Counting Vector Bundles

Counting vector bundles

Aravind Asok (USC)

March 9, 2017

Aravind Asok (USC) Counting vector bundles Vector bundles and projective modules

Aravind Asok (USC) Counting vector bundles Definition An R-module P is called projective if it is a direct summand of a free R-module. Equivalently, P is projective if: (lifting property) given an R-module map f : P → M, and a surjective ˜ R-module map N M, we may always find f : P → N. (linear algebraic) if P is also finitely generated, then there exist an integer ⊕n 2 ⊕n n, and ∈ EndR(R ) such that = and P = R . From now on, all projective modules will be assumed finitely generated (f.g.)

Throughout the talk: R is a commutative (unital) ring.

Aravind Asok (USC) Counting vector bundles Equivalently, P is projective if: (lifting property) given an R-module map f : P → M, and a surjective ˜ R-module map N M, we may always find f : P → N. (linear algebraic) if P is also finitely generated, then there exist an integer ⊕n 2 ⊕n n, and ∈ EndR(R ) such that = and P = R . From now on, all projective modules will be assumed finitely generated (f.g.)

Throughout the talk: R is a commutative (unital) ring. Deﬁnition An R-module P is called projective if it is a direct summand of a free R-module.

Aravind Asok (USC) Counting vector bundles (lifting property) given an R-module map f : P → M, and a surjective ˜ R-module map N M, we may always find f : P → N. (linear algebraic) if P is also finitely generated, then there exist an integer ⊕n 2 ⊕n n, and ∈ EndR(R ) such that = and P = R . From now on, all projective modules will be assumed finitely generated (f.g.)

Throughout the talk: R is a commutative (unital) ring. Deﬁnition An R-module P is called projective if it is a direct summand of a free R-module. Equivalently, P is projective if:

Aravind Asok (USC) Counting vector bundles (linear algebraic) if P is also ﬁnitely generated, then there exist an integer ⊕n 2 ⊕n n, and ∈ EndR(R ) such that = and P = R . From now on, all projective modules will be assumed ﬁnitely generated (f.g.)

Throughout the talk: R is a commutative (unital) ring. Deﬁnition An R-module P is called projective if it is a direct summand of a free R-module. Equivalently, P is projective if: (lifting property) given an R-module map f : P → M, and a surjective ˜ R-module map N M, we may always ﬁnd f : P → N.

Aravind Asok (USC) Counting vector bundles Throughout the talk: R is a commutative (unital) ring. Definition An R-module P is called projective if it is a direct summand of a free R-module. Equivalently, P is projective if: (lifting property) given an R-module map f : P → M, and a surjective ˜ R-module map N M, we may always find f : P → N. (linear algebraic) if P is also finitely generated, then there exist an integer ⊕n 2 ⊕n n, and ∈ EndR(R ) such that = and P = R . From now on, all projective modules will be assumed finitely generated (f.g.)

Aravind Asok (USC) Counting vector bundles f.g. projective modules are “locally free” modules

Algebraically: P a f.g. projective R-module; we can ﬁnd elements f1,..., fr ∈ R 1 1 such that fi generate the unit ideal and such that P[ ] is a free R[ ]-module of ﬁnite fi fi rank Geometrically: we associate with R its prime spectrum Spec R, and Spec R[ 1 ] fi forms an open cover of Spec R on which the bundle corresponding to P may be trivialized f.g. projective modules have a rank if Spec R is connected, then this is just an integer

Projective modules behave like vector bundles:

Aravind Asok (USC) Counting vector bundles Geometrically: we associate with R its prime spectrum Spec R, and Spec R[ 1 ] fi forms an open cover of Spec R on which the bundle corresponding to P may be trivialized f.g. projective modules have a rank if Spec R is connected, then this is just an integer

Projective modules behave like vector bundles: f.g. projective modules are “locally free” modules

Algebraically: P a f.g. projective R-module; we can ﬁnd elements f1,..., fr ∈ R 1 1 such that fi generate the unit ideal and such that P[ ] is a free R[ ]-module of ﬁnite fi fi rank

Aravind Asok (USC) Counting vector bundles f.g. projective modules have a rank if Spec R is connected, then this is just an integer

Projective modules behave like vector bundles: f.g. projective modules are “locally free” modules

Aravind Asok (USC) Counting vector bundles Projective modules behave like vector bundles: f.g. projective modules are “locally free” modules

Aravind Asok (USC) Counting vector bundles Analogously: Serre’s dictionary If R is a ring, then

{ ﬁnite rank v.b. over Spec R} ←→ { f.g. projective R − modules };

Using this dictionary, one transplants intuition from geometry to algebra

Serre–Swan correspondence

{ ﬁnite rank v.b. over M} ←→ { f.g. projective C(M) − modules };

M a (say) compact manifold; C(M) = ring of continuous real-valued functions on M

Aravind Asok (USC) Counting vector bundles Using this dictionary, one transplants intuition from geometry to algebra

Serre–Swan correspondence

{ ﬁnite rank v.b. over M} ←→ { f.g. projective C(M) − modules };

M a (say) compact manifold; C(M) = ring of continuous real-valued functions on M

Analogously: Serre’s dictionary If R is a ring, then

{ ﬁnite rank v.b. over Spec R} ←→ { f.g. projective R − modules };

Aravind Asok (USC) Counting vector bundles Serre–Swan correspondence

{ ﬁnite rank v.b. over M} ←→ { f.g. projective C(M) − modules };

M a (say) compact manifold; C(M) = ring of continuous real-valued functions on M

Analogously: Serre’s dictionary If R is a ring, then

{ ﬁnite rank v.b. over Spec R} ←→ { f.g. projective R − modules };

Using this dictionary, one transplants intuition from geometry to algebra

Aravind Asok (USC) Counting vector bundles Example

If K is a number ﬁeld, and OK is the ring of integers in K, then there are at most ﬁnitely many projective OK-modules of a given rank.

OK is a Dedekind domain (in particular, it has Krull dimension 1)

By Serre’s theorem, a f.g. projective OK-module of rank r can be written r−1 L ⊕ OK where L has rank 1 Rank 1 projective modules form an abelian group (the Picard group) under tensor product Minkowski’s theorem implies that the Picard group is ﬁnite

Theorem (Serre’s splitting theorem ’58) Suppose R is a Noetherian commutative ring of Krull dimension d. If P is a projective R-module of rank r > d, then there exists a projective R-module Q of rank d and an isomorphism P =∼ Q ⊕ R⊕r−d.

Aravind Asok (USC) Counting vector bundles OK is a Dedekind domain (in particular, it has Krull dimension 1)

Example

If K is a number ﬁeld, and OK is the ring of integers in K, then there are at most ﬁnitely many projective OK-modules of a given rank.

Aravind Asok (USC) Counting vector bundles By Serre’s theorem, a f.g. projective OK-module of rank r can be written r−1 L ⊕ OK where L has rank 1 Rank 1 projective modules form an abelian group (the Picard group) under tensor product Minkowski’s theorem implies that the Picard group is ﬁnite

Example

If K is a number ﬁeld, and OK is the ring of integers in K, then there are at most ﬁnitely many projective OK-modules of a given rank.

OK is a Dedekind domain (in particular, it has Krull dimension 1)

Aravind Asok (USC) Counting vector bundles Rank 1 projective modules form an abelian group (the Picard group) under tensor product Minkowski’s theorem implies that the Picard group is ﬁnite

Example

If K is a number ﬁeld, and OK is the ring of integers in K, then there are at most ﬁnitely many projective OK-modules of a given rank.

OK is a Dedekind domain (in particular, it has Krull dimension 1)

By Serre’s theorem, a f.g. projective OK-module of rank r can be written r−1 L ⊕ OK where L has rank 1

Aravind Asok (USC) Counting vector bundles Minkowski’s theorem implies that the Picard group is ﬁnite

Example

If K is a number ﬁeld, and OK is the ring of integers in K, then there are at most ﬁnitely many projective OK-modules of a given rank.

OK is a Dedekind domain (in particular, it has Krull dimension 1)

By Serre’s theorem, a f.g. projective OK-module of rank r can be written r−1 L ⊕ OK where L has rank 1 Rank 1 projective modules form an abelian group (the Picard group) under tensor product

Aravind Asok (USC) Counting vector bundles Theorem (Serre’s splitting theorem ’58) Suppose R is a Noetherian commutative ring of Krull dimension d. If P is a projective R-module of rank r > d, then there exists a projective R-module Q of rank d and an isomorphism P =∼ Q ⊕ R⊕r−d.

Example

If K is a number ﬁeld, and OK is the ring of integers in K, then there are at most ﬁnitely many projective OK-modules of a given rank.

