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Home , Andrei Suslin, Chow group, Ivan Fesenko, Milnor conjecture, Motivic cohomology

AN OVERVIEW OF MOTIVIC

PETER J. HAINE

Abstract. In this talk we give an overview of some of the motivations behind and applica- tions of . We first present a 퐾-theoretic timeline converging on motivic cohomology, then briefly discuss algebraic 퐾-theory and its more concrete cousin Milnor 퐾-theory. We then present a geometric definition of motivic cohomology via Bloch’s higher Chow groups and outline the main features of motivic cohomology, namely, its relation to Milnor 퐾-theory. In the last part of the talk we discuss the role of motivic cohomology in Voevodsky’s proof of the Bloch–Kato conjecture. The statement of the Bloch–Kato conjec- ture is elementary and predates motivic cohomolgy, but its proof heavily relies on motivic cohomology and motivic homotopy theory.

Contents 1. A Timeline Converging to Motivic Cohomology 1 2. A Taste of Algebraic 퐾-Theory 3 3. Milnor 퐾-theory 4 4. Bloch’s Higher Chow Groups 4 5. Properties of Motivic Cohomology 6 6. The Bloch–Kato Conjecture 7 6.1. The Kummer Sequence 7 References 10

1. A Timeline Converging to Motivic Cohomology In this section we give a timeline of events converging on motivic cohomology. We first say a few words about what motivic cohomology is. 푝 op • Motivic cohomology is a bigraded cohomology theory H (−; 퐙(푞))∶ Sm/푘 → Ab. • Voevodsky won the because “he defined and developed motivic coho- mology and the 퐀1-homotopy theory of algebraic varieties; he proved the Milnor conjectures on the 퐾-theory of fields.” Moreover, Voevodsky’s proof of the makes extensive use of motivic cohomology. • Later Voevodsky went on to prove the (Milnor 퐾-theory) Bloch–Kato conjecture using motivic homotopy theory. • There are many applications of motivic cohomology, even in classical settings. For example, motivic cohomology has been used by Asok–Fasel [1] to classify vector bundles. • Motivic cohomology is relatively new, so there is a lot of work to be done in this and there are many conjectures.

Date: September 10, 2017. Acknowledgments: I am indebted to Marc Hoyois’ wonderful talk [13]. 1 2 PETER J. HAINE

1.1. Timeline (퐾-theoretic timeline converging to motivic cohomology [13, 28]). There were various motivations for and timelines converging to motivic cohomology. We present only one of them to remain concise. 1957: Grothendieck defines algebraic 퐾-theory to prove the Grotendieck–Riemann– Roch theorem. One of Grothendieck’s motivations for defining 퐾-theory was his dislike for the Chow groups and 1.2. Definition. Let 푘 be a field and 푋 be a smooth 푘-variety†. The 0th algebraic 퐾- 퐾0(푋) of 푋 is the quotient of the on the set of algebraic vector bundles on 푋 by the relation [푉] = [푉′] + [푉″] if there is a short of algebraic vector bundles on 푋 0 푉′ 푉 푉″ 0 .

Note that the of algebraic vector bundles endows 퐾0(푋) with the structure of a commutative . 1.3. Remarks.

(1.3.a) Historically 퐾0 was simply written 퐾 and 퐾0(푋) was called the algebraic 퐾-theory of 푋. (1.3.b) The letter “K” stands for the German word for “class”. 1.4. Example. If 푘 is a field, then the category of algebraic vector bundles on 푘 is equivalent to the category Vect푘 of finite-dimensional 푘-vector spaces. Since every short exact sequence of vector bundles splits, if 푉 is an 푛-dimensional 푘-vector space, we see (by induction) that [푉] = 푛[푘]. From this observation it easily follows that 퐾0(Spec(푘)) ≅ 퐙. A consequence of the Grothendieck–Riemann–Roch Theorem is the following. 1.5. Corollary. Let 푋 be a smooth quasiprojective variety. Then the reational Chern character ∗ ch ⊗ 퐐∶ 퐾0(푋) ⊗퐙 퐐 → CH (푋) ⊗퐙 퐐 is an isomorphism of rings, where CH∗(푋) denotes the Chow ring of 푋 (see Definition 4.4). 0 1959: • Atiyah and Hirzebruch define complex topological 퐾-theory 퐾top(푋) for a space 푋. • Bott proves Bott periodicity (i.e., that the homotopy groups of the infinite uni- tary group are 2-periodic, or that 훺2푈 ≃ 푈, or that 훺2퐵푈 ≃ 퐙 × 퐵푈). 0 • Combining their definition of 퐾top with Bott’s periodicity theorem, Atiyah and 0 Hirzebruch extended 퐾top to a generalized cohomology theory 퐾top. This co- ∗ ±1 2 theory has the property that 퐾top(∗) ≅ 퐙[훽 ], where 훽 ∈ 퐾top(∗) is an element called the Bott element. • One immediately gets the Atiyah–Hirzebruch ∗ ∗ ∗ Hsing(푋; 퐾top(∗)) ⟹ 퐾top(푋) , an extremely powerful tool for computing topological 퐾-theory.

