−1) dim H (X , ΩX ) i p,q

i Het (X , Q`) are the `-adic cohomology groups introduced by p q Grothendieck. H (X , ΩX ) sheaf cohomology for the Zariski topology. i p q Warning: In general Het (X , Q`) and ⊕p+q=i H (X , ΩX ) dont have the same dimension ! Algebraic geometry, categories and trace formula 7 / 32 [Γf .∆X ] is the intersection number of the graph of f with the diagonal inside X × X . The case f = id recovers the formula for χ(X ).

The trace formula for algebraic varieties

The formula is a special case of the trace formula: f  X algebraic endomorphism of X .

X i i (−1) Trace(f : Het (X , Q`))= [Γ f .∆X ] i

Algebraic geometry, categories and trace formula 8 / 32 The trace formula for algebraic varieties

The formula is a special case of the trace formula: f  X algebraic endomorphism of X .

X i i (−1) Trace(f : Het (X , Q`))= [Γ f .∆X ] i

[Γf .∆X ] is the intersection number of the graph of f with the diagonal inside X × X . The case f = id recovers the formula for χ(X ).

Algebraic geometry, categories and trace formula 8 / 32 Moduli theory: moduli of varieties are non-compact. Compactification by adding singular varieties at infinity (e.g. Mg,n ⊂ Mg,n). Arithmetic geometry: algebraic varieties defined over rings of integers in number fields. Bad reduction at some prime.

Topology of degeneration of algebraic varieties

Algebraic varieties naturally arise in family, and tend to degenerate to varieties with singularities.

Algebraic geometry, categories and trace formula 9 / 32 Topology of degeneration of algebraic varieties

Algebraic varieties naturally arise in family, and tend to degenerate to varieties with singularities.

Moduli theory: moduli of varieties are non-compact. Compactification by adding singular varieties at infinity (e.g. Mg,n ⊂ Mg,n). Arithmetic geometry: algebraic varieties defined over rings of integers in number fields. Bad reduction at some prime.

Algebraic geometry, categories and trace formula 9 / 32 Topology of degeneration of algebraic varieties

Homogeneous polynomials F1,..., Fp ∈ A[X0,..., Xn] where A = O(S)=ring of functions on the parameter space S.

n X := {(x0,..., xn, s)/Fi (x, s) = 0} ⊂ P × S. Family of algebraic varieties parametrized by S. For a point n s ∈ S, we have Xs ⊂ Pk(s) an algebraic variety defined over k(s)=residue field of s.

Algebraic geometry, categories and trace formula 10 / 32 S = Spec A for A a complete d.v.r. Two points in S: The special point o ∈ S (k := k(s) = A/m is the residue field). The generic point η ∈ S (K := k(η) = Frac(A) the fraction field).

Topology of degeneration of algebraic varieties S is taken to be a ”small disk” := henselian trait. Typical examples:

S = Spec C[[t]] formal holomorphic disk. S = Spec k[[t]] formal disk over an algebraically closed field.

S = Spec Zp formal p-adic disk.

Algebraic geometry, categories and trace formula 11 / 32 Topology of degeneration of algebraic varieties S is taken to be a ”small disk” := henselian trait. Typical examples:

S = Spec C[[t]] formal holomorphic disk. S = Spec k[[t]] formal disk over an algebraically closed field.

S = Spec Zp formal p-adic disk. S = Spec A for A a complete d.v.r. Two points in S: The special point o ∈ S (k := k(s) = A/m is the residue field). The generic point η ∈ S (K := k(η) = Frac(A) the fraction field).

Algebraic geometry, categories and trace formula 11 / 32 Variational problem: understand the change of topology

between XK¯ and Xk¯.

Topology of degeneration of algebraic varieties

n X := {(x0,..., xn, s)/Fi (x, s) = 0} ⊂ P × S. Two algebraic varieties:

The special fiber Xk¯.

The generic fiber XK¯ .

Algebraic geometry, categories and trace formula 12 / 32 Topology of degeneration of algebraic varieties

n X := {(x0,..., xn, s)/Fi (x, s) = 0} ⊂ P × S. Two algebraic varieties:

The special fiber Xk¯.

The generic fiber XK¯ .

Variational problem: understand the change of topology between XK¯ and Xk¯.