OK is a Dedekind domain (in particular, it has Krull dimension 1)

Aravind Asok (USC) Counting vector bundles Vr(M) - set of isomorphism classes of rank r (real) vector bundles on M

Vr(M) → Vr(M × I) is a bijection (homotopy invariance)

Grr - Grassmannian of r-dimensional subspaces of an inﬁnite dimensional real vector space

Grr carries a rank r “tautological” vector bundle

Continuous maps M → Grr yield rank r vector bundle on M by pullback; homotopic maps yield isomorphic bundles

Theorem (Pontryagin–Steenrod) Pulling back the tautological bundle determines a bijection:

∼ [M, Grr]−→Vr(M).

n Real vector bundles on any contractible manifold (e.g., R ) are trivial

The problem of describing vector bundles is really homotopy-theoretic:

Aravind Asok (USC) Counting vector bundles Vr(M) → Vr(M × I) is a bijection (homotopy invariance)

Grr - Grassmannian of r-dimensional subspaces of an inﬁnite dimensional real vector space

Grr carries a rank r “tautological” vector bundle

Continuous maps M → Grr yield rank r vector bundle on M by pullback; homotopic maps yield isomorphic bundles

Theorem (Pontryagin–Steenrod) Pulling back the tautological bundle determines a bijection:

∼ [M, Grr]−→Vr(M).

n Real vector bundles on any contractible manifold (e.g., R ) are trivial

The problem of describing vector bundles is really homotopy-theoretic:

Vr(M) - set of isomorphism classes of rank r (real) vector bundles on M

Aravind Asok (USC) Counting vector bundles Grr - Grassmannian of r-dimensional subspaces of an inﬁnite dimensional real vector space

Grr carries a rank r “tautological” vector bundle

Continuous maps M → Grr yield rank r vector bundle on M by pullback; homotopic maps yield isomorphic bundles

Theorem (Pontryagin–Steenrod) Pulling back the tautological bundle determines a bijection:

∼ [M, Grr]−→Vr(M).

n Real vector bundles on any contractible manifold (e.g., R ) are trivial

The problem of describing vector bundles is really homotopy-theoretic:

Vr(M) - set of isomorphism classes of rank r (real) vector bundles on M

Vr(M) → Vr(M × I) is a bijection (homotopy invariance)

Aravind Asok (USC) Counting vector bundles Grr carries a rank r “tautological” vector bundle

Continuous maps M → Grr yield rank r vector bundle on M by pullback; homotopic maps yield isomorphic bundles

Theorem (Pontryagin–Steenrod) Pulling back the tautological bundle determines a bijection:

∼ [M, Grr]−→Vr(M).

n Real vector bundles on any contractible manifold (e.g., R ) are trivial

The problem of describing vector bundles is really homotopy-theoretic:

Vr(M) - set of isomorphism classes of rank r (real) vector bundles on M

Vr(M) → Vr(M × I) is a bijection (homotopy invariance)

Grr - Grassmannian of r-dimensional subspaces of an inﬁnite dimensional real vector space

Aravind Asok (USC) Counting vector bundles Continuous maps M → Grr yield rank r vector bundle on M by pullback; homotopic maps yield isomorphic bundles

Theorem (Pontryagin–Steenrod) Pulling back the tautological bundle determines a bijection:

∼ [M, Grr]−→Vr(M).

n Real vector bundles on any contractible manifold (e.g., R ) are trivial

The problem of describing vector bundles is really homotopy-theoretic:

Vr(M) - set of isomorphism classes of rank r (real) vector bundles on M

Vr(M) → Vr(M × I) is a bijection (homotopy invariance)

Grr - Grassmannian of r-dimensional subspaces of an inﬁnite dimensional real vector space

Grr carries a rank r “tautological” vector bundle

Aravind Asok (USC) Counting vector bundles Theorem (Pontryagin–Steenrod) Pulling back the tautological bundle determines a bijection:

∼ [M, Grr]−→Vr(M).

n Real vector bundles on any contractible manifold (e.g., R ) are trivial

The problem of describing vector bundles is really homotopy-theoretic:

Vr(M) - set of isomorphism classes of rank r (real) vector bundles on M

Vr(M) → Vr(M × I) is a bijection (homotopy invariance)

Grr - Grassmannian of r-dimensional subspaces of an inﬁnite dimensional real vector space

Grr carries a rank r “tautological” vector bundle

Continuous maps M → Grr yield rank r vector bundle on M by pullback; homotopic maps yield isomorphic bundles

Aravind Asok (USC) Counting vector bundles n Real vector bundles on any contractible manifold (e.g., R ) are trivial

The problem of describing vector bundles is really homotopy-theoretic:

Vr(M) - set of isomorphism classes of rank r (real) vector bundles on M

Vr(M) → Vr(M × I) is a bijection (homotopy invariance)

Grr - Grassmannian of r-dimensional subspaces of an inﬁnite dimensional real vector space

Grr carries a rank r “tautological” vector bundle

Continuous maps M → Grr yield rank r vector bundle on M by pullback; homotopic maps yield isomorphic bundles

Theorem (Pontryagin–Steenrod) Pulling back the tautological bundle determines a bijection:

∼ [M, Grr]−→Vr(M).

Aravind Asok (USC) Counting vector bundles The problem of describing vector bundles is really homotopy-theoretic:

Vr(M) - set of isomorphism classes of rank r (real) vector bundles on M

Vr(M) → Vr(M × I) is a bijection (homotopy invariance)

Grr - Grassmannian of r-dimensional subspaces of an inﬁnite dimensional real vector space

Grr carries a rank r “tautological” vector bundle

Continuous maps M → Grr yield rank r vector bundle on M by pullback; homotopic maps yield isomorphic bundles

Theorem (Pontryagin–Steenrod) Pulling back the tautological bundle determines a bijection:

∼ [M, Grr]−→Vr(M).

n Real vector bundles on any contractible manifold (e.g., R ) are trivial

Aravind Asok (USC) Counting vector bundles n = 1: yes, structure theorem for f.g. modules over a PID n = 2: yes, Seshadri ’58 yes if r > n, Bass ’64 n = 3: yes if k algebraically closed, Murthy–Towber ’74 n = 3, 4, 5: yes, Suslin–Vaserstein ’73/’74

Theorem (Quillen–Suslin ’76)

If R is a PID, then every f.g. projective R[x1,..., xn]-module is free.

Question (Serre ’55)

If k is a ﬁeld, is every f.g. projective k[x1,..., xn]-module free?

Aravind Asok (USC) Counting vector bundles n = 2: yes, Seshadri ’58 yes if r > n, Bass ’64 n = 3: yes if k algebraically closed, Murthy–Towber ’74 n = 3, 4, 5: yes, Suslin–Vaserstein ’73/’74

Theorem (Quillen–Suslin ’76)

If R is a PID, then every f.g. projective R[x1,..., xn]-module is free.

Question (Serre ’55)

If k is a ﬁeld, is every f.g. projective k[x1,..., xn]-module free?

n = 1: yes, structure theorem for f.g. modules over a PID

Aravind Asok (USC) Counting vector bundles yes if r > n, Bass ’64 n = 3: yes if k algebraically closed, Murthy–Towber ’74 n = 3, 4, 5: yes, Suslin–Vaserstein ’73/’74

Theorem (Quillen–Suslin ’76)

If R is a PID, then every f.g. projective R[x1,..., xn]-module is free.

Question (Serre ’55)

If k is a ﬁeld, is every f.g. projective k[x1,..., xn]-module free?

n = 1: yes, structure theorem for f.g. modules over a PID n = 2: yes, Seshadri ’58

Aravind Asok (USC) Counting vector bundles n = 3: yes if k algebraically closed, Murthy–Towber ’74 n = 3, 4, 5: yes, Suslin–Vaserstein ’73/’74

Theorem (Quillen–Suslin ’76)

If R is a PID, then every f.g. projective R[x1,..., xn]-module is free.

Question (Serre ’55)

If k is a ﬁeld, is every f.g. projective k[x1,..., xn]-module free?

n = 1: yes, structure theorem for f.g. modules over a PID n = 2: yes, Seshadri ’58 yes if r > n, Bass ’64

Aravind Asok (USC) Counting vector bundles n = 3, 4, 5: yes, Suslin–Vaserstein ’73/’74

Theorem (Quillen–Suslin ’76)

If R is a PID, then every f.g. projective R[x1,..., xn]-module is free.

Question (Serre ’55)

If k is a ﬁeld, is every f.g. projective k[x1,..., xn]-module free?

n = 1: yes, structure theorem for f.g. modules over a PID n = 2: yes, Seshadri ’58 yes if r > n, Bass ’64 n = 3: yes if k algebraically closed, Murthy–Towber ’74

Aravind Asok (USC) Counting vector bundles Theorem (Quillen–Suslin ’76)

If R is a PID, then every f.g. projective R[x1,..., xn]-module is free.

Question (Serre ’55)

If k is a ﬁeld, is every f.g. projective k[x1,..., xn]-module free?

n = 1: yes, structure theorem for f.g. modules over a PID n = 2: yes, Seshadri ’58 yes if r > n, Bass ’64 n = 3: yes if k algebraically closed, Murthy–Towber ’74 n = 3, 4, 5: yes, Suslin–Vaserstein ’73/’74

Aravind Asok (USC) Counting vector bundles Question (Serre ’55)

If k is a ﬁeld, is every f.g. projective k[x1,..., xn]-module free?

Theorem (Quillen–Suslin ’76)

If R is a PID, then every f.g. projective R[x1,..., xn]-module is free.

Aravind Asok (USC) Counting vector bundles Question If R is a ring, is Vr(Spec R) −→ Vr(Spec R[t]) a bijection?

False for r = 1 without additional hypotheses on R (e.g., R = k[x, y]/(y2 − x3)) We will assume R is regular (an analog of smoothness)

Vr(Spec R) = {isomorphism classes of rank r v.b. on Spec R}

Aravind Asok (USC) Counting vector bundles False for r = 1 without additional hypotheses on R (e.g., R = k[x, y]/(y2 − x3)) We will assume R is regular (an analog of smoothness)

Vr(Spec R) = {isomorphism classes of rank r v.b. on Spec R}

Question If R is a ring, is Vr(Spec R) −→ Vr(Spec R[t]) a bijection?

