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fixed an inaccuracy in the application of Serre's conjecture to abelian varieties. added remarks on rigidity
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Matthias Wendt
Matthias Wendt
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  • First step: if $X/F$ is a smooth projective variety over a number field $F$, then the function field $F(X)$ is finitely generated and therefore embeds into $\mathbb{C}$.

  • Second, the rational motivic cohomology of a function field embeds into the rational cohomology of $\mathbb{C}$. This follows from the existence of transfers on motivic cohomology; in a finite field extension, composition of the transfer with the natural presheaf restriction is multiplication by the degree and therefore an isomorphism for rational coefficients. So we cannot kill any motivic cohomology by finite extensions. On the other hand, finite transcendental extensions of algebraically closed fields also induce split injections (by evaluating on a rational point of the respective affine space). So, no rational motivic cohomology is lost by passing from the function field to $\mathbb{C}$.

  • Third, (in the range we are interested in) the motivic cohomology of a smooth projective variety $X/F$ embeds into the motivic cohomology of its function field. This is essentially a form of the Gersten resolution. For ease of citation, I would replace motivic cohomology by Nisnevich cohomology of the Milnor K-sheaf (see my answer to this MO-question: Motivic cohomology and cohomology of Milnor K-theory sheaf) and use the Gersten resolution of the latter.

  • Fourth, we now employ the conjectures of Beilinson and Serre (but note that all reduction steps so far were unconditional). For more information, see Schneider's survey "Introduction to the Beilinson conjectures" (see here) or this great MO-answer: Beilinson conjectures. If we have a smooth projective variety $X/F$, Beilinson's conjecture predicts an isomorphism between "the integral part" of motivic cohomology of $X$ with Deligne-Beilinson cohomology of $X(\mathbb{R})$ (under suitable condition on the indices). Put differently, the rank of the integral part of motivic cohomology (a subspace in ${\rm H}^{i+1}(X,\mathbb{Q}(n))$) should be equal to the order of vanishing of the L-function at $s=i+1-n$, assuming that $i/2+1<n$. Note that this condition is satisfied for our range of interest $i\geq 1$ because $i/2+1<i+1\leq n$. Now in Schneider's survey paper we also find (p.6) that - assuming conjectures of Serre on the L-functions of varieties over number fields - this order of vanishing is given by $\dim {\rm H}^i(X(\mathbb{C}),\mathbb{C})^{(-1)^{i+1-n}}$. For odd $i$, we can replace this by $1/2\dim {\rm H}^i(X(\mathbb{C}),\mathbb{C})$.

  • Summing up, the conjectures of Beilinson and Serre imply that the motivic cohomology group ${\rm H}^{i+1}({\rm Spec}\mathbb{C},\mathbb{Q}(n))$ is at least as big as $\dim {\rm H}^i(X(\mathbb{C}),\mathbb{C})^{(-1)^{i+1-n}}$ (resp. $1/2\dim{\rm H}^i(X(\mathbb{C}),\mathbb{C})$ for odd $i$) for any smooth projective variety $X$ over a number field. Now we can just take abelian varieties over number fields: the dimensions of the cohomology in a fixed degree $i$ grow(and then also the $\pm 1$ eigenspaces for even $i$) grow like a binomial coefficient times dimension of the abelian variety times the degree of the number field. So, we can find abelian varieties where (conjecturally, Serre) the order of vanishing of the L-function is arbitrarily big and consequently (conjecturally, Beilinson) the dimension of the motivic cohomology group ${\rm H}^{i,n}$ is arbitrarily big. (Actually, for high-enough dimension, we only need the conjecture of Serre and can replace the application of Beilinson's conjecture by Soule's result mentioned above.) Since all these motivic cohomology groups of abelian varieties embed into the motivic cohomology of $\mathbb{C}$, we get an infinite-dimensional vector space.

As a final remark, there is another possible conjectural way to deduce statements about the motivic cohomology of $\mathbb{C}$. In the papers of Goncharov (in particular the trilogarithm paper in the Motives Proceedings), there is a conjectural description of the K-theory of fields. The motivation for this is that K-theory of a field $F$ should be the cohomology of the motivic Lie algebra associated to the category of mixed Tate motives over the field (again requires Beilinson-Soulé conjecture). The conjecture says that the motivic Lie algebra should be as simple as possible: the abelianization is related to $F^\times$, and the commutator subalgebra is a free Lie algebra related to higher Bloch groups. This gives rise to explicit complexes conjecturally computing rational K-theory of $F$ in symbol-like terms. However, I am unaware that anyone has used this to produce large subspaces in the K-theory of $\mathbb{C}$. Goncharov has used the presentation to produce elements in K-theory of elliptic curves.