Algebraic geometry, categories and trace formula 12 / 32 When X → S is a submersion, Xk¯ and XK¯ smooth and the topology is constant:

χ(Xk¯) − χ(XK¯ ) = 0.

We assume XK¯ is smooth and Xk¯ possibly singular.

Euler characteristic of degeneration of algebraic varieties

Variational problem: evaluate χ(Xk¯) − χ(XK¯ ) in terms of an algebraic construction.

Algebraic geometry, categories and trace formula 13 / 32 We assume XK¯ is smooth and Xk¯ possibly singular.

Euler characteristic of degeneration of algebraic varieties

Variational problem: evaluate χ(Xk¯) − χ(XK¯ ) in terms of an algebraic construction.

When X → S is a submersion, Xk¯ and XK¯ smooth and the topology is constant:

χ(Xk¯) − χ(XK¯ ) = 0.

Algebraic geometry, categories and trace formula 13 / 32 Euler characteristic of degeneration of algebraic varieties

Variational problem: evaluate χ(Xk¯) − χ(XK¯ ) in terms of an algebraic construction.

When X → S is a submersion, Xk¯ and XK¯ smooth and the topology is constant:

χ(Xk¯) − χ(XK¯ ) = 0.

We assume XK¯ is smooth and Xk¯ possibly singular.

Algebraic geometry, categories and trace formula 13 / 32 ∗ df 1 ∧df 2 ∧df (ΩX , ∧df ): OX / ΩX / ΩX / ...

Case where Xk¯ has only isolated singularities: Milnor formula (dimension of Jacobian ring).

Euler characteristic of degeneration of algebraic varieties: charac. 0

Easy solution for characteristic zero case (A = C[[t]]): twisted de Rham complex. t defines a function on X and we have Theorem (Milnor, Kapranov, Saito, . . . )

X i i ∗ χ(Xk¯) − χ(XK¯ )= (−1) dimH (X , (ΩX , ∧df )). i

Algebraic geometry, categories and trace formula 14 / 32 Euler characteristic of degeneration of algebraic varieties: charac. 0

Easy solution for characteristic zero case (A = C[[t]]): twisted de Rham complex. t defines a function on X and we have Theorem (Milnor, Kapranov, Saito, . . . )

X i i ∗ χ(Xk¯) − χ(XK¯ )= (−1) dimH (X , (ΩX , ∧df )). i

∗ df 1 ∧df 2 ∧df (ΩX , ∧df ): OX / ΩX / ΩX / ...

Case where Xk¯ has only isolated singularities: Milnor formula (dimension of Jacobian ring).

Algebraic geometry, categories and trace formula 14 / 32 [∆X .∆X ]0 is the degree of a 0-cycle on Xk¯ that measures the singularities (Bloch’s localized intersection number). Generalization of the different in algebraic number theory.

Sw(XK¯ ) is the Swan conductor which measures wild ¯ ∗ ramifications: action of Gal(K/K) on Het (XK¯ ).

Euler characteristic of degeneration of algebraic varieties: Bloch’s formula Without characteristic zero: more complicated due to arithmetic aspects. Conjecture (Deligne 67, Bloch 85) If the scheme X is regular and k perfect.

χ(Xk¯) − χ(XK¯ )= [∆ X .∆X ]0 + Sw(XK¯ ).

Algebraic geometry, categories and trace formula 15 / 32 Sw(XK¯ ) is the Swan conductor which measures wild ¯ ∗ ramifications: action of Gal(K/K) on Het (XK¯ ).

Euler characteristic of degeneration of algebraic varieties: Bloch’s formula Without characteristic zero: more complicated due to arithmetic aspects. Conjecture (Deligne 67, Bloch 85) If the scheme X is regular and k perfect.

χ(Xk¯) − χ(XK¯ )= [∆ X .∆X ]0 + Sw(XK¯ ).

[∆X .∆X ]0 is the degree of a 0-cycle on Xk¯ that measures the singularities (Bloch’s localized intersection number). Generalization of the different in algebraic number theory.

Algebraic geometry, categories and trace formula 15 / 32 Euler characteristic of degeneration of algebraic varieties: Bloch’s formula Without characteristic zero: more complicated due to arithmetic aspects. Conjecture (Deligne 67, Bloch 85) If the scheme X is regular and k perfect.