Aravind Asok (USC) Counting vector bundles We will assume R is regular (an analog of smoothness)

Vr(Spec R) = {isomorphism classes of rank r v.b. on Spec R}

Question If R is a ring, is Vr(Spec R) −→ Vr(Spec R[t]) a bijection?

False for r = 1 without additional hypotheses on R (e.g., R = k[x, y]/(y2 − x3))

Aravind Asok (USC) Counting vector bundles Vr(Spec R) = {isomorphism classes of rank r v.b. on Spec R}

Question If R is a ring, is Vr(Spec R) −→ Vr(Spec R[t]) a bijection?

False for r = 1 without additional hypotheses on R (e.g., R = k[x, y]/(y2 − x3)) We will assume R is regular (an analog of smoothness)

Aravind Asok (USC) Counting vector bundles Quillen’s solution to Serre’s problem actually shows that the Bass-Quillen conjecture holds for R a polynomial ring over a Dedekind domain.

Theorem (Lindel ’81) The Bass-Quillen conjecture is true if R contains a ﬁeld.

Popescu ’89 extended the Lindel’s theorem to some arithmetic situations (e.g., R is regular over a Dedekind domain with perfect residue ﬁelds) Still open in completely generality!

Conjecture (Bass–Quillen ’72) If R is a regular ring of ﬁnite Krull dimension, then for any r ≥ 0

Vr(Spec R) −→ Vr(Spec R[t]) is a bijection.

Aravind Asok (USC) Counting vector bundles Theorem (Lindel ’81) The Bass-Quillen conjecture is true if R contains a ﬁeld.

Popescu ’89 extended the Lindel’s theorem to some arithmetic situations (e.g., R is regular over a Dedekind domain with perfect residue ﬁelds) Still open in completely generality!

Conjecture (Bass–Quillen ’72) If R is a regular ring of ﬁnite Krull dimension, then for any r ≥ 0

Vr(Spec R) −→ Vr(Spec R[t]) is a bijection.

Quillen’s solution to Serre’s problem actually shows that the Bass-Quillen conjecture holds for R a polynomial ring over a Dedekind domain.

Aravind Asok (USC) Counting vector bundles Popescu ’89 extended the Lindel’s theorem to some arithmetic situations (e.g., R is regular over a Dedekind domain with perfect residue ﬁelds) Still open in completely generality!

Conjecture (Bass–Quillen ’72) If R is a regular ring of ﬁnite Krull dimension, then for any r ≥ 0

Vr(Spec R) −→ Vr(Spec R[t]) is a bijection.

Quillen’s solution to Serre’s problem actually shows that the Bass-Quillen conjecture holds for R a polynomial ring over a Dedekind domain.

Theorem (Lindel ’81) The Bass-Quillen conjecture is true if R contains a ﬁeld.

Aravind Asok (USC) Counting vector bundles Still open in completely generality!

Conjecture (Bass–Quillen ’72) If R is a regular ring of ﬁnite Krull dimension, then for any r ≥ 0

Vr(Spec R) −→ Vr(Spec R[t]) is a bijection.

Quillen’s solution to Serre’s problem actually shows that the Bass-Quillen conjecture holds for R a polynomial ring over a Dedekind domain.

Theorem (Lindel ’81) The Bass-Quillen conjecture is true if R contains a ﬁeld.

Popescu ’89 extended the Lindel’s theorem to some arithmetic situations (e.g., R is regular over a Dedekind domain with perfect residue ﬁelds)

Aravind Asok (USC) Counting vector bundles Conjecture (Bass–Quillen ’72) If R is a regular ring of ﬁnite Krull dimension, then for any r ≥ 0

Vr(Spec R) −→ Vr(Spec R[t]) is a bijection.

Quillen’s solution to Serre’s problem actually shows that the Bass-Quillen conjecture holds for R a polynomial ring over a Dedekind domain.

Theorem (Lindel ’81) The Bass-Quillen conjecture is true if R contains a ﬁeld.

Popescu ’89 extended the Lindel’s theorem to some arithmetic situations (e.g., R is regular over a Dedekind domain with perfect residue ﬁelds) Still open in completely generality!

Aravind Asok (USC) Counting vector bundles Vector bundles and motivic homotopy theory

Aravind Asok (USC) Counting vector bundles A map Spec R → Grr corresponds to a vector bundle on R and a collection of generating sections We may speak of “naive” homotopies between such maps, i.e., two maps f , g : Spec R → Grr are naively homotopic if there exists a map H : Spec R[t] → Grr with H(0) = f and H(1) = g.

Theorem If k is a ﬁeld, and R is a regular k-algebra, then

∼ [Spec R, Grr]naive−→Vr(Spec R);

Unfortunately, naive homotopy is not a “good” notion (e.g., it is not an equivalence relation).

Grr is an (inﬁnite-dimensional) algebraic variety

Aravind Asok (USC) Counting vector bundles We may speak of “naive” homotopies between such maps, i.e., two maps f , g : Spec R → Grr are naively homotopic if there exists a map H : Spec R[t] → Grr with H(0) = f and H(1) = g.

Theorem If k is a ﬁeld, and R is a regular k-algebra, then

∼ [Spec R, Grr]naive−→Vr(Spec R);

Unfortunately, naive homotopy is not a “good” notion (e.g., it is not an equivalence relation).

Grr is an (inﬁnite-dimensional) algebraic variety

A map Spec R → Grr corresponds to a vector bundle on R and a collection of generating sections

Aravind Asok (USC) Counting vector bundles Theorem If k is a ﬁeld, and R is a regular k-algebra, then

∼ [Spec R, Grr]naive−→Vr(Spec R);

Unfortunately, naive homotopy is not a “good” notion (e.g., it is not an equivalence relation).

Grr is an (inﬁnite-dimensional) algebraic variety

A map Spec R → Grr corresponds to a vector bundle on R and a collection of generating sections We may speak of “naive” homotopies between such maps, i.e., two maps f , g : Spec R → Grr are naively homotopic if there exists a map H : Spec R[t] → Grr with H(0) = f and H(1) = g.

Aravind Asok (USC) Counting vector bundles Unfortunately, naive homotopy is not a “good” notion (e.g., it is not an equivalence relation).

Grr is an (inﬁnite-dimensional) algebraic variety

Theorem If k is a ﬁeld, and R is a regular k-algebra, then

∼ [Spec R, Grr]naive−→Vr(Spec R);

Aravind Asok (USC) Counting vector bundles Grr is an (inﬁnite-dimensional) algebraic variety

Theorem If k is a ﬁeld, and R is a regular k-algebra, then

∼ [Spec R, Grr]naive−→Vr(Spec R);

Unfortunately, naive homotopy is not a “good” notion (e.g., it is not an equivalence relation).

Aravind Asok (USC) Counting vector bundles k a ﬁxed base-ring; Smk, the category of smooth k-varieties

Smk is not “big enough” to do homotopy theory (e.g., Grr is not in this category, cannot form all quotients, etc.)

Spck - “spaces” an enlargement of Smk where we can perform all the constructions we will want later (simplicial presheaves on Smk) We now force two kinds of maps to be “weak-equivalences”:

Nisnevich local weak equivalences: roughly, if X can be covered by {Ui}, then we may build a Cech object C(u) → X, and we force C(u) → X to be an isomorphism A1-weak equivalences: X × A1 → X Halg(k) for the category obtained by inverting Nisnevich local weak equivalences and

Hmot(k) for the category obtained by inverting both Nisnevich local and A1-weak equivalences (this is the Morel–Voevodsky A1-homotopy category)

We restricted attention to regular rings in order to have an analog of homotopy invariance.

Aravind Asok (USC) Counting vector bundles Smk is not “big enough” to do homotopy theory (e.g., Grr is not in this category, cannot form all quotients, etc.)

Spck - “spaces” an enlargement of Smk where we can perform all the constructions we will want later (simplicial presheaves on Smk) We now force two kinds of maps to be “weak-equivalences”:

Hmot(k) for the category obtained by inverting both Nisnevich local and A1-weak equivalences (this is the Morel–Voevodsky A1-homotopy category)

We restricted attention to regular rings in order to have an analog of homotopy invariance. k a ﬁxed base-ring; Smk, the category of smooth k-varieties

Aravind Asok (USC) Counting vector bundles Spck - “spaces” an enlargement of Smk where we can perform all the constructions we will want later (simplicial presheaves on Smk) We now force two kinds of maps to be “weak-equivalences”:

Hmot(k) for the category obtained by inverting both Nisnevich local and A1-weak equivalences (this is the Morel–Voevodsky A1-homotopy category)

We restricted attention to regular rings in order to have an analog of homotopy invariance. k a ﬁxed base-ring; Smk, the category of smooth k-varieties

Smk is not “big enough” to do homotopy theory (e.g., Grr is not in this category, cannot form all quotients, etc.)

Aravind Asok (USC) Counting vector bundles We now force two kinds of maps to be “weak-equivalences”:

Hmot(k) for the category obtained by inverting both Nisnevich local and A1-weak equivalences (this is the Morel–Voevodsky A1-homotopy category)

We restricted attention to regular rings in order to have an analog of homotopy invariance. k a ﬁxed base-ring; Smk, the category of smooth k-varieties

Smk is not “big enough” to do homotopy theory (e.g., Grr is not in this category, cannot form all quotients, etc.)