Here is some more information on the rigidity question, as requested. Explicitly, this is the question if for any field $F$ with field of constants $F_0$ (i.e. algebraic closure of prime field in $F$) the inclusion $F_0\to F$ induces an isomorphism ${\rm H}^1({\rm Spec}F,\mathbb{Q}(n))\to {\rm H}^1({\rm Spec}F_0,\mathbb{Q}(n))$ for any $n\geq 2$.

For $n=2$, this is a question about indecomposable $K_3$. In this case, we can express the regulator in terms of the dilogarithm. If we knew injectivity of the regulator (conjectures of Milnor and Ramakrishnan), then rigidity would follow from the rigidity of the dilogarithm, cf. Sections 6 and 7 of Bloch's Irvine notes on K-theory of elliptic curves. In Sah's paper "Homology of Lie groups made discrete III", JPAA 56 (1989), pp. 269-312, Conjecture 4.6, the rigidity conjecture for $K_3$ is attributed to Suslin. It appeared in scissors congruence literature because rigidity would complete the scissors congruence classification of 3-dimensional hyperbolic polytopes.

In Goncharov's paper in the Motives proceedings, the rigidity is formulated as Conjecture 1.7 and attributed to Beilinson. Indeed, it's a consequence of the Beilinson conjectures as follows. Any class in ${\rm H}^1({\rm Spec}\mathbb{C},\mathbb{Q}(n))$ comes from a smooth projective variety $X/F$ over a number field $F$. The map induced on motivic cohomology is injective, so it suffices to compute dimensions. If we assume the vanishing conjecture, then the Gersten resolution also tells us that we get an isomorphism ${\rm H}^1({\rm Spec} F(X),\mathbb{Q}(n))\cong {\rm H}^1(X,\mathbb{Q}(n))$. (There is a slight gap here; actually, I want to put the integral subspace there, but that requires a slightly subtler argument.) Again, if we assume the conjectures of Beilinson and Serre, we can determine the dimension of the vector space ${\rm H}^1(X,\mathbb{Q}(n))_{\mathbb{Z}}$ as $\dim {\rm H}^0(X(\mathbb{C}),\mathbb{C})^{(-1)^{1-n}}$. This is exactly the same dimension as for the vector space ${\rm H}^1({\rm Spec} F_0,\mathbb{Q}(n))$ (in some sense, ${\rm Spec}F_0$ is the scheme of connected components of $X$) hence we get an isomorphism in motivic cohomology.

  • First step: if $X/F$ is a smooth projective variety over a number field $F$, then the function field $F(X)$ is finitely generated and therefore embeds into $\mathbb{C}$.

  • Second, the rational motivic cohomology of a function field embeds into the rational cohomology of $\mathbb{C}$. This follows from the existence of transfers on motivic cohomology; in a finite field extension, composition of the transfer with the natural presheaf restriction is multiplication by the degree and therefore an isomorphism for rational coefficients. So we cannot kill any motivic cohomology by finite extensions. On the other hand, finite transcendental extensions of algebraically closed fields also induce split injections (by evaluating on a rational point of the respective affine space). So, no rational motivic cohomology is lost by passing from the function field to $\mathbb{C}$.

  • Third, (in the range we are interested in) the motivic cohomology of a smooth projective variety $X/F$ embeds into the motivic cohomology of its function field. This is essentially a form of the Gersten resolution. For ease of citation, I would replace motivic cohomology by Nisnevich cohomology of the Milnor K-sheaf (see my answer to this MO-question: Motivic cohomology and cohomology of Milnor K-theory sheaf) and use the Gersten resolution of the latter.