χ(Xk¯) − χ(XK¯ )= [∆ X .∆X ]0 + Sw(XK¯ ).

[∆X .∆X ]0 is the degree of a 0-cycle on Xk¯ that measures the singularities (Bloch’s localized intersection number). Generalization of the different in algebraic number theory.

Sw(XK¯ ) is the Swan conductor which measures wild ¯ ∗ ramifications: action of Gal(K/K) on Het (XK¯ ). Algebraic geometry, categories and trace formula 15 / 32 Mixed characteristic case (e.g. A = Zp) is open: in particular isolated singularities (Deligne-Milnor formula).

Euler characteristic of degeneration of algebraic varieties: Bloch’s formula The Bloch’s formula is a theorem when: Characteristic 0: Milnor, Kapranov, M. Saito. Equicharacteristic p > 0 (A = k[[t]]): T. Saito (2017).

Semi-stable case: Xk¯ ⊂ X is (supported by) a simple normal crossing divisor (Kato-Saito 2001).

Degenerate case: Xη¯ = ∅ (recovers formula for χ(X )). Family of curves (Bloch), and finite ramified cover X → S (standard formula for the different).

Algebraic geometry, categories and trace formula 16 / 32 Euler characteristic of degeneration of algebraic varieties: Bloch’s formula The Bloch’s formula is a theorem when: Characteristic 0: Milnor, Kapranov, M. Saito. Equicharacteristic p > 0 (A = k[[t]]): T. Saito (2017).

Semi-stable case: Xk¯ ⊂ X is (supported by) a simple normal crossing divisor (Kato-Saito 2001).

Degenerate case: Xη¯ = ∅ (recovers formula for χ(X )). Family of curves (Bloch), and finite ramified cover X → S (standard formula for the different).

Mixed characteristic case (e.g. A = Zp) is open: in particular isolated singularities (Deligne-Milnor formula). Algebraic geometry, categories and trace formula 16 / 32 We want to make progress on this conjecture by introducing a new point of view: non-commutative geometry.

Degenerate case Xη¯ = ∅ is the commutative case. General case involves a quantum parameter: the function π ∈ A, a choice of uniformizer on S.

For this: find a non-commutative variety Xπ such that χ(Xπ) = χ(Xk¯) − χ(XK¯ ) and apply a non-commutative trace formula for Xπ.

Bloch’s conductor formula

Conjecture : χ(Xk¯) − χ(XK¯ )= [∆ X .∆X ]0 + Sw(XK¯ ).

Algebraic geometry, categories and trace formula 17 / 32 For this: find a non-commutative variety Xπ such that χ(Xπ) = χ(Xk¯) − χ(XK¯ ) and apply a non-commutative trace formula for Xπ.

Bloch’s conductor formula

Conjecture : χ(Xk¯) − χ(XK¯ )= [∆ X .∆X ]0 + Sw(XK¯ ).

We want to make progress on this conjecture by introducing a new point of view: non-commutative geometry.

Degenerate case Xη¯ = ∅ is the commutative case. General case involves a quantum parameter: the function π ∈ A, a choice of uniformizer on S.

Algebraic geometry, categories and trace formula 17 / 32 Bloch’s conductor formula

Conjecture : χ(Xk¯) − χ(XK¯ )= [∆ X .∆X ]0 + Sw(XK¯ ).

We want to make progress on this conjecture by introducing a new point of view: non-commutative geometry.

Degenerate case Xη¯ = ∅ is the commutative case. General case involves a quantum parameter: the function π ∈ A, a choice of uniformizer on S.

For this: find a non-commutative variety Xπ such that χ(Xπ) = χ(Xk¯) − χ(XK¯ ) and apply a non-commutative trace formula for Xπ.

Algebraic geometry, categories and trace formula 17 / 32 Reminder: a (dg-)category is

a set of objects

for two objects x and y a (complex of) k-module Hom(x, y)

compositions Hom(x, y) ⊗k Hom(y, z) → Hom(x, z) Dg-categories are considered up to Morita equivalences (”generate the same triangulated categories”).

Non-commutative varieties

Definition A non-commutative variety over some base commutative ring k is a k-linear (dg-)category.