Spck - “spaces” an enlargement of Smk where we can perform all the constructions we will want later (simplicial presheaves on Smk)

Aravind Asok (USC) Counting vector bundles Nisnevich local weak equivalences: roughly, if X can be covered by {Ui}, then we may build a Cech object C(u) → X, and we force C(u) → X to be an isomorphism A1-weak equivalences: X × A1 → X Halg(k) for the category obtained by inverting Nisnevich local weak equivalences and

Hmot(k) for the category obtained by inverting both Nisnevich local and A1-weak equivalences (this is the Morel–Voevodsky A1-homotopy category)

We restricted attention to regular rings in order to have an analog of homotopy invariance. k a ﬁxed base-ring; Smk, the category of smooth k-varieties

Smk is not “big enough” to do homotopy theory (e.g., Grr is not in this category, cannot form all quotients, etc.)

Spck - “spaces” an enlargement of Smk where we can perform all the constructions we will want later (simplicial presheaves on Smk) We now force two kinds of maps to be “weak-equivalences”:

Aravind Asok (USC) Counting vector bundles A1-weak equivalences: X × A1 → X Halg(k) for the category obtained by inverting Nisnevich local weak equivalences and

Hmot(k) for the category obtained by inverting both Nisnevich local and A1-weak equivalences (this is the Morel–Voevodsky A1-homotopy category)

We restricted attention to regular rings in order to have an analog of homotopy invariance. k a ﬁxed base-ring; Smk, the category of smooth k-varieties

Smk is not “big enough” to do homotopy theory (e.g., Grr is not in this category, cannot form all quotients, etc.)

Spck - “spaces” an enlargement of Smk where we can perform all the constructions we will want later (simplicial presheaves on Smk) We now force two kinds of maps to be “weak-equivalences”:

Nisnevich local weak equivalences: roughly, if X can be covered by {Ui}, then we may build a Cech object C(u) → X, and we force C(u) → X to be an isomorphism

Aravind Asok (USC) Counting vector bundles Halg(k) for the category obtained by inverting Nisnevich local weak equivalences and

Hmot(k) for the category obtained by inverting both Nisnevich local and A1-weak equivalences (this is the Morel–Voevodsky A1-homotopy category)

We restricted attention to regular rings in order to have an analog of homotopy invariance. k a ﬁxed base-ring; Smk, the category of smooth k-varieties

Smk is not “big enough” to do homotopy theory (e.g., Grr is not in this category, cannot form all quotients, etc.)

Spck - “spaces” an enlargement of Smk where we can perform all the constructions we will want later (simplicial presheaves on Smk) We now force two kinds of maps to be “weak-equivalences”:

Nisnevich local weak equivalences: roughly, if X can be covered by {Ui}, then we may build a Cech object C(u) → X, and we force C(u) → X to be an isomorphism A1-weak equivalences: X × A1 → X

Aravind Asok (USC) Counting vector bundles and

Hmot(k) for the category obtained by inverting both Nisnevich local and A1-weak equivalences (this is the Morel–Voevodsky A1-homotopy category)

We restricted attention to regular rings in order to have an analog of homotopy invariance. k a ﬁxed base-ring; Smk, the category of smooth k-varieties

Smk is not “big enough” to do homotopy theory (e.g., Grr is not in this category, cannot form all quotients, etc.)

Spck - “spaces” an enlargement of Smk where we can perform all the constructions we will want later (simplicial presheaves on Smk) We now force two kinds of maps to be “weak-equivalences”:

Aravind Asok (USC) Counting vector bundles We restricted attention to regular rings in order to have an analog of homotopy invariance. k a ﬁxed base-ring; Smk, the category of smooth k-varieties

Smk is not “big enough” to do homotopy theory (e.g., Grr is not in this category, cannot form all quotients, etc.)

Spck - “spaces” an enlargement of Smk where we can perform all the constructions we will want later (simplicial presheaves on Smk) We now force two kinds of maps to be “weak-equivalences”:

Hmot(k) for the category obtained by inverting both Nisnevich local and A1-weak equivalences (this is the Morel–Voevodsky A1-homotopy category)

Aravind Asok (USC) Counting vector bundles Example (Non-abelian sheaf cohomology) Rank r vector bundles are the same thing as principal bundles under the general linear group GLr; If X is a smooth scheme, then the sheaf cohomology pointed set 1 H (X, GLr) classiﬁes GLr-torsors (can be computed using Cech cohomology)

We may deﬁne a space BGLr

··· // GL × GL / GL / ∗ / r r / r /

∼ Then, for any smooth scheme X, HomHalg(k)(X, BGLr) = Vr(X)

Note: BGLr and Grr are not isomorphic in Halg(k)!

Computing maps in Halg(k) or Hmot(k) is hard (not just a quotient of morphisms by an equivalence relation)

Aravind Asok (USC) Counting vector bundles If X is a smooth scheme, then the sheaf cohomology pointed set 1 H (X, GLr) classiﬁes GLr-torsors (can be computed using Cech cohomology)

We may deﬁne a space BGLr

··· // GL × GL / GL / ∗ / r r / r /

∼ Then, for any smooth scheme X, HomHalg(k)(X, BGLr) = Vr(X)

Note: BGLr and Grr are not isomorphic in Halg(k)!

Aravind Asok (USC) Counting vector bundles We may deﬁne a space BGLr

··· // GL × GL / GL / ∗ / r r / r /

∼ Then, for any smooth scheme X, HomHalg(k)(X, BGLr) = Vr(X)

Note: BGLr and Grr are not isomorphic in Halg(k)!

Computing maps in Halg(k) or Hmot(k) is hard (not just a quotient of morphisms by an equivalence relation) Example (Non-abelian sheaf cohomology) Rank r vector bundles are the same thing as principal bundles under the general linear group GLr; If X is a smooth scheme, then the sheaf cohomology pointed set 1 H (X, GLr) classiﬁes GLr-torsors (can be computed using Cech cohomology)

Aravind Asok (USC) Counting vector bundles ∼ Then, for any smooth scheme X, HomHalg(k)(X, BGLr) = Vr(X)

Note: BGLr and Grr are not isomorphic in Halg(k)!

We may deﬁne a space BGLr

··· // GL × GL / GL / ∗ / r r // r /

Aravind Asok (USC) Counting vector bundles Note: BGLr and Grr are not isomorphic in Halg(k)!

We may deﬁne a space BGLr

··· // GL × GL / GL / ∗ / r r // r /

∼ Then, for any smooth scheme X, HomHalg(k)(X, BGLr) = Vr(X)

Aravind Asok (USC) Counting vector bundles Computing maps in Halg(k) or Hmot(k) is hard (not just a quotient of morphisms by an equivalence relation) Example (Non-abelian sheaf cohomology) Rank r vector bundles are the same thing as principal bundles under the general linear group GLr; If X is a smooth scheme, then the sheaf cohomology pointed set 1 H (X, GLr) classiﬁes GLr-torsors (can be computed using Cech cohomology)

We may deﬁne a space BGLr

··· // GL × GL / GL / ∗ / r r // r /

∼ Then, for any smooth scheme X, HomHalg(k)(X, BGLr) = Vr(X)

Note: BGLr and Grr are not isomorphic in Halg(k)!

Aravind Asok (USC) Counting vector bundles Example If π : Y → X is Zariski locally trivial with affine space fibers, then π is an A1-weak equivalence (e.g., if π is a vector bundle): 2 1 SL2 → A \ 0 (project a matrix onto its first column) is an A -weak equivalence; the complement of the incidence divisor in P1 × P1 is a smooth affine 1 1 quadric Q2 that is A -weakly equivalent to P

Example ∞ A \ 0 is A1-contractible: the “shift map” is naively homotopic to the identity ∞ ∞ 1 Gr1 = P = A \ 0/GL1 → BGL1 is an A -weak equivalence More generally, Grn → BGLn corresponding to the tautological vector bundle is an A1-weak equivalence

1 Isomorphisms in Hmot(k) are called A -weak equivalences

Aravind Asok (USC) Counting vector bundles 2 1 SL2 → A \ 0 (project a matrix onto its ﬁrst column) is an A -weak equivalence; the complement of the incidence divisor in P1 × P1 is a smooth afﬁne 1 1 quadric Q2 that is A -weakly equivalent to P

Aravind Asok (USC) Counting vector bundles the complement of the incidence divisor in P1 × P1 is a smooth afﬁne 1 1 quadric Q2 that is A -weakly equivalent to P

1 Isomorphisms in Hmot(k) are called A -weak equivalences Example If π : Y → X is Zariski locally trivial with affine space fibers, then π is an A1-weak equivalence (e.g., if π is a vector bundle): 2 1 SL2 → A \ 0 (project a matrix onto its first column) is an A -weak equivalence;

Aravind Asok (USC) Counting vector bundles Example ∞ A \ 0 is A1-contractible: the “shift map” is naively homotopic to the identity ∞ ∞ 1 Gr1 = P = A \ 0/GL1 → BGL1 is an A -weak equivalence More generally, Grn → BGLn corresponding to the tautological vector bundle is an A1-weak equivalence

Aravind Asok (USC) Counting vector bundles ∞ ∞ 1 Gr1 = P = A \ 0/GL1 → BGL1 is an A -weak equivalence More generally, Grn → BGLn corresponding to the tautological vector bundle is an A1-weak equivalence

Example ∞ A \ 0 is A1-contractible: the “shift map” is naively homotopic to the identity

Aravind Asok (USC) Counting vector bundles More generally, Grn → BGLn corresponding to the tautological vector bundle is an A1-weak equivalence

Example ∞ A \ 0 is A1-contractible: the “shift map” is naively homotopic to the identity ∞ ∞ 1 Gr1 = P = A \ 0/GL1 → BGL1 is an A -weak equivalence

Aravind Asok (USC) Counting vector bundles 1 Isomorphisms in Hmot(k) are called A -weak equivalences Example If π : Y → X is Zariski locally trivial with affine space fibers, then π is an A1-weak equivalence (e.g., if π is a vector bundle): 2 1 SL2 → A \ 0 (project a matrix onto its first column) is an A -weak equivalence; the complement of the incidence divisor in P1 × P1 is a smooth affine 1 1 quadric Q2 that is A -weakly equivalent to P

Aravind Asok (USC) Counting vector bundles Unfortunately, this fails for BGLr with r ≥ 2. Example (A., B. Doran ’08)

Set Q4 = Spec k[x1, x2, x3, x4, z]/hx1x2 − x3x4 = z(z + 1)i 2 E2 ⊂ Q4 deﬁned by x1 = x3 = z + 1 = 0 is isomorphic to A 1 X4 = Q4 \ E2 is actually A -contractible: there is an explicit morphism 5 1 A → X4 that is Zariski locally trivial with ﬁbers A Q4 carries an explicit non-trivial rank 2 bundle (the Hopf bundle); 1 this bundle restricts non-trivially to X4, i.e., A -contractible varieties may carry non-trivial vector bundles!