  • Fourth, we now employ the conjectures of Beilinson and Serre (but note that all reduction steps so far were unconditional). For more information, see Schneider's survey "Introduction to the Beilinson conjectures" (see here) or this great MO-answer: Beilinson conjectures. If we have a smooth projective variety $X/F$, Beilinson's conjecture predicts an isomorphism between "the integral part" of motivic cohomology of $X$ with Deligne-Beilinson cohomology of $X(\mathbb{R})$ (under suitable condition on the indices). Put differently, the rank of the integral part of motivic cohomology (a subspace in ${\rm H}^{i+1}(X,\mathbb{Q}(n))$) should be equal to the order of vanishing of the L-function at $s=i+1-n$, assuming that $i/2+1<n$. Note that this condition is satisfied for our range of interest $i\geq 1$ because $i/2+1<i+1\leq n$. Now in Schneider's survey paper we also find (p.6) that - assuming conjectures of Serre on the L-functions of varieties over number fields - this order of vanishing is given by $1/2\dim {\rm H}^i(X(\mathbb{C}),\mathbb{C})$.

  • Summing up, the conjectures of Beilinson and Serre imply that the motivic cohomology group ${\rm H}^{i+1}({\rm Spec}\mathbb{C},\mathbb{Q}(n))$ is at least as big as $1/2\dim{\rm H}^i(X(\mathbb{C}),\mathbb{C})$ for any smooth projective variety $X$ over a number field. Now we can just take abelian varieties over number fields: the dimensions of the cohomology in a fixed degree $i$ grow like a binomial coefficient times dimension of the abelian variety times the degree of the number field. So, we can find abelian varieties where (conjecturally, Serre) the order of vanishing of the L-function is arbitrarily big and consequently (conjecturally, Beilinson) the dimension of the motivic cohomology group ${\rm H}^{i,n}$ is arbitrarily big. (Actually, for high-enough dimension, we only need the conjecture of Serre and can replace the application of Beilinson's conjecture by Soule's result mentioned above.) Since all these motivic cohomology groups of abelian varieties embed into the motivic cohomology of $\mathbb{C}$, we get an infinite-dimensional vector space.

As a final remark, there is another possible conjectural way to deduce statements about the motivic cohomology of $\mathbb{C}$. In the papers of Goncharov (in particular the trilogarithm paper in the Motives Proceedings), there is a conjectural description of the K-theory of fields. The motivation for this is that K-theory of a field $F$ should be the cohomology of the motivic Lie algebra associated to the category of mixed Tate motives over the field (again requires Beilinson-Soulé conjecture). The conjecture says that the motivic Lie algebra should be as simple as possible: the abelianization is related to $F^\times$, and the commutator subalgebra is a free Lie algebra related to higher Bloch groups. This gives rise to explicit complexes conjecturally computing rational K-theory of $F$ in symbol-like terms. However, I am unaware that anyone has used this to produce large subspaces in the K-theory of $\mathbb{C}$. Goncharov has used the presentation to produce elements in K-theory of elliptic curves.

  • First step: if $X/F$ is a smooth projective variety over a number field $F$, then the function field $F(X)$ is finitely generated and therefore embeds into $\mathbb{C}$.

  • Second, the rational motivic cohomology of a function field embeds into the rational cohomology of $\mathbb{C}$. This follows from the existence of transfers on motivic cohomology; in a finite field extension, composition of the transfer with the natural presheaf restriction is multiplication by the degree and therefore an isomorphism for rational coefficients. So we cannot kill any motivic cohomology by finite extensions. On the other hand, finite transcendental extensions of algebraically closed fields also induce split injections (by evaluating on a rational point of the respective affine space). So, no rational motivic cohomology is lost by passing from the function field to $\mathbb{C}$.

  • Third, (in the range we are interested in) the motivic cohomology of a smooth projective variety $X/F$ embeds into the motivic cohomology of its function field. This is essentially a form of the Gersten resolution. For ease of citation, I would replace motivic cohomology by Nisnevich cohomology of the Milnor K-sheaf (see my answer to this MO-question: Motivic cohomology and cohomology of Milnor K-theory sheaf) and use the Gersten resolution of the latter.