Algebraic geometry, categories and trace formula 18 / 32 Dg-categories are considered up to Morita equivalences (”generate the same triangulated categories”).

Non-commutative varieties

Definition A non-commutative variety over some base commutative ring k is a k-linear (dg-)category.

Reminder: a (dg-)category is

a set of objects

for two objects x and y a (complex of) k-module Hom(x, y)

compositions Hom(x, y) ⊗k Hom(y, z) → Hom(x, z)

Algebraic geometry, categories and trace formula 18 / 32 Non-commutative varieties

Definition A non-commutative variety over some base commutative ring k is a k-linear (dg-)category.

Reminder: a (dg-)category is

a set of objects

for two objects x and y a (complex of) k-module Hom(x, y)

compositions Hom(x, y) ⊗k Hom(y, z) → Hom(x, z) Dg-categories are considered up to Morita equivalences (”generate the same triangulated categories”).

Algebraic geometry, categories and trace formula 18 / 32 Non-commutative varieties: examples

Very weak definition ⇒ plenty of examples !

1 Algebras: A a k-algebra, D(A) = dg-category of complexes of A-modules.

2 Schemes over k: as above + gluing ⇒ D(X ).

3 Other examples: Quivers, topology, symplectic manifolds.

Algebraic geometry, categories and trace formula 19 / 32 Good surprise: non-commutative schemes have reasonable notions of

differential forms (Hochschild homology)

`-adic cohomology (recent construction, see below).

Geometry of non-commutative varieties

Very weak definition ⇒ hard to believe the notion is interesting ! No true geometry (no notions of opens, no topology, no points).

Algebraic geometry, categories and trace formula 20 / 32 Geometry of non-commutative varieties

Very weak definition ⇒ hard to believe the notion is interesting ! No true geometry (no notions of opens, no topology, no points). Good surprise: non-commutative schemes have reasonable notions of

differential forms (Hochschild homology)

`-adic cohomology (recent construction, see below).

Algebraic geometry, categories and trace formula 20 / 32 This extends to dg-algebras and dg-categories: for any non-commutative scheme T we have a Hochschild complex HH(T ).

Hochschild homology (differential forms) For a k-algebra B we have a Hochschild complex

HH(B) := ... B⊗ n / B⊗ (n−1) / ... / B⊗ 2 / B with differential

X i−1 d(b1 ⊗ · · · ⊗ bn) = (−1) b1 ⊗ ... bi−1 ⊗ bi bi+1 ⊗ · · · ⊗ bn i

n +(−1) bnb1 ⊗ b2 ⊗ · · · ⊗ bn−1.

Algebraic geometry, categories and trace formula 21 / 32 Hochschild homology (differential forms) For a k-algebra B we have a Hochschild complex

HH(B) := ... B⊗ n / B⊗ (n−1) / ... / B⊗ 2 / B with differential

X i−1 d(b1 ⊗ · · · ⊗ bn) = (−1) b1 ⊗ ... bi−1 ⊗ bi bi+1 ⊗ · · · ⊗ bn i

n +(−1) bnb1 ⊗ b2 ⊗ · · · ⊗ bn−1. This extends to dg-algebras and dg-categories: for any non-commutative scheme T we have a Hochschild complex HH(T ). Algebraic geometry, categories and trace formula 21 / 32 In particular we have

X p+q p q χ(HH(D(X ))) = (−1) dim H (X , ΩX ). p,q

Hochschild homology (differential forms)

Theorem (Hochschild-Kostant-Rosenberg, Keller) For X a smooth algebraic variety over k, considered as a non-commutative scheme D(X ) we have

M p q HHn(D(X )) ' H (X , ΩX ). p−q=n

Algebraic geometry, categories and trace formula 22 / 32 Hochschild homology (differential forms)

Theorem (Hochschild-Kostant-Rosenberg, Keller) For X a smooth algebraic variety over k, considered as a non-commutative scheme D(X ) we have

M p q HHn(D(X )) ' H (X , ΩX ). p−q=n

In particular we have

X p+q p q χ(HH(D(X ))) = (−1) dim H (X , ΩX ). p,q

Algebraic geometry, categories and trace formula 22 / 32 In particular

χ(HH(D(X ), f )) = [Γf .∆X ].