1.6. Question. What about in ? We have only defined 퐾0 so far.

1964: Bass defined 퐾1 for rings using matrices. Later some generalizations were made to the settings of symmetric monoidal or abelian categories. 1967: Milnor defines 퐾2 for rings, again using matrices.

†This assumption is a technical one used to insure that this definition agrees with a less naïve definition using perfect complexes (see [25, §2]). We could also require 푋 to be a quasiprojective over a ring. AN OVERVIEW OF MOTIVIC COHOMOLOGY 3

1971: Quillen gives definition of 퐾푛(푅) for all 푛 ≥ 0 and rings 푅 using homotopy theory that agrees with the previous definitions of 퐾0, 퐾1, and 퐾2. 1972: Quillen computes 퐾∗(퐅푞) [21]; this is essentially the only “complete” 퐾-theory computation. 1973: Quillen defines 퐾푛 for all 푛 ≥ 0 for schemes, and much more generally [22]. Again this requires homotopy theory.

1.7. Question. Now what about an Atiyah–Hirzebruch spectral sequence ? ⟹ 퐾∗(푋) in algebraic 퐾-theory? • First, for this to make sense, we remark that the algebraic 퐾-theory of a scheme has a natural filtration, called the 훾-filtration. • People called whatever should be on the 퐸2-page of an algebraic Atiyah–Hirzebruch spectral sequence converging to algebraic 퐾-theory motivic cohomology. 푝 op • Thus motivic cohomology groups should be 퐻 (−; 퐴(푞))∶ Sm/푘 → Ab, defined for 푝, 푞 ∈ 퐙 and 퐴 ∈ Ab. 1980s: Beilinson, Lichtenbaum, and others make bold conjectures about motivic co- homology and its relation to . 1986: Bloch defines motivic cohomology in the guise of higher Chow groups [4], but could neither prove all of Beilinson and Lichtenbaum’s conjectures nor produce an Atiyah–Hirzebruch spectral sequence. Suslin later found a crucial mistake in Bloch’s paper, which Bloch later fixed in 1994 [5]. ∼1999: Bloch, Deligne, Friedland, Lichtenbaum, and Deligne construct the Atiyah– Hirzebruch spectral sequence (see [9, Thm. 13.6]). Early 2000s: Voevodsky completed the proofs of the main bold conjectures that Beilin- son and Lichtenbaum made in the 1980s.

2. A Taste of Algebraic 퐾-Theory 2.1. Idea. “Algebraic 퐾-theory is the universal invariant of categories equipped with a suit- able notion of exact sequences that splits short exact sequences.” For precise universal properties of algebraic 퐾-theory, see [2, Props. 10.2 & 10.8; 7, Thm. 1.3].

2.2. Example. The category Fin∗ of finite pointed sets has a reasonable notion of exact se- th quences, and 퐾푛(Fin∗) is the 푛 stable homotopy group of spheres. op For our purposes, algebraic 퐾-theory is a collection of functors 퐾푛 ∶ Sch → Ab for 푛 ≥ 0 satisfying a number of properties. We are mostly interested in the affine case, and can restrict to 퐾푛 ∶ CRing → Ab. The following property suggests that algebraic 퐾-theory should have something to do with the homotopy theory of schemes. 2.3. Property (퐀1-homotopy invariance [29, Ch. v Thm. 6.3]). If 푅 is a regular ring, then the inclusion 푅 ↪ 푅[푡] induces an isomorphism 퐾∗(푅) ⥲ 퐾∗(푅[푡]). Algebraic 퐾-theory has numerous deep relations to algebraic geometry and number the- ory. The Kummer–Vandiver conjecture gives a taste of the deep relationship between alge- braic 퐾-theory and number theory. 2.4. Conjecture (Kummer–Vandiver). For all primes 푝, the prime 푝 does not divide the class th number of the maximal real subfield of the 푝 cyclotomic field 퐐(휁푝).