If k is regular, one may show HomHmot(k)(X, BGL1) = Pic(X) for any smooth k-scheme X.

Aravind Asok (USC) Counting vector bundles Example (A., B. Doran ’08)

If k is regular, one may show HomHmot(k)(X, BGL1) = Pic(X) for any smooth k-scheme X. Unfortunately, this fails for BGLr with r ≥ 2.

Aravind Asok (USC) Counting vector bundles 2 E2 ⊂ Q4 deﬁned by x1 = x3 = z + 1 = 0 is isomorphic to A 1 X4 = Q4 \ E2 is actually A -contractible: there is an explicit morphism 5 1 A → X4 that is Zariski locally trivial with ﬁbers A Q4 carries an explicit non-trivial rank 2 bundle (the Hopf bundle); 1 this bundle restricts non-trivially to X4, i.e., A -contractible varieties may carry non-trivial vector bundles!

If k is regular, one may show HomHmot(k)(X, BGL1) = Pic(X) for any smooth k-scheme X. Unfortunately, this fails for BGLr with r ≥ 2. Example (A., B. Doran ’08)

Set Q4 = Spec k[x1, x2, x3, x4, z]/hx1x2 − x3x4 = z(z + 1)i

Aravind Asok (USC) Counting vector bundles 1 X4 = Q4 \ E2 is actually A -contractible: there is an explicit morphism 5 1 A → X4 that is Zariski locally trivial with ﬁbers A Q4 carries an explicit non-trivial rank 2 bundle (the Hopf bundle); 1 this bundle restricts non-trivially to X4, i.e., A -contractible varieties may carry non-trivial vector bundles!

If k is regular, one may show HomHmot(k)(X, BGL1) = Pic(X) for any smooth k-scheme X. Unfortunately, this fails for BGLr with r ≥ 2. Example (A., B. Doran ’08)

Set Q4 = Spec k[x1, x2, x3, x4, z]/hx1x2 − x3x4 = z(z + 1)i 2 E2 ⊂ Q4 deﬁned by x1 = x3 = z + 1 = 0 is isomorphic to A

Aravind Asok (USC) Counting vector bundles Q4 carries an explicit non-trivial rank 2 bundle (the Hopf bundle); 1 this bundle restricts non-trivially to X4, i.e., A -contractible varieties may carry non-trivial vector bundles!

If k is regular, one may show HomHmot(k)(X, BGL1) = Pic(X) for any smooth k-scheme X. Unfortunately, this fails for BGLr with r ≥ 2. Example (A., B. Doran ’08)

Aravind Asok (USC) Counting vector bundles 1 this bundle restricts non-trivially to X4, i.e., A -contractible varieties may carry non-trivial vector bundles!

If k is regular, one may show HomHmot(k)(X, BGL1) = Pic(X) for any smooth k-scheme X. Unfortunately, this fails for BGLr with r ≥ 2. Example (A., B. Doran ’08)

Aravind Asok (USC) Counting vector bundles If k is regular, one may show HomHmot(k)(X, BGL1) = Pic(X) for any smooth k-scheme X. Unfortunately, this fails for BGLr with r ≥ 2. Example (A., B. Doran ’08)

Aravind Asok (USC) Counting vector bundles Morel ’06 if r 6= 2 and k a perfect ﬁeld Schlichting ’15 arbitrary r, k perfect; simpliﬁes part of Morel’s argument A.–M. Hoyois–M. Wendt ’15 (essentially self-contained: in essence, the theorem is equivalent to the Bass–Quillen conjecture for all smooth algebras over k)

New goal: effectively describe [Spec R, Grr]A1 .

Nevertheless: Theorem If k is a ﬁeld or Z, then for any smooth afﬁne k-scheme X = Spec R, ∼ [Spec R, Grr]naive = [Spec R, Grr]A1 −→Vr(Spec R).

Aravind Asok (USC) Counting vector bundles Schlichting ’15 arbitrary r, k perfect; simpliﬁes part of Morel’s argument A.–M. Hoyois–M. Wendt ’15 (essentially self-contained: in essence, the theorem is equivalent to the Bass–Quillen conjecture for all smooth algebras over k)

New goal: effectively describe [Spec R, Grr]A1 .

Nevertheless: Theorem If k is a ﬁeld or Z, then for any smooth afﬁne k-scheme X = Spec R, ∼ [Spec R, Grr]naive = [Spec R, Grr]A1 −→Vr(Spec R).

Morel ’06 if r 6= 2 and k a perfect ﬁeld

Aravind Asok (USC) Counting vector bundles A.–M. Hoyois–M. Wendt ’15 (essentially self-contained: in essence, the theorem is equivalent to the Bass–Quillen conjecture for all smooth algebras over k)

New goal: effectively describe [Spec R, Grr]A1 .

Nevertheless: Theorem If k is a ﬁeld or Z, then for any smooth afﬁne k-scheme X = Spec R, ∼ [Spec R, Grr]naive = [Spec R, Grr]A1 −→Vr(Spec R).

Morel ’06 if r 6= 2 and k a perfect ﬁeld Schlichting ’15 arbitrary r, k perfect; simpliﬁes part of Morel’s argument

Aravind Asok (USC) Counting vector bundles New goal: effectively describe [Spec R, Grr]A1 .

Nevertheless: Theorem If k is a ﬁeld or Z, then for any smooth afﬁne k-scheme X = Spec R, ∼ [Spec R, Grr]naive = [Spec R, Grr]A1 −→Vr(Spec R).

Morel ’06 if r 6= 2 and k a perfect ﬁeld Schlichting ’15 arbitrary r, k perfect; simpliﬁes part of Morel’s argument A.–M. Hoyois–M. Wendt ’15 (essentially self-contained: in essence, the theorem is equivalent to the Bass–Quillen conjecture for all smooth algebras over k)

Aravind Asok (USC) Counting vector bundles Nevertheless: Theorem If k is a ﬁeld or Z, then for any smooth afﬁne k-scheme X = Spec R, ∼ [Spec R, Grr]naive = [Spec R, Grr]A1 −→Vr(Spec R).

New goal: effectively describe [Spec R, Grr]A1 .

Aravind Asok (USC) Counting vector bundles If (X , x) is a pointed space, we may deﬁne A1-homotopy sheaves A1 πi (X , x). A1-Postnikov tower: given a pointed A1-connected space, we can build X inductively out of Eilenberg-Mac Lane spaces K(π, n); these have exactly 1 non-trivial A1-homotopy sheaf in degree n

We can inductively describe the set of maps [U, X ]A1 using sheaf cohomology with coefﬁcients in A1-homotopy sheaves

Aravind Asok (USC) Counting vector bundles A1-Postnikov tower: given a pointed A1-connected space, we can build X inductively out of Eilenberg-Mac Lane spaces K(π, n); these have exactly 1 non-trivial A1-homotopy sheaf in degree n

We can inductively describe the set of maps [U, X ]A1 using sheaf cohomology with coefﬁcients in A1-homotopy sheaves

Aravind Asok (USC) Counting vector bundles We can inductively describe the set of maps [U, X ]A1 using sheaf cohomology with coefﬁcients in A1-homotopy sheaves

Classical homotopy theory gives techniques for providing a “cohomological” description of homotopy classes: one factors a space into homotopically simple spaces (Eilenberg–Mac Lane spaces). F. Morel developed these ideas in algebraic geometry. If (X , x) is a pointed space, we may deﬁne A1-homotopy sheaves A1 πi (X , x). A1-Postnikov tower: given a pointed A1-connected space, we can build X inductively out of Eilenberg-Mac Lane spaces K(π, n); these have exactly 1 non-trivial A1-homotopy sheaf in degree n

Aravind Asok (USC) Counting vector bundles Classical homotopy theory gives techniques for providing a “cohomological” description of homotopy classes: one factors a space into homotopically simple spaces (Eilenberg–Mac Lane spaces). F. Morel developed these ideas in algebraic geometry. If (X , x) is a pointed space, we may deﬁne A1-homotopy sheaves A1 πi (X , x). A1-Postnikov tower: given a pointed A1-connected space, we can build X inductively out of Eilenberg-Mac Lane spaces K(π, n); these have exactly 1 non-trivial A1-homotopy sheaf in degree n

We can inductively describe the set of maps [U, X ]A1 using sheaf cohomology with coefﬁcients in A1-homotopy sheaves