Hochschild homology (differential forms)

There is a version with coefficients in f  THH (T , f ). Hochschild-Kostant-Rosenberg becomes

L HH(D(X ), f ) ' RΓ(X × X , OΓf ⊗ O∆X ).

Algebraic geometry, categories and trace formula 23 / 32 Hochschild homology (differential forms)

There is a version with coefficients in f  THH (T , f ). Hochschild-Kostant-Rosenberg becomes

L HH(D(X ), f ) ' RΓ(X × X , OΓf ⊗ O∆X ). In particular

χ(HH(D(X ), f )) = [Γf .∆X ].

Algebraic geometry, categories and trace formula 23 / 32 Construction: approximation of T by commutative algebraic varieties + homotopy theory of schemes of Voevodsky-Morel. In particular, we can define Euler characteristic in the non-commutative situation

0 1 χ(T ) := dimH (T , Q`) − dimH (T , Q`) ∈ Z.

`-adic cohomology

Theorem (Blanc, Robalo, T., Vezzosi) For all non-commutative scheme T , and of a prime ` invertible ∗ in k, it is possible to define a Q`-vector spaceH (T , Q`). For T = D(X ) we have

n M 2i+n H (T , Q`) ' H (X , Q`(i)). i

Algebraic geometry, categories and trace formula 24 / 32 In particular, we can define Euler characteristic in the non-commutative situation

0 1 χ(T ) := dimH (T , Q`) − dimH (T , Q`) ∈ Z.

`-adic cohomology

Theorem (Blanc, Robalo, T., Vezzosi) For all non-commutative scheme T , and of a prime ` invertible ∗ in k, it is possible to define a Q`-vector spaceH (T , Q`). For T = D(X ) we have

n M 2i+n H (T , Q`) ' H (X , Q`(i)). i Construction: approximation of T by commutative algebraic varieties + homotopy theory of schemes of Voevodsky-Morel.

Algebraic geometry, categories and trace formula 24 / 32 `-adic cohomology

Theorem (Blanc, Robalo, T., Vezzosi) For all non-commutative scheme T , and of a prime ` invertible ∗ in k, it is possible to define a Q`-vector spaceH (T , Q`). For T = D(X ) we have

n M 2i+n H (T , Q`) ' H (X , Q`(i)). i Construction: approximation of T by commutative algebraic varieties + homotopy theory of schemes of Voevodsky-Morel. In particular, we can define Euler characteristic in the non-commutative situation

0 1 χ(T ) := dimH (T , Q`) − dimH (T , Q`) ∈ Z.

Algebraic geometry, categories and trace formula 24 / 32 By definition, the dual T ∨ comes equipped with

coev : 1 −→ T ⊗ T ∨ ev : T ⊗ T ∨ −→ 1

+ usual properties ( T / T ⊗ T ∨ ⊗ T / T = id). D(X ) is saturated ⇐⇒ X is proper and smooth. Then D(X )∨ 'D(X ) (Poincar´eduality). ev and coev are non-commutative maps which do not exists in the commutative setting !

The non-commutative trace formula

Definition A non-commutative scheme T is saturated (also smooth and proper) if it has a dual T ∨.

Algebraic geometry, categories and trace formula 25 / 32 D(X ) is saturated ⇐⇒ X is proper and smooth. Then D(X )∨ 'D(X ) (Poincar´eduality). ev and coev are non-commutative maps which do not exists in the commutative setting !

The non-commutative trace formula

Definition A non-commutative scheme T is saturated (also smooth and proper) if it has a dual T ∨.

By definition, the dual T ∨ comes equipped with

coev : 1 −→ T ⊗ T ∨ ev : T ⊗ T ∨ −→ 1

+ usual properties ( T / T ⊗ T ∨ ⊗ T / T = id).

Algebraic geometry, categories and trace formula 25 / 32 The non-commutative trace formula

Definition A non-commutative scheme T is saturated (also smooth and proper) if it has a dual T ∨.