2.5. Proposition (Kurihara [16]). We have that 퐾4푛(퐙) = 0 for all positive 푛 if and only if the Kummer–Vandiver Conjecture 2.4 holds. 4 PETER J. HAINE

2.6. Remark. The 퐾-theory of 퐙 is hair-raisingly difficult to compute — a summary of many of the known computations can be found in [29, Ch. 6 §10].

3. Milnor 퐾-theory The following is a classical computation of low 퐾-groups for fields, done before Quillen defined 퐾푛 for 푛 ≥ 3. 3.1. Computation ([11, p. 1]). If 푘 is a field, then we have 퐾 (푘) ≅ 퐙 { 0 퐾 (푘) ≅ 푘× { 1 × × × {퐾2(푘) ≅ (푘 ⊗퐙 푘 )/⟨푎 ⊗ (1 − 푎) | 푎, 1 − 푎 ∈ 푘 ⟩ . 3.2. Question (Milnor, before higher 퐾-theory was defined). What if the relations appear- ing in Computation 3.1 are the only relations in 퐾-theory?

⊗푛 3.3. Notation. For an abelian group 퐺, write 푇(퐺) for the tensor algebra 푇(퐺) = ⨁푛≥0 퐺 of 퐺 (as a 퐙-module). 3.4. Definition (Milnor 1970 [20, §1]). The Milnor 퐾-theory of a commutative ring 퐴 is the graded abelian group 푀 × × K∗ (퐴) ≔ 푇(퐴 )/⟨푎 ⊗ (1 − 푎) | 푎, 1 − 푎 ∈ 퐴 ⟩ , where we write ⟨푎 ⊗ (1 − 푎) | 푎, 1 − 푎 ∈ 퐴×⟩ for the homogeneous ideal of 푇(퐴×) generated by 푎 ⊗ (1 − 푎), where 푎, 1 − 푎 ∈ 퐴×. One of the main results of involves Milnor 퐾-theory. 3.5. Theorem ([15]). Let 푘 be an 푛-dimensional local field. Then there exists a canonical ho- momorphism 푀 ab K푛 (푘) → Gal(푘ab/푘) ≅ Gal(푘sep/푘) 푀 푀 inducing an isomorphism K푛 (푘)/푁퐿/푘(K푛 (퐿)) ⥲ Gal(퐿/푘) for every finite abelian extension 푀 푀 푘 ⊂ 퐿. Here 푁퐿/푘 ∶ K푛 (퐿) → K푛 (푘) is a norm map on Milnor 퐾-theory. 3.6. Remark. Though Milnor 퐾-theory may seem somewhat ad hoc, there are good reasons to care about Milnor 퐾-theory. From the perspective of a number theorist, Milnor 퐾-theory is crucial in Kato’s higher class field theory [8, 15].

4. Bloch’s Higher Chow Groups Now that we have said a few things about 퐾-theory, we need to introduce the other key player — motivic cohomology. As motivation, we first remark that on the following topolog- ical generalization of due to Atiyah and Hirzebruch. 4.1. Theorem (Atiyah–Hirzebruch). Let 푋 be a space with the homotopy type of a finite CW-complex. Then there exists a topological Chern character giving a natural isomorphism of graded rings −푛 2푞−푛 chtop ∶ ⨁ 퐾top(푋) ⊗퐙 퐐 ⥲ ⨁ Hsing (푋; 퐐) . 푛∈퐙 푛,푞∈퐙 We now recall the definition of Chow groups from algebraic geometry. AN OVERVIEW OF MOTIVIC COHOMOLOGY 5