Aravind Asok (USC) Counting vector bundles Example

A1 For any integer n ≥ 1, π0 (SLn) = 1 Turns out it sufﬁces to check this on sections over ﬁelds

For any ﬁeld F, any matrix in SLn(F) may be factored as a product of elementary (shearing) matrices Any elementary shearing matrix is A1-homotopic: if a ∈ F then use 1 at 0 1

1 Any matrix in SLn(F) is naively A -homotopic to the identity.

Intuition: a space should be A1-connected if points can be connected by chains of afﬁne lines

Aravind Asok (USC) Counting vector bundles Turns out it sufﬁces to check this on sections over ﬁelds

For any ﬁeld F, any matrix in SLn(F) may be factored as a product of elementary (shearing) matrices Any elementary shearing matrix is A1-homotopic: if a ∈ F then use 1 at 0 1

1 Any matrix in SLn(F) is naively A -homotopic to the identity.

Intuition: a space should be A1-connected if points can be connected by chains of afﬁne lines Example

A1 For any integer n ≥ 1, π0 (SLn) = 1

Aravind Asok (USC) Counting vector bundles For any ﬁeld F, any matrix in SLn(F) may be factored as a product of elementary (shearing) matrices Any elementary shearing matrix is A1-homotopic: if a ∈ F then use 1 at 0 1

1 Any matrix in SLn(F) is naively A -homotopic to the identity.

Intuition: a space should be A1-connected if points can be connected by chains of afﬁne lines Example

A1 For any integer n ≥ 1, π0 (SLn) = 1 Turns out it sufﬁces to check this on sections over ﬁelds

Aravind Asok (USC) Counting vector bundles Any elementary shearing matrix is A1-homotopic: if a ∈ F then use 1 at 0 1

1 Any matrix in SLn(F) is naively A -homotopic to the identity.

Intuition: a space should be A1-connected if points can be connected by chains of afﬁne lines Example

A1 For any integer n ≥ 1, π0 (SLn) = 1 Turns out it sufﬁces to check this on sections over ﬁelds

For any ﬁeld F, any matrix in SLn(F) may be factored as a product of elementary (shearing) matrices

Aravind Asok (USC) Counting vector bundles 1 Any matrix in SLn(F) is naively A -homotopic to the identity.

Intuition: a space should be A1-connected if points can be connected by chains of afﬁne lines Example

A1 For any integer n ≥ 1, π0 (SLn) = 1 Turns out it sufﬁces to check this on sections over ﬁelds

For any ﬁeld F, any matrix in SLn(F) may be factored as a product of elementary (shearing) matrices Any elementary shearing matrix is A1-homotopic: if a ∈ F then use 1 at 0 1

Aravind Asok (USC) Counting vector bundles Intuition: a space should be A1-connected if points can be connected by chains of afﬁne lines Example

A1 For any integer n ≥ 1, π0 (SLn) = 1 Turns out it sufﬁces to check this on sections over ﬁelds

For any ﬁeld F, any matrix in SLn(F) may be factored as a product of elementary (shearing) matrices Any elementary shearing matrix is A1-homotopic: if a ∈ F then use 1 at 0 1

1 Any matrix in SLn(F) is naively A -homotopic to the identity.

Aravind Asok (USC) Counting vector bundles A1 GL1 is discrete, i.e., π0 (GL1) = GL1: there are no non-constant 1 algebraic maps A → GL1 ∞ the map A \ 0 → BGL1 is a principal GL1-bundle and this yields an A1-ﬁber sequence ∞ since A \ 0 is A1-contractible, the result follows from the long exact sequence in homotopy

Example

A1 A1 For any n ≥ 2, π1 (BSLn) = 1; one identiﬁes π1 (BSLn) = π0(SLn) using a ﬁber sequence.

Example

For any n ≥ 2, the map BGLn → BGL1 coming from det : GLn → GL1 A1 induces an isomorphism π1 (BGLn) = GL1.

Example

A1 π1 (BGL1) = GL1

Aravind Asok (USC) Counting vector bundles ∞ the map A \ 0 → BGL1 is a principal GL1-bundle and this yields an A1-ﬁber sequence ∞ since A \ 0 is A1-contractible, the result follows from the long exact sequence in homotopy

Example

A1 A1 For any n ≥ 2, π1 (BSLn) = 1; one identiﬁes π1 (BSLn) = π0(SLn) using a ﬁber sequence.

Example

For any n ≥ 2, the map BGLn → BGL1 coming from det : GLn → GL1 A1 induces an isomorphism π1 (BGLn) = GL1.

Example

A1 π1 (BGL1) = GL1 A1 GL1 is discrete, i.e., π0 (GL1) = GL1: there are no non-constant 1 algebraic maps A → GL1

Aravind Asok (USC) Counting vector bundles ∞ since A \ 0 is A1-contractible, the result follows from the long exact sequence in homotopy

Example

A1 A1 For any n ≥ 2, π1 (BSLn) = 1; one identiﬁes π1 (BSLn) = π0(SLn) using a ﬁber sequence.

Example

For any n ≥ 2, the map BGLn → BGL1 coming from det : GLn → GL1 A1 induces an isomorphism π1 (BGLn) = GL1.

Example

Aravind Asok (USC) Counting vector bundles Example

A1 A1 For any n ≥ 2, π1 (BSLn) = 1; one identiﬁes π1 (BSLn) = π0(SLn) using a ﬁber sequence.

Example

For any n ≥ 2, the map BGLn → BGL1 coming from det : GLn → GL1 A1 induces an isomorphism π1 (BGLn) = GL1.

Example

A1 π1 (BGL1) = GL1 A1 GL1 is discrete, i.e., π0 (GL1) = GL1: there are no non-constant 1 algebraic maps A → GL1 ∞ the map A \ 0 → BGL1 is a principal GL1-bundle and this yields an A1-ﬁber sequence ∞ since A \ 0 is A1-contractible, the result follows from the long exact sequence in homotopy

Aravind Asok (USC) Counting vector bundles Example

For any n ≥ 2, the map BGLn → BGL1 coming from det : GLn → GL1 A1 induces an isomorphism π1 (BGLn) = GL1.

Example

A1 A1 For any n ≥ 2, π1 (BSLn) = 1; one identiﬁes π1 (BSLn) = π0(SLn) using a ﬁber sequence.

Aravind Asok (USC) Counting vector bundles Example

Example

A1 A1 For any n ≥ 2, π1 (BSLn) = 1; one identiﬁes π1 (BSLn) = π0(SLn) using a ﬁber sequence.

Example

For any n ≥ 2, the map BGLn → BGL1 coming from det : GLn → GL1 A1 induces an isomorphism π1 (BGLn) = GL1.

Aravind Asok (USC) Counting vector bundles M K2 is the second Milnor K-theory sheaf 1 A the map BSLn → BGL∞ induces an isomorphism on π2 (−) for n ≥ 3 and the latter represents Quillen’s algebraic K-theory 1 M A K2 = π1 (SLn), n ≥ 3 and can be thought of as “non-trivial relations among elementary matrices” (classic presentation of Milnor K2) MW K2 is the second Milnor–Witt K-theory sheaf 1 A SL2 = Sp2 and the map BSL2 → BSp∞ is an isomorphism on π2 (−) the latter represents symplectic K-theory and includes information about symplectic forms over our base MW M the map BSp∞ → BGL∞ yields a map K2 → K2 ; this map is an epimorphism of sheaves and its kernel may be described via the “fundamental ideal” in the Witt ring (A. Suslin)

Example (F. Morel) There are isomorphisms

( MW 1 K if n = 2 πA (BSL )−→∼ 2 2 n M K2 if n ≥ 3.

Aravind Asok (USC) Counting vector bundles 1 A the map BSLn → BGL∞ induces an isomorphism on π2 (−) for n ≥ 3 and the latter represents Quillen’s algebraic K-theory 1 M A K2 = π1 (SLn), n ≥ 3 and can be thought of as “non-trivial relations among elementary matrices” (classic presentation of Milnor K2) MW K2 is the second Milnor–Witt K-theory sheaf 1 A SL2 = Sp2 and the map BSL2 → BSp∞ is an isomorphism on π2 (−) the latter represents symplectic K-theory and includes information about symplectic forms over our base MW M the map BSp∞ → BGL∞ yields a map K2 → K2 ; this map is an epimorphism of sheaves and its kernel may be described via the “fundamental ideal” in the Witt ring (A. Suslin)

Example (F. Morel) There are isomorphisms

( MW 1 K if n = 2 πA (BSL )−→∼ 2 2 n M K2 if n ≥ 3.

M K2 is the second Milnor K-theory sheaf

Aravind Asok (USC) Counting vector bundles 1 M A K2 = π1 (SLn), n ≥ 3 and can be thought of as “non-trivial relations among elementary matrices” (classic presentation of Milnor K2) MW K2 is the second Milnor–Witt K-theory sheaf 1 A SL2 = Sp2 and the map BSL2 → BSp∞ is an isomorphism on π2 (−) the latter represents symplectic K-theory and includes information about symplectic forms over our base MW M the map BSp∞ → BGL∞ yields a map K2 → K2 ; this map is an epimorphism of sheaves and its kernel may be described via the “fundamental ideal” in the Witt ring (A. Suslin)

Example (F. Morel) There are isomorphisms

( MW 1 K if n = 2 πA (BSL )−→∼ 2 2 n M K2 if n ≥ 3.