By definition, the dual T ∨ comes equipped with

coev : 1 −→ T ⊗ T ∨ ev : T ⊗ T ∨ −→ 1

+ usual properties ( T / T ⊗ T ∨ ⊗ T / T = id). D(X ) is saturated ⇐⇒ X is proper and smooth. Then D(X )∨ 'D(X ) (Poincar´eduality). ev and coev are non-commutative maps which do not exists in the commutative setting !

Algebraic geometry, categories and trace formula 25 / 32 For T = D(X ) gives back the trace formula of Grothendieck.

Also extends to non-commutative schemes T over non-commutative base B.

The non-commutative trace formula

Theorem (Vezzosi, T.) Let T be a saturated non-commutative variety T + technical condition called admissibility. For f  T

∗ Tr(f : H (T , Q`))= χ(HH∗(T , f )).

Algebraic geometry, categories and trace formula 26 / 32 The non-commutative trace formula

Theorem (Vezzosi, T.) Let T be a saturated non-commutative variety T + technical condition called admissibility. For f  T

∗ Tr(f : H (T , Q`))= χ(HH∗(T , f )).

For T = D(X ) gives back the trace formula of Grothendieck.

Also extends to non-commutative schemes T over non-commutative base B.

Algebraic geometry, categories and trace formula 26 / 32 A matrix factorisation for π consists of

two vector bundles E0, E1 on X two morphisms ∂ ∂ E0 / E1 / E0 such that ∂2 = ×π. Matrix factorizations form a (dg-)category MF (X , π). Definition The non-commutative scheme of singularities of X /S is defined to be Xπ := MF (X , π).

Bloch’s formula and non-commutative schemes Back to X ⊂ Pn × S. We fix a uniformizer π of A, which defines a function π on X .

Algebraic geometry, categories and trace formula 27 / 32 Matrix factorizations form a (dg-)category MF (X , π). Definition The non-commutative scheme of singularities of X /S is defined to be Xπ := MF (X , π).

Bloch’s formula and non-commutative schemes Back to X ⊂ Pn × S. We fix a uniformizer π of A, which defines a function π on X . A matrix factorisation for π consists of

two vector bundles E0, E1 on X two morphisms ∂ ∂ E0 / E1 / E0 such that ∂2 = ×π.

Algebraic geometry, categories and trace formula 27 / 32 Bloch’s formula and non-commutative schemes Back to X ⊂ Pn × S. We fix a uniformizer π of A, which defines a function π on X . A matrix factorisation for π consists of

two vector bundles E0, E1 on X two morphisms ∂ ∂ E0 / E1 / E0 such that ∂2 = ×π. Matrix factorizations form a (dg-)category MF (X , π). Definition The non-commutative scheme of singularities of X /S is defined to be Xπ := MF (X , π).

Algebraic geometry, categories and trace formula 27 / 32 2 Over C, and Xo with an isolated singularity we have (Dyckerhoff)

HH0(Xπ) ' Jac(π) = OX ,x /(∂π/∂xi ).

Xπ sees algebraic aspects.

In general MF (X , π) concentrated on singularities MF (X , f ) = 0 when X is smooth over S.

Two examples

1 X a vector space, π = q a quadratic form on X .

Xπ 'D(Ciff (q))

where Ciff (q) is the Clifford algebra of (X , q). Xπ sees arithmetic aspects.

Algebraic geometry, categories and trace formula 28 / 32 In general MF (X , π) concentrated on singularities MF (X , f ) = 0 when X is smooth over S.

Two examples

1 X a vector space, π = q a quadratic form on X .

Xπ 'D(Ciff (q))

where Ciff (q) is the Clifford algebra of (X , q). Xπ sees arithmetic aspects.

2 Over C, and Xo with an isolated singularity we have (Dyckerhoff)

HH0(Xπ) ' Jac(π) = OX ,x /(∂π/∂xi ).

Xπ sees algebraic aspects.

Algebraic geometry, categories and trace formula 28 / 32 Two examples

1 X a vector space, π = q a quadratic form on X .

Xπ 'D(Ciff (q))

where Ciff (q) is the Clifford algebra of (X , q). Xπ sees arithmetic aspects.

2 Over C, and Xo with an isolated singularity we have (Dyckerhoff)

HH0(Xπ) ' Jac(π) = OX ,x /(∂π/∂xi ).

Xπ sees algebraic aspects.