  • Fourth, we now employ the conjectures of Beilinson and Serre (but note that all reduction steps so far were unconditional). For more information, see Schneider's survey "Introduction to the Beilinson conjectures" (see here) or this great MO-answer: Beilinson conjectures. If we have a smooth projective variety $X/F$, Beilinson's conjecture predicts an isomorphism between "the integral part" of motivic cohomology of $X$ with Deligne-Beilinson cohomology of $X(\mathbb{R})$ (under suitable condition on the indices). Put differently, the rank of the integral part of motivic cohomology (a subspace in ${\rm H}^{i+1}(X,\mathbb{Q}(n))$) should be equal to the order of vanishing of the L-function at $s=i+1-n$, assuming that $i/2+1<n$. Note that this condition is satisfied for our range of interest $i\geq 1$ because $i/2+1<i+1\leq n$. Now in Schneider's survey paper we also find (p.6) that - assuming conjectures of Serre on the L-functions of varieties over number fields - this order of vanishing is given by $\dim {\rm H}^i(X(\mathbb{C}),\mathbb{C})^{(-1)^{i+1-n}}$. For odd $i$, we can replace this by $1/2\dim {\rm H}^i(X(\mathbb{C}),\mathbb{C})$.

  • Summing up, the conjectures of Beilinson and Serre imply that the motivic cohomology group ${\rm H}^{i+1}({\rm Spec}\mathbb{C},\mathbb{Q}(n))$ is at least as big as $\dim {\rm H}^i(X(\mathbb{C}),\mathbb{C})^{(-1)^{i+1-n}}$ (resp. $1/2\dim{\rm H}^i(X(\mathbb{C}),\mathbb{C})$ for odd $i$) for any smooth projective variety $X$ over a number field. Now we can just take abelian varieties over number fields: the dimensions of the cohomology in a fixed degree $i$ (and then also the $\pm 1$ eigenspaces for even $i$) grow like a binomial coefficient times dimension of the abelian variety times the degree of the number field. So, we can find abelian varieties where (conjecturally, Serre) the order of vanishing of the L-function is arbitrarily big and consequently (conjecturally, Beilinson) the dimension of the motivic cohomology group ${\rm H}^{i,n}$ is arbitrarily big. (Actually, for high-enough dimension, we only need the conjecture of Serre and can replace the application of Beilinson's conjecture by Soule's result mentioned above.) Since all these motivic cohomology groups of abelian varieties embed into the motivic cohomology of $\mathbb{C}$, we get an infinite-dimensional vector space.

As a final remark, there is another possible conjectural way to deduce statements about the motivic cohomology of $\mathbb{C}$. In the papers of Goncharov (in particular the trilogarithm paper in the Motives Proceedings), there is a conjectural description of the K-theory of fields. The motivation for this is that K-theory of a field $F$ should be the cohomology of the motivic Lie algebra associated to the category of mixed Tate motives over the field (again requires Beilinson-Soulé conjecture). The conjecture says that the motivic Lie algebra should be as simple as possible: the abelianization is related to $F^\times$, and the commutator subalgebra is a free Lie algebra related to higher Bloch groups. This gives rise to explicit complexes conjecturally computing rational K-theory of $F$ in symbol-like terms. However, I am unaware that anyone has used this to produce large subspaces in the K-theory of $\mathbb{C}$. Goncharov has used the presentation to produce elements in K-theory of elliptic curves.

Here is some more information on the rigidity question, as requested. Explicitly, this is the question if for any field $F$ with field of constants $F_0$ (i.e. algebraic closure of prime field in $F$) the inclusion $F_0\to F$ induces an isomorphism ${\rm H}^1({\rm Spec}F,\mathbb{Q}(n))\to {\rm H}^1({\rm Spec}F_0,\mathbb{Q}(n))$ for any $n\geq 2$.

For $n=2$, this is a question about indecomposable $K_3$. In this case, we can express the regulator in terms of the dilogarithm. If we knew injectivity of the regulator (conjectures of Milnor and Ramakrishnan), then rigidity would follow from the rigidity of the dilogarithm, cf. Sections 6 and 7 of Bloch's Irvine notes on K-theory of elliptic curves. In Sah's paper "Homology of Lie groups made discrete III", JPAA 56 (1989), pp. 269-312, Conjecture 4.6, the rigidity conjecture for $K_3$ is attributed to Suslin. It appeared in scissors congruence literature because rigidity would complete the scissors congruence classification of 3-dimensional hyperbolic polytopes.