M K2 is the second Milnor K-theory sheaf 1 A the map BSLn → BGL∞ induces an isomorphism on π2 (−) for n ≥ 3 and the latter represents Quillen’s algebraic K-theory

Aravind Asok (USC) Counting vector bundles MW K2 is the second Milnor–Witt K-theory sheaf 1 A SL2 = Sp2 and the map BSL2 → BSp∞ is an isomorphism on π2 (−) the latter represents symplectic K-theory and includes information about symplectic forms over our base MW M the map BSp∞ → BGL∞ yields a map K2 → K2 ; this map is an epimorphism of sheaves and its kernel may be described via the “fundamental ideal” in the Witt ring (A. Suslin)

Example (F. Morel) There are isomorphisms

( MW 1 K if n = 2 πA (BSL )−→∼ 2 2 n M K2 if n ≥ 3.

Aravind Asok (USC) Counting vector bundles 1 A SL2 = Sp2 and the map BSL2 → BSp∞ is an isomorphism on π2 (−) the latter represents symplectic K-theory and includes information about symplectic forms over our base MW M the map BSp∞ → BGL∞ yields a map K2 → K2 ; this map is an epimorphism of sheaves and its kernel may be described via the “fundamental ideal” in the Witt ring (A. Suslin)

Example (F. Morel) There are isomorphisms

( MW 1 K if n = 2 πA (BSL )−→∼ 2 2 n M K2 if n ≥ 3.

Aravind Asok (USC) Counting vector bundles the latter represents symplectic K-theory and includes information about symplectic forms over our base MW M the map BSp∞ → BGL∞ yields a map K2 → K2 ; this map is an epimorphism of sheaves and its kernel may be described via the “fundamental ideal” in the Witt ring (A. Suslin)

Example (F. Morel) There are isomorphisms

( MW 1 K if n = 2 πA (BSL )−→∼ 2 2 n M K2 if n ≥ 3.

Aravind Asok (USC) Counting vector bundles MW M the map BSp∞ → BGL∞ yields a map K2 → K2 ; this map is an epimorphism of sheaves and its kernel may be described via the “fundamental ideal” in the Witt ring (A. Suslin)

Example (F. Morel) There are isomorphisms

( MW 1 K if n = 2 πA (BSL )−→∼ 2 2 n M K2 if n ≥ 3.

M K2 is the second Milnor K-theory sheaf 1 A the map BSLn → BGL∞ induces an isomorphism on π2 (−) for n ≥ 3 and the latter represents Quillen’s algebraic K-theory 1 M A K2 = π1 (SLn), n ≥ 3 and can be thought of as “non-trivial relations among elementary matrices” (classic presentation of Milnor K2) MW K2 is the second Milnor–Witt K-theory sheaf 1 A SL2 = Sp2 and the map BSL2 → BSp∞ is an isomorphism on π2 (−) the latter represents symplectic K-theory and includes information about symplectic forms over our base

Aravind Asok (USC) Counting vector bundles Example (F. Morel) There are isomorphisms

( MW 1 K if n = 2 πA (BSL )−→∼ 2 2 n M K2 if n ≥ 3.

Aravind Asok (USC) Counting vector bundles Counting vector bundles with motivic homotopy theory (based on joint work with J. Fasel, M. Hopkins)

Aravind Asok (USC) Counting vector bundles 1 No, not even in dimension 2. Indeed, Pic(Q2) = Pic(P ) = Z, which is not ﬁnite. This problem persists in higher rank, since we may take direct sums of line bundles. Fix the determinant, a.k.a., the ﬁrst Chern class.

Restrict attention to smooth affine varieties over finite fields. Question Are there finitely many vector bundles of a given rank on a smooth affine variety over a finite field?

Aravind Asok (USC) Counting vector bundles This problem persists in higher rank, since we may take direct sums of line bundles. Fix the determinant, a.k.a., the ﬁrst Chern class.

Restrict attention to smooth affine varieties over finite fields. Question Are there finitely many vector bundles of a given rank on a smooth affine variety over a finite field?

1 No, not even in dimension 2. Indeed, Pic(Q2) = Pic(P ) = Z, which is not ﬁnite.

Aravind Asok (USC) Counting vector bundles Fix the determinant, a.k.a., the ﬁrst Chern class.

Restrict attention to smooth affine varieties over finite fields. Question Are there finitely many vector bundles of a given rank on a smooth affine variety over a finite field?

1 No, not even in dimension 2. Indeed, Pic(Q2) = Pic(P ) = Z, which is not ﬁnite. This problem persists in higher rank, since we may take direct sums of line bundles.

Aravind Asok (USC) Counting vector bundles Restrict attention to smooth affine varieties over finite fields. Question Are there finitely many vector bundles of a given rank on a smooth affine variety over a finite field?

1 No, not even in dimension 2. Indeed, Pic(Q2) = Pic(P ) = Z, which is not ﬁnite. This problem persists in higher rank, since we may take direct sums of line bundles. Fix the determinant, a.k.a., the ﬁrst Chern class.

Restrict attention to smooth affine varieties over finite fields. Question Are there finitely many vector bundles of a given rank with fixed determinant on a smooth affine variety over a finite field?

Aravind Asok (USC) Counting vector bundles Proof (trivial determinant case). Serre’s splitting theorem =⇒ suffices to prove finiteness in rank 2 2 MW Since X has dimension 2, we can identify [X, BSL2]A1 = H (X, K2 ) 2 MW 2 M ∼ The canonical map H (X, K2 ) → H (X, K2 ) is surjective, and an = since we work over a finite field (uses Merkurjev–Suslin theorem) 2 M When working over a finite field, H (X, K2 ) is finite by higher-dimensional class field theory (Kato–Saito)

Theorem (Bloch, Mohan Kumar–Murthy-Roy, Parshin) If X is a smooth affine surface over a finite field, then there are finitely many isomorphism classes of vector bundles with a given rank and determinant.

Aravind Asok (USC) Counting vector bundles 2 MW Since X has dimension 2, we can identify [X, BSL2]A1 = H (X, K2 ) 2 MW 2 M ∼ The canonical map H (X, K2 ) → H (X, K2 ) is surjective, and an = since we work over a finite field (uses Merkurjev–Suslin theorem) 2 M When working over a finite field, H (X, K2 ) is finite by higher-dimensional class field theory (Kato–Saito)

Proof (trivial determinant case). Serre’s splitting theorem =⇒ sufﬁces to prove ﬁniteness in rank 2

Aravind Asok (USC) Counting vector bundles 2 MW 2 M ∼ The canonical map H (X, K2 ) → H (X, K2 ) is surjective, and an = since we work over a finite field (uses Merkurjev–Suslin theorem) 2 M When working over a finite field, H (X, K2 ) is finite by higher-dimensional class field theory (Kato–Saito)

Proof (trivial determinant case). Serre’s splitting theorem =⇒ sufﬁces to prove ﬁniteness in rank 2 2 MW Since X has dimension 2, we can identify [X, BSL2]A1 = H (X, K2 )

Aravind Asok (USC) Counting vector bundles 2 M When working over a finite field, H (X, K2 ) is finite by higher-dimensional class field theory (Kato–Saito)

Proof (trivial determinant case). Serre’s splitting theorem =⇒ suffices to prove finiteness in rank 2 2 MW Since X has dimension 2, we can identify [X, BSL2]A1 = H (X, K2 ) 2 MW 2 M ∼ The canonical map H (X, K2 ) → H (X, K2 ) is surjective, and an = since we work over a finite field (uses Merkurjev–Suslin theorem)

Aravind Asok (USC) Counting vector bundles Theorem (Bloch, Mohan Kumar–Murthy-Roy, Parshin) If X is a smooth affine surface over a finite field, then there are finitely many isomorphism classes of vector bundles with a given rank and determinant.

Aravind Asok (USC) Counting vector bundles We may define higher Chern classes in Chow theory as in topology: the Chow ring of the Grr may be computed to be a polynomial ring on generators c1,..., cr 2 M 2 The class in H (X, K2 ) = CH (X) (Kato’s formula) described above is precisely the second Chern class of the vector bundle The argument actually shows that there are precisely |CH2(X)| vector bundles with a fixed rank and determinant The result actually holds for a “regular affine arithmetic surface” (without using any A1-homotopy theory), but the A1-homotopy theoretic argument generalizes.

Remark The determinant of a vector bundle is a class in Pic(X); this is the ﬁrst Chern class in Chow-theory

Aravind Asok (USC) Counting vector bundles 2 M 2 The class in H (X, K2 ) = CH (X) (Kato’s formula) described above is precisely the second Chern class of the vector bundle The argument actually shows that there are precisely |CH2(X)| vector bundles with a ﬁxed rank and determinant The result actually holds for a “regular afﬁne arithmetic surface” (without using any A1-homotopy theory), but the A1-homotopy theoretic argument generalizes.

Aravind Asok (USC) Counting vector bundles The argument actually shows that there are precisely |CH2(X)| vector bundles with a ﬁxed rank and determinant The result actually holds for a “regular afﬁne arithmetic surface” (without using any A1-homotopy theory), but the A1-homotopy theoretic argument generalizes.

Remark The determinant of a vector bundle is a class in Pic(X); this is the ﬁrst Chern class in Chow-theory We may deﬁne higher Chern classes in Chow theory as in topology: the Chow ring of the Grr may be computed to be a polynomial ring on generators c1,..., cr 2 M 2 The class in H (X, K2 ) = CH (X) (Kato’s formula) described above is precisely the second Chern class of the vector bundle

Aravind Asok (USC) Counting vector bundles The result actually holds for a “regular afﬁne arithmetic surface” (without using any A1-homotopy theory), but the A1-homotopy theoretic argument generalizes.