In general MF (X , π) concentrated on singularities MF (X , f ) = 0 when X is smooth over S. Algebraic geometry, categories and trace formula 28 / 32 3 i If the monodromy acts unipotently on Het (XK¯ , Q`) then

∗ χ(H (Xπ, Q`)) = χ(Xk¯) − χ(XK¯ ).

Corollary The Bloch’s conductor conjecture is true when the monodromy is unipotent.

Bloch’s formula and non-commutative schemes Theorem (T., Vezzosi)

1 The non-commutative scheme Xπ is saturated over some funny non-commutative base ring B.

2 χ(HH(Xπ)) = [∆X .∆X ]0

Algebraic geometry, categories and trace formula 29 / 32 Corollary The Bloch’s conductor conjecture is true when the monodromy is unipotent.

Bloch’s formula and non-commutative schemes Theorem (T., Vezzosi)

1 The non-commutative scheme Xπ is saturated over some funny non-commutative base ring B.

2 χ(HH(Xπ)) = [∆X .∆X ]0

3 i If the monodromy acts unipotently on Het (XK¯ , Q`) then

∗ χ(H (Xπ, Q`)) = χ(Xk¯) − χ(XK¯ ).

Algebraic geometry, categories and trace formula 29 / 32 Bloch’s formula and non-commutative schemes Theorem (T., Vezzosi)

1 The non-commutative scheme Xπ is saturated over some funny non-commutative base ring B.

2 χ(HH(Xπ)) = [∆X .∆X ]0

3 i If the monodromy acts unipotently on Het (XK¯ , Q`) then

∗ χ(H (Xπ, Q`)) = χ(Xk¯) − χ(XK¯ ).

Corollary The Bloch’s conductor conjecture is true when the monodromy is unipotent.

Algebraic geometry, categories and trace formula 29 / 32 Appears in 0 Xπ ⊗B Sπ for S 0/S ramified finite covering of S. How this implies the general case of the Bloch’s formula is still under investigation.

Bloch’s formula and non-commutative schemes

What about Sw(XK¯ )?

Algebraic geometry, categories and trace formula 30 / 32 How this implies the general case of the Bloch’s formula is still under investigation.

Bloch’s formula and non-commutative schemes

What about Sw(XK¯ )? Appears in 0 Xπ ⊗B Sπ for S 0/S ramified finite covering of S.

Algebraic geometry, categories and trace formula 30 / 32 Bloch’s formula and non-commutative schemes

What about Sw(XK¯ )? Appears in 0 Xπ ⊗B Sπ for S 0/S ramified finite covering of S. How this implies the general case of the Bloch’s formula is still under investigation.

Algebraic geometry, categories and trace formula 30 / 32 The (3) above recovers the whole action of Galois group GK ∗ on Het (X , ν) = vanishing cohomology. Xπ is surely useful beyond the Bloch’s formula.

A final comment

The non-commutative scheme Xπ sees many interesting aspects

∗ 1 Topological: H (Xπ, Q`).

2 Algebraic: HH(Xπ).

0 3 Arithmetic: Xπ ⊗B Sπ.

Algebraic geometry, categories and trace formula 31 / 32 Xπ is surely useful beyond the Bloch’s formula.

A final comment

The non-commutative scheme Xπ sees many interesting aspects

∗ 1 Topological: H (Xπ, Q`).

2 Algebraic: HH(Xπ).

0 3 Arithmetic: Xπ ⊗B Sπ.

The (3) above recovers the whole action of Galois group GK ∗ on Het (X , ν) = vanishing cohomology.

Algebraic geometry, categories and trace formula 31 / 32 A final comment

The non-commutative scheme Xπ sees many interesting aspects

∗ 1 Topological: H (Xπ, Q`).

2 Algebraic: HH(Xπ).

0 3 Arithmetic: Xπ ⊗B Sπ.

The (3) above recovers the whole action of Galois group GK ∗ on Het (X , ν) = vanishing cohomology. Xπ is surely useful beyond the Bloch’s formula.

Algebraic geometry, categories and trace formula 31 / 32 Algebraic geometry, categories and trace formula

Bertrand To¨en (CNRS, Toulouse)

Clay Research Conference, Oxford, September 2017

Algebraic geometry, categories and trace formula 32 / 32