In Goncharov's paper in the Motives proceedings, the rigidity is formulated as Conjecture 1.7 and attributed to Beilinson. Indeed, it's a consequence of the Beilinson conjectures as follows. Any class in ${\rm H}^1({\rm Spec}\mathbb{C},\mathbb{Q}(n))$ comes from a smooth projective variety $X/F$ over a number field $F$. The map induced on motivic cohomology is injective, so it suffices to compute dimensions. If we assume the vanishing conjecture, then the Gersten resolution also tells us that we get an isomorphism ${\rm H}^1({\rm Spec} F(X),\mathbb{Q}(n))\cong {\rm H}^1(X,\mathbb{Q}(n))$. (There is a slight gap here; actually, I want to put the integral subspace there, but that requires a slightly subtler argument.) Again, if we assume the conjectures of Beilinson and Serre, we can determine the dimension of the vector space ${\rm H}^1(X,\mathbb{Q}(n))_{\mathbb{Z}}$ as $\dim {\rm H}^0(X(\mathbb{C}),\mathbb{C})^{(-1)^{1-n}}$. This is exactly the same dimension as for the vector space ${\rm H}^1({\rm Spec} F_0,\mathbb{Q}(n))$ (in some sense, ${\rm Spec}F_0$ is the scheme of connected components of $X$) hence we get an isomorphism in motivic cohomology.

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Matthias Wendt
Matthias Wendt
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This is going to be a slightly extended explanation, I apologize. The short version is basically that little is actually known (and even that is hard to prove), but conjecturally everything permitted by Beilinson-Soulé vanishing should be infinite-dimensional.

What is known unconditionally (I bet everybody knows the first three items and I make no claim that this is an exhaustive list):

  • the groups ${\rm H}^i({\rm Spec} \mathbb{C},\mathbb{Q}(n))$ vanish for $i>n$ (follows from Nisnevich cohomology dimension reasons).

  • the groups ${\rm H}^n({\rm Spec} \mathbb{C},\mathbb{Q}(n))$ are uniquely divisible groups, uncountably-dimensional as $\mathbb{Q}$-vector spaces. (Milnor K-theory, related to elements in $\mathbb{C}^\times$)

  • the groups ${\rm H}^{1}({\rm Spec} \mathbb{C},\mathbb{Q}(n))$ (corresponding to weight $n$ part of $K_{2n-1}$) are non-trivial infinite-dimensional $\mathbb{Q}$-vector spaces. The K-theory of $\mathbb{C}$ contains the K-theory of $\overline{\mathbb{Q}}$ as direct summand, and the groups ${\rm H}^{1}({\rm Spec}\overline{\mathbb{Q}},\mathbb{Q}(n))$ are vector spaces of countable dimension, detected by Borel's regulator (this computation is due to Borel). The rigidity conjecture voices the expectation that the inclusion $\overline{\mathbb{Q}}\subset \overline{\mathbb{C}}$ induces an isomorphism on ${\rm H}^{1}(-,\mathbb{Q}(n))$ for $n\geq 2$.

  • C. Soulé. $p$-adic K-theory of elliptic curves. Duke Math. J. 54 (1987), 249--269. His Theorem 2.1 provides unconditional lower bounds for high-dimensional K-groups (related to the argument given below). From curves over number fields we would get contributions in ${\rm H}^2({\rm Spec}\mathbb{C},\mathbb{Q}(n))$, which is just above the Borel case above. For example, Soule's unconditional result tells us that weight 82 in $K_{162}(\mathbb{C})$ is unbounded (as well as all the bigger weights in the appropriate K-groups...).

What is expected: (I think)

Of course, the first thing to mention is that the Beilinson-Soulé vanishing conjecture predicts the vanishing of ${\rm H}^i({\rm Spec}\mathbb{C},\mathbb{Q}(n))$ for any $i\leq 0$ except for $i=n=0$. In particular, it would be conjectured that the motivic cohomology of $\mathbb{C}$ can only be nontrivial for $1\leq i\leq n$.

Now I would like to claim that whatever nontrivial motivic cohomology groups are allowed by Beilinson-Soulé vanishing also contain nontrivial elements and are in fact infinite-dimensional vector spaces, depending on conjectures of Beilinson and Serre.

  • First step: if $X/F$ is a smooth projective variety over a number field $F$, then the function field $F(X)$ is finitely generated and therefore embeds into $\mathbb{C}$.