4.2. Definition. Let 푋 be a variety. The group of cycles on 푋 is the free abelian group 푍∗(푋) ≔ 퐙⟨closed subvarieties 푌 ⊂ 푋⟩ , which is naturally a graded group, graded by . Elements of 푍∗(푋) are called cycles on 푋. 4.3. Idea. We would like for 푍∗(푋) to be a under intersection, but two cycles may not intersect in the correct codimension: in general ′ ′ codim푋(푌 ∩ 푌 ) ≠ codim푋(푌) + codim푋(푌 ) . 4.4. Definition. Let 푋 be a 푘-variety. A rational equivalence between cycles 훼, 훽 ∈ 푍∗(푋) ∗ 1 is a cycle 훾 ∈ 푍 (푋 × 퐀푘) such that 훾 ∩ (푋 × {0}) = 훼 and 훾 ∩ (푋 × {1}) = 훽. The of 푋 is the graded group ∗ ∗ CH (푋) ≔ 푍 (푋)/∼rat of cycles modulo rational equivalence. 4.5. Remark. The Chow group CH∗(푋) is a ring under intersection of rational equivalence classes of cycles. To prove this, one has to prove a moving lemma, showing that given a pair ′ of cycles 훼 and 훽, there is a rationally equivalent cycle 훽 ∼rat 훽 so that the terms in 훼 and 훽 intersect in the correct , i.e., we can “move” cycles to rationally equivalent ones that intersect properly. 4.6. Idea (of higher Chow groups). In homotopy theory we have the following general prin- ciple

“Quotienting by an equivalence relation ⟺ Taking 휋0 of some space” (4.6.a) If 훼, 훽 ∈ 푍∗(푋) and 훾 is a rational equivalence from 훼 to 훽, we should think “훾 is a path from 훼 to 훽.” (4.6.b) We could also consider “homotopies between paths” and so on. (4.6.c) Bloch [4, §1]: there is a space (i.e., simplicial abelian group) 푍∗(푋, ⋅) of cycles on 푋 with a natural identification ∗ ∗ 휋0푍 (푋, ⋅) ≅ CH (푋) . 4.7. Definition. Let 푋 be a variety. The higher Chow groups of 푋 are the homotopy groups ∗ ∗ CH (푋, 푖) ≔ 휋푖푍 (푋, ⋅) of the space of cycles on 푋 (defined for 푖 ≥ 0). 4.8. Remark (on notation). Each higher Chow group CH∗(푋, 푖) has a natural grading com- ing from a grading on 푍∗(푋, ⋅) given by codimension of cycles, which is why we write “CH∗(푋, 푖)” rather than “CH(푋, 푖)”. 4.9. Upshot. The higher Chow groups of a scheme 푋 are geometrically-defined groups that can be computed as the homology of a (graded) chain complex which we abusively also write as 푍∗(푋, ⋅), with 푍∗(푋, 푛) defined as a the subgroup of the group of cycles 푍∗(푋 × 훥푛) generated by those cycles which meet all faces 푋 × 훥푚 ⊂ 푋 × 훥푛 properly (i.e., in the correct codimension), where 푛 훥 ≔ Spec(푘[푥0, …, 푥푛]/⟨푥0 + ⋯ + 푥푛 − 1⟩) is the standard algebraic 푛-. The boundary maps are defined via the Dold–Kan cor- respondence (see [17, §1.2.3] for the relation between simplicial abelian groups and chain complexes). 6 PETER J. HAINE

A theorem of Voevodsky [26, Cor. 2] allows us to take the following as a definition of motivic cohomology. 4.10. Definition. Let 푋 be a smooth variety. The motivic cohomology groups H푝(푋; 퐙(푞)) for 푝, 푞 ∈ 퐙 are given by CH푞(푋, 2푞 − 푝), 푞 ≥ 0 and 2푞 − 푝 ≥ 0 H푝(푋; 퐙(푞)) ≔ { 0, otherwise . The index 푞 in H푝(푋; 퐙(푞)) is referred to as the weight of the motivic cohomology group. 4.11. Remark. Here are some reasons for the choice of grading relationship between the higher Chow groups and motivic cohomology: first, the higher Chow groups have their own natural grading, and second, we want to have a rational isomorphism between algebraic 퐾-theory and motivic cohomology that resembles the Atiyah–Hirzebruch isomorphism of Theorem 4.1. To relate motivic cohomology to something more concrete, we recall the definition of the zeta function of a finite type 퐙-scheme. 4.12. Definition. Let 푋 be a finite type 퐙-scheme. The zeta function of 푋 is the Euler product 1 휁푋(푠) ≔ ∏ −푠 , 푥∈푋 1 − |휅(푥)| closed which converges when Re(푠) > dim(푋). 4.13. Examples.