Aravind Asok (USC) Counting vector bundles Remark The determinant of a vector bundle is a class in Pic(X); this is the first Chern class in Chow-theory We may define higher Chern classes in Chow theory as in topology: the Chow ring of the Grr may be computed to be a polynomial ring on generators c1,..., cr 2 M 2 The class in H (X, K2 ) = CH (X) (Kato’s formula) described above is precisely the second Chern class of the vector bundle The argument actually shows that there are precisely |CH2(X)| vector bundles with a fixed rank and determinant The result actually holds for a “regular affine arithmetic surface” (without using any A1-homotopy theory), but the A1-homotopy theoretic argument generalizes.

Aravind Asok (USC) Counting vector bundles In particular, there are ﬁnitely many isomorphism classes of vector bundles with given rank and determinant if and only if CH2(X) is ﬁnite.

Proof. Sufﬁces to establish the result for ranks 2 and 3 by Serre’s splitting A1 Working up the Postnikov tower requires computation of π3 (GL2) and π3(BGL3) (A.-Fasel) Then, use known ﬁniteness results for certain motivic cohomology groups (Kato–Saito, Kerz–Saito)

Aravind Asok (USC) Counting vector bundles Proof. Sufﬁces to establish the result for ranks 2 and 3 by Serre’s splitting A1 Working up the Postnikov tower requires computation of π3 (GL2) and π3(BGL3) (A.-Fasel) Then, use known ﬁniteness results for certain motivic cohomology groups (Kato–Saito, Kerz–Saito)

Theorem If F is a finite field, characteristic unequal to 2, and X is a smooth affine 3-fold over F, then there are finitely many isomorphism classes of vector i bundles with given ci ∈ CH (X), i = 1, 2. In particular, there are finitely many isomorphism classes of vector bundles with given rank and determinant if and only if CH2(X) is finite.

Aravind Asok (USC) Counting vector bundles A1 Working up the Postnikov tower requires computation of π3 (GL2) and π3(BGL3) (A.-Fasel) Then, use known ﬁniteness results for certain motivic cohomology groups (Kato–Saito, Kerz–Saito)

Proof. Sufﬁces to establish the result for ranks 2 and 3 by Serre’s splitting

Aravind Asok (USC) Counting vector bundles Then, use known ﬁniteness results for certain motivic cohomology groups (Kato–Saito, Kerz–Saito)

Proof. Sufﬁces to establish the result for ranks 2 and 3 by Serre’s splitting A1 Working up the Postnikov tower requires computation of π3 (GL2) and π3(BGL3) (A.-Fasel)

Aravind Asok (USC) Counting vector bundles Theorem If F is a finite field, characteristic unequal to 2, and X is a smooth affine 3-fold over F, then there are finitely many isomorphism classes of vector i bundles with given ci ∈ CH (X), i = 1, 2. In particular, there are finitely many isomorphism classes of vector bundles with given rank and determinant if and only if CH2(X) is finite.

Aravind Asok (USC) Counting vector bundles A variant of the Beilinson–Tate conjecture implies that CH2(X) is always ﬁnite under the above hypotheses.

Conjecture If X is a smooth affine threefold over a finite field, then there are always finitely many isomorphism classes of vector bundles with a given rank and determinant.

One may check CH2(X) is ﬁnite in many examples.

Aravind Asok (USC) Counting vector bundles Conjecture If X is a smooth affine threefold over a finite field, then there are always finitely many isomorphism classes of vector bundles with a given rank and determinant.

One may check CH2(X) is ﬁnite in many examples. A variant of the Beilinson–Tate conjecture implies that CH2(X) is always ﬁnite under the above hypotheses.

Aravind Asok (USC) Counting vector bundles One may check CH2(X) is ﬁnite in many examples. A variant of the Beilinson–Tate conjecture implies that CH2(X) is always ﬁnite under the above hypotheses.

Conjecture If X is a smooth affine threefold over a finite field, then there are always finitely many isomorphism classes of vector bundles with a given rank and determinant.

Aravind Asok (USC) Counting vector bundles There are smooth affine 4-folds over a finite field that have infinitely many isomorphism classes of rank 2 vector bundles with fixed rank and determinant (e.g., take the complement of the incidence divisor in P2 × P2). Similar counterexamples suggest that the following is the best-possible statement. Conjecture If X is a smooth affine variety of dimension d over a finite field, then there are finitely many isomorphism classes of vector bundles with fixed Chern clases i d ci ∈ CH (X), 1 ≤ i ≤ b 2 c.

What should we expect in higher dimensions?

Aravind Asok (USC) Counting vector bundles Similar counterexamples suggest that the following is the best-possible statement. Conjecture If X is a smooth affine variety of dimension d over a finite field, then there are finitely many isomorphism classes of vector bundles with fixed Chern clases i d ci ∈ CH (X), 1 ≤ i ≤ b 2 c.

What should we expect in higher dimensions? There are smooth affine 4-folds over a finite field that have infinitely many isomorphism classes of rank 2 vector bundles with fixed rank and determinant (e.g., take the complement of the incidence divisor in P2 × P2).

Aravind Asok (USC) Counting vector bundles Conjecture If X is a smooth affine variety of dimension d over a finite field, then there are finitely many isomorphism classes of vector bundles with fixed Chern clases i d ci ∈ CH (X), 1 ≤ i ≤ b 2 c.

Aravind Asok (USC) Counting vector bundles What should we expect in higher dimensions? There are smooth affine 4-folds over a finite field that have infinitely many isomorphism classes of rank 2 vector bundles with fixed rank and determinant (e.g., take the complement of the incidence divisor in P2 × P2). Similar counterexamples suggest that the following is the best-possible statement. Conjecture If X is a smooth affine variety of dimension d over a finite field, then there are finitely many isomorphism classes of vector bundles with fixed Chern clases i d ci ∈ CH (X), 1 ≤ i ≤ b 2 c.

Aravind Asok (USC) Counting vector bundles Jannsen’s version of Beilinson–Tate conjecture, resolution of singularities in positive characteristic and the motivic Bass conjecture on finite generation of motivic cohomology guarantee that CHi(X) is finite d for i > b 2 c, together with finiteness of a host of other motivic cohomology groups Thus, the conjecture follows if we know that we can always express

maps into [X, BGLn]A1 purely in terms of motivic cohomology The latter follows from Hopkins’ “Wilson splitting hypothesis”; loosely the classifying space for algebraic cobordism is “even”; this guarantees that we may write nice “resolutions” of BGLn

Why should we believe this? Panglossian optimism:

Aravind Asok (USC) Counting vector bundles Thus, the conjecture follows if we know that we can always express

Why should we believe this? Panglossian optimism: Jannsen’s version of Beilinson–Tate conjecture, resolution of singularities in positive characteristic and the motivic Bass conjecture on finite generation of motivic cohomology guarantee that CHi(X) is finite d for i > b 2 c, together with finiteness of a host of other motivic cohomology groups

Aravind Asok (USC) Counting vector bundles The latter follows from Hopkins’ “Wilson splitting hypothesis”; loosely the classifying space for algebraic cobordism is “even”; this guarantees that we may write nice “resolutions” of BGLn

maps into [X, BGLn]A1 purely in terms of motivic cohomology

Aravind Asok (USC) Counting vector bundles Why should we believe this? Panglossian optimism: Jannsen’s version of Beilinson–Tate conjecture, resolution of singularities in positive characteristic and the motivic Bass conjecture on finite generation of motivic cohomology guarantee that CHi(X) is finite d for i > b 2 c, together with finiteness of a host of other motivic cohomology groups Thus, the conjecture follows if we know that we can always express

Aravind Asok (USC) Counting vector bundles Question If X is a smooth affine threefold over a finite field, then how many vector bundles are there with a given rank and determinant?

This number may depend on the determinant. What do these numbers mean, what do they measure? We might think of them as some higher rank/higher dimensional version of the class number What happens for general regular rings of Krull dimension d ≥ 3 that are ﬁnitely generated as Z-algebras?

Given ﬁniteness, we may actually count vector bundles. For concreteness:

Aravind Asok (USC) Counting vector bundles This number may depend on the determinant. What do these numbers mean, what do they measure? We might think of them as some higher rank/higher dimensional version of the class number What happens for general regular rings of Krull dimension d ≥ 3 that are ﬁnitely generated as Z-algebras?

Given finiteness, we may actually count vector bundles. For concreteness: Question If X is a smooth affine threefold over a finite field, then how many vector bundles are there with a given rank and determinant?

Aravind Asok (USC) Counting vector bundles What do these numbers mean, what do they measure? We might think of them as some higher rank/higher dimensional version of the class number What happens for general regular rings of Krull dimension d ≥ 3 that are ﬁnitely generated as Z-algebras?

This number may depend on the determinant.

Aravind Asok (USC) Counting vector bundles What happens for general regular rings of Krull dimension d ≥ 3 that are ﬁnitely generated as Z-algebras?

This number may depend on the determinant. What do these numbers mean, what do they measure? We might think of them as some higher rank/higher dimensional version of the class number

Aravind Asok (USC) Counting vector bundles Given finiteness, we may actually count vector bundles. For concreteness: Question If X is a smooth affine threefold over a finite field, then how many vector bundles are there with a given rank and determinant?

Aravind Asok (USC) Counting vector bundles Thank you!

Aravind Asok (USC) Counting vector bundles