  • Second, the rational motivic cohomology of a function field embeds into the rational cohomology of $\mathbb{C}$. This follows from the existence of transfers on motivic cohomology; in a finite field extension, composition of the transfer with the natural presheaf restriction is multiplication by the degree and therefore an isomorphism for rational coefficients. So we cannot kill any motivic cohomology by finite extensions. On the other hand, finite transcendental extensions of algebraically closed fields also induce split injections (by evaluating on a rational point of the respective affine space). So, no rational motivic cohomology is lost by passing from the function field to $\mathbb{C}$.

  • Third, (in the range we are interested in) the motivic cohomology of a smooth projective variety $X/F$ embeds into the motivic cohomology of its function field. This is essentially a form of the Gersten resolution. For ease of citation, I would replace motivic cohomology by Nisnevich cohomology of the Milnor K-sheaf (see my answer to this MO-question: Motivic cohomology and cohomology of Milnor K-theory sheaf) and use the Gersten resolution of the latter.

  • Fourth, we now employ the conjectures of Beilinson and Serre (but note that all reduction steps so far were unconditional). For more information, see Schneider's survey "Introduction to the Beilinson conjectures" (see here) or this great MO-answer: Beilinson conjectures. If we have a smooth projective variety $X/F$, Beilinson's conjecture predicts an isomorphism between "the integral part" of motivic cohomology of $X$ with Deligne-Beilinson cohomology of $X(\mathbb{R})$ (under suitable condition on the indices). Put differently, the rank of the integral part of motivic cohomology (a subspace in ${\rm H}^{i+1}(X,\mathbb{Q}(n))$) should be equal to the order of vanishing of the L-function at $s=i+1-n$, assuming that $i/2+1<n$. Note that this condition is satisfied for our range of interest $i\geq 1$ because $i/2+1<i+1\leq n$. Now in Schneider's survey paper we also find (p.6) that - assuming conjectures of Serre on the L-functions of varieties over number fields - this order of vanishing is given by $1/2\dim {\rm H}^i(X(\mathbb{C}),\mathbb{C})$.

  • Summing up, the conjectures of Beilinson and Serre imply that the motivic cohomology group ${\rm H}^{i+1}({\rm Spec}\mathbb{C},\mathbb{Q}(n))$ is at least as big as $1/2\dim{\rm H}^i(X(\mathbb{C}),\mathbb{C})$ for any smooth projective variety $X$ over a number field. Now we can just take abelian varieties over number fields: the dimensions of the cohomology in a fixed degree $i$ grow like a binomial coefficient times dimension of the abelian variety times the degree of the number field. So, we can find abelian varieties where (conjecturally, Serre) the order of vanishing of the L-function is arbitrarily big and consequently (conjecturally, Beilinson) the dimension of the motivic cohomology group ${\rm H}^{i,n}$ is arbitrarily big. (Actually, for high-enough dimension, we only need the conjecture of Serre and can replace the application of Beilinson's conjecture by Soule's result mentioned above.) Since all these motivic cohomology groups of abelian varieties embed into the motivic cohomology of $\mathbb{C}$, we get an infinite-dimensional vector space.

So the motivic cohomology of the complex numbers should be big because it incorporates motivic cohomology of all arithmetic schemes. At this point, though, I don't know if the conjectures would make a prediction as to countable vs uncountable dimension. (Of course, that rational cohomology has to be so big if it's non-vanishing is sort of clear: if we had finite-dimensional vector spaces, there would have to be residual torsion but this can't exist by Suslin's rigidity.)

As a final remark, there is another possible conjectural way to deduce statements about the motivic cohomology of $\mathbb{C}$. In the papers of Goncharov (in particular the trilogarithm paper in the Motives Proceedings), there is a conjectural description of the K-theory of fields. The motivation for this is that K-theory of a field $F$ should be the cohomology of the motivic Lie algebra associated to the category of mixed Tate motives over the field (again requires Beilinson-Soulé conjecture). The conjecture says that the motivic Lie algebra should be as simple as possible: the abelianization is related to $F^\times$, and the commutator subalgebra is a free Lie algebra related to higher Bloch groups. This gives rise to explicit complexes conjecturally computing rational K-theory of $F$ in symbol-like terms. However, I am unaware that anyone has used this to produce large subspaces in the K-theory of $\mathbb{C}$. Goncharov has used the presentation to produce elements in K-theory of elliptic curves.