(4.13) If 푋 = Spec(퐙), then 휁푋 is the classical Riemann zeta function. (4.13) More generally, if 퐾 is an , 푂퐾 ⊂ 퐾 is the ring of integers of 퐾, and 푋 = Spec(푂퐾), then 휁푋 is the Dedekind zeta function of 퐾. 4.14. Conjecture (Soulé, but in more generality [4, p. 271; 24, Conj. 2.2]). If 푋 is regular and proper over Spec(퐙), then for an 푛 ∈ 퐙 we have 푖 ord푠=푛 휁푋(푠) = − ∑(−1) rank(H푖(푋; 퐙(푛))) , 푖≥0 where the groups H푖(푋; 퐙(푛)) are motivic homology groups, which by a Poincaré duality result are dual to (specific) motivic cohomology groups. That is, the orders of the zero of 휁푋 at 푛 is given (up to a sign) by the weight 푛 motivic Euler characteristic of 푋. 4.15. Remark. A special case of Soulé’s conjecture for elliptic curves implies part of the Birch–Swinnerton-Dyer conjecture. 4.16. Idea. Euler characteristics are tools for decatigorification. The classical Euler character- istic decategorifies homology groups. Since zeta functions are much more elementary than motivic cohomology, we might want to think of motivic cohomology as a categorification of the zeta function.

5. Properties of Motivic Cohomology 5.1. Properties (properties of motivic cohomology [18, p. viii]). Let 푘 be a field, 퐴 an abelian group, and 푋 ∈ Sm/푘. Motivic cohomology for smooth 푘-schemes satisfies the following properties. (5.1.a) We have H푝(푋; 퐴(푞)) = 0 if 푞 < 0. AN OVERVIEW OF MOTIVIC COHOMOLOGY 7

(5.1.b) If 푋 is connected, 퐴, 푝 = 0 H푝(푋; 퐴(0)) ≅ { 0, 푝 ≠ 0 . (5.1.c) We have 푂× (푋), 푝 = 1 { 푋 H푝(푋; 퐙(1)) ≅ Pic(푋), 푝 = 2 { {0, 푝 ≠ 1, 2 . 푝 푀 (5.1.d) We have H (Spec(푘); 퐴(푝)) ≅ K푝 (푘) ⊗퐙 퐴. (5.1.e) If 푋 is a strictly Henselian local 푘-scheme and 푛 ∈ 퐙 is prime to char(푘), we have ⊗푞 휇푛 (푆), 푝 = 0 H푝(푋; (퐙/푛)(푞)) ≅ { 0, 푝 ≠ 0 .

6. The Bloch–Kato Conjecture 6.1. The Kummer Sequence. If 푋 is a scheme and ℓ ∈ 퐙 is relatively prime to char(푋), then the we have a Kummer exact sequence [19, Ch. ii Ex. 2.18(b) & Ch. iii Prop. 4.11; 29, Ch. iii Ex. 6.10.1]

(−)ℓ (6.1) 0 흁ℓ 퐆푚 퐆푚 0 of étale sheaves on 푋 (though usually not an exact sequence of Zariski sheaves). Let us examine the Kummer sequence in the case that 푋 is the spectrum of a field — in this case the Kummer sequence has an especially nice description. Let 푘 be a field and we write 퐺푘 ≔ Gal(푘sep/푘) for the absolute Galois group of 푘, then the category of discrete 퐺푘-modules is equivalent to the category of étale sheaves on Spec(푘) [19, Ch. ii Thm. 1.9].

6.2. Notation. For a field 푘 we write 퐺푘 ≔ Gal(푘sep/푘) for the absolute Galois group of 푘. 6.3. Recollection. Let 푘 be a field. The absolute Galois group is a profinite group, with pre- sentation given by the diagram

퐺푘 ≅ lim Gal(푘sep/퐾) . 푘⊂퐾⊂푘sep finite 6.4. Recollection ([23, Ch. i §2.1]). Let 퐺 be a profinite group. A 퐺-module 푀 is discrete if the action map 퐺 × 푀 → 푀 is continuous, where the abelian group 푀 is given the discrete disc topology. Write Mod퐺 ⊂ Mod퐺 for the full subcategory of Mod퐺 spanned by the discrete 퐺-modules. 6.5. Recollection ([19, Ch. ii Thm. 1.9 & Ch. iii Ex. 1.7]). Let 푘 be a field. There is an explicit equivalence of categories Moddisc ≃ Shv (Spec(푘); Ab) , 퐺푘 ét with explicit quasi-inverse. Moreover, this equivalence identifies the 퐺푘-invariants disc (−)퐺푘 ∶ Mod → Ab with the global sections functor 퐺푘

훤(Spec(푘); −)∶ Shvét(Spec(푘); Ab) → Ab . Since étale cohomology is the right derived functor cohomology of the left exact functor 훤(Spec(푘); −) and is the right derived functor cohomology of the left 퐺푘 exact functor (−) , if F is an étale on Spec(푘) with corresponding 퐺푘-module 푀F , then for all 푛 ≥ 0 we have an identification 푛 푛 Hét(Spec(푘); F) ≅ HGal(푘; 푀F ) . 8 PETER J. HAINE

6.6. Remark. Galois cohomology is very classical and can be computed by an explicit cochain complex (see [23, Ch. i §2.2 & Ch. ii §1.1]). If one writes out this equivalence explicitly for, the exactness of the Kummer sequence for Spec(푘) is equivalent to the exactness of the sequence

ℓ × (−) × 0 흁ℓ(푘sep) 푘sep 푘sep 0 of discrete 퐺푘-modules. 6.7. Proposition (Kummer isomorphism). Let 푘 be a field and ℓ an invertible in 푘. Then × 1 there is a canonical isomorphism 휕∶ 푘 /ℓ → Hét(Spec(푘); 흁ℓ) is an isomorphism. Proof. The Kummer sequence (6.1) gives rise to a long exact sequence in étale cohomology

0 0 ⋅ℓ 0 휕 1 1 (6.8) 0 Hét(흁ℓ) Hét(퐆푚) Hét(퐆푚) Hét(흁ℓ) Hét(퐆푚) ⋯ , ∗ ∗ where we have written Hét(−) for Hét(Spec(푘); −). By Hilbert’s Theorem 90, we have 1 Hét(Spec(푘); 퐆푚) ≅ Pic(푘) , and Pic(푘) = 0 since every 1-dimensional 푘-vector space is isomorphic to 푘. Hence the long exact sequence (6.8) shows that the boundary map 휕 gives an identification 0 1 Hét(Spec(푘); 퐆푚)/ℓ ≅ Hét(Spec(푘); 흁ℓ) . 0 × □ To conclude, note that Hét(Spec(푘); 퐆푚) = 퐆푚(푘) = 푘 . 6.9. Definition. Let 푘 be a field and ℓ an integer invertible in 푘. The first norm residue map × 1 or Galois symbol 휕∶ 푘 /ℓ → Hét(Spec(푘); 흁ℓ) is the map induced by the boundary map × 0 1 푘 = Hét(Spec(푘); 퐆푚) → Hét(Spec(푘); 흁ℓ) in the long exact sequence in étale cohomology (6.8) coming from the Kummer sequence (6.1). 6.10. Observation. Let 푘 be a field and ℓ an integer invertible in 푘. Taking the tensor power of the first norm residue map and applying the cup product on étale cohomology defines a map ⊗푛 ⌣푛 푛 × ⊗푛 휕 1 ⊗푛 (−) 푛 ⊗푛 휕 ∶ (푘 ) Hét(Spec(푘); 흁ℓ) Hét(푘; 흁ℓ ) . 푛 th 푀 One can easily check that the map 휕 factors through the 푛 Milnor 퐾-group K푛 (푘), i.e., that if 푎, 1 − 푎 ∈ 푘×, then 푛 휕 (푏1 ⊗ ⋯ ⊗ 푎 ⊗ ⋯ ⊗ (1 − 푎) ⊗ ⋯ ⊗ 푏푛−2) = 0 × for 푏1, …, 푏푛−2 ∈ 푘 . Moreover, because étale cohomology with coefficients in a 퐙/ℓ-module is an ℓ-torsion 푛 푀 th group, the map 휕 factors through K푛 (푘)/ℓ, defining the 푛 norm residue map map 푛 푀 푛 ⊗푛 휕 ∶ K푛 (푘)/ℓ → Hét(Spec(푘); 흁ℓ ) . 6.11. Conjecture (Bloch–Kato). Let 푘 be a field. If ℓ is a prime that is invertible in 푘, then for 푛 푀 푛 ⊗푛 all 푛 ≥ 0 the norm residue map 휕 ∶ K푛 (푘)/ℓ → Hét(푘; 흁ℓ ) is an isomorphism. 6.12. Remarks. (6.12.a) The case where ℓ = 2 is known as the Milnor conjecture. Voevodsky’s proof of the Milnor conjecture was part of his fields medal citation. (6.12.b) The case where 푛 = 2 is known as the Merkurjev–Suslin theorem. AN OVERVIEW OF MOTIVIC COHOMOLOGY 9

6.13. Conjecture (Beilinson–Lichtenbaum). Let 푘 be a field. If ℓ is a prime that is invertible in 푘, then for all smooth 푘-varieties 푋, for 푝 ≤ 푞 there is a naturally-defined isomorphism 푝 푝 ⊗푞 H (푋; (퐙/ℓ)(푞)) ≅ Hét(푋; 흁ℓ ) . 6.14. Theorem (Voevodsky). The Bloch–Kato Conjecture 6.11 is equivalent to the Beilinson– Lichtenbaum Conjecture 6.13. 6.15. Theorem (Voevodsky, Rost, Weibel, Haesemeyer, …[12, Thm. A]). The Bloch–Kato Conjecture 6.11 is true. This theorem is incredibly difficult and involves deep theorems in homotopy theory and algebraic geometry (namely, resolution of singularities in characteristic 0). See [12] for an expository account. We give a sweeping overview of the proof of the Bloch–Kato Conjecture 6.11. Sweeping overview of the proof of the Bloch–Kato Conjecture 6.11 (see [12, §§1.1–1.2]). (6.15.a) Using a transfer argument, reduce the problem to fields containing all ℓth roots of unity. (6.15.b) Using a transfer argument, reduce the proving the result for fields of characteristic 0. (6.15.c) Translate the problem into motivic language. 6.16. Definition. Let 푛 and ℓ ≠ 0 be integers. We say that H90(푛) holds if 푛+1 Hét (Spec(푘); 퐙⟨ℓ⟩(푛)) = 0 for every field 푘 with ℓ invertible. 6.17. Remark. Note that H90(1) holds because H90(1) is equivalent to the classical Hilbert’s Theorem 90. 6.18. Theorem ([12, Thms. 1.6 & 2.38]). Let 푛 be a nonnegative integer and ℓ a prime. Then 푛 푀 푛 ⊗푛 휕 ∶ K푛 (푘)/ℓ ⥲ Hét(Spec(푘); 흁ℓ ) for all fields 푘 with ℓ invertible if and only if H90(푛) holds. (6.15.e) Induct on 푛. • Todo the induction, one needs to construct a Rost variety, a smooth whose function field satisfies certain specific conditions. Thisis very difficult. 6.19. Theorem ([12, Thm. 1.10]). Suppose that H90(푛 − 1) holds. Then for every field 푘 of 푀 characteristic 0 and every nonzero element 풂 ∈ K푛 (푘)/ℓ there is a smooth projective variety 푋풂 whose function field 퐾풂 ≔ 휅(푋풂) satisfies the following: 푀 (6.19.a) The element 풂 vanishes in K푛 (퐾풂)/ℓ. 푛+1 푛+1 (6.19.b) The map Hét (Spec(푘); 퐙⟨ℓ⟩(푛)) → Hét (퐾풂; 퐙⟨ℓ⟩(푛)) is an injection. • The Rost variety needs to have an additional property, namely that it has an associated Rost motive. As input, this requires a motivic cohomology class 휇, and to show that everything works out, we need to show that a certain motivic cohomology operation associated with 휇 agrees with 훽푃푖 (where 푖 is an integer related to the bidegree of 휇). This requires a complete understanding of the motivic Steenrod algebra (in characteristic 0), as per the following theorem. 6.20. Theorem (Voevodsky in characteristic 0 [27], Hoyois–Kelly–Østvær in general [14, Thm. 1.1]). Let 푘 be a field and ℓ a prime different from char(푘). Then ∗,∗ ∗ ∗ □ 퐴mot,퐙/ℓ ≅ 퐴퐙/ℓ ⊗ H (Spec(푘); (퐙/ℓ)(∗)) . 10 PETER J. HAINE

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