# What are the consequences of the finite generation of $\operatorname{Ext}^1_{\mathcal{O}_F}(\mathbb{1},M)$?

Let $$F$$ be a number fields. Conjecturally, there is a rigid $$\mathbb{Q}$$-linear abelian category of mixed motives over $$F$$. Let $$\mathbb{1}$$ denotes the unit object of this category. Given a mixed motive $$M$$, we can consider the extension group $$\operatorname{Ext}^1_{F}(\mathbb{1},M)$$ which is a vector space over $$\mathbb{Q}$$. According to Scholl, we can define a subspace $$\operatorname{Ext}^1_{\mathcal{O}_F}(\mathbb{1},M)$$ of extensions having everywhere good reduction. Scholl conjectures (loc. cit.) that $$\operatorname{Ext}^1_{\mathcal{O}_F}(\mathbb{1},M)$$ is finite dimensional.

1. The conjecture is stated inside a paragraph in Scholl's paper, so it seems that it comes from elsewhere. Was Scholl the first to formulate this conjecture?
2. What are the arithmetic consequences of such a conjecture?
• For instance, does the finite generation of class groups, the fact that $$\mathcal{O}_F^{\times}$$ is finitely generated, or the Mordel-Weil Theorem follow from it? Are there more consequences?
• More generally, is the conjecture equivalent to the finite generation of some algebraic $$K$$-groups?
3. I believe that, in most cases, $$\operatorname{Ext}^1_{F}(\mathbb{1},M)$$ is not finitely generated. Is this known or at least conjectured somewhere?

Sorry for keeping this a bit vague. Many thanks in advance!

The idea of defining motivic cohomology in terms of Ext-groups in an hypothetical category $$\mathcal{MM}_\mathbb{Q}$$ of mixed motives over $$\mathbb{Q}$$ dates back at least to Beilinson and Deligne, see Nekovar's survey on the Beilinson conjectures, (2.6) and section 3. So for pure motives, the Beilinson conjectures predict finite dimensionality.
For question 2., in order to prove finite generation you need to assume that motivic cohomology is indeed given by these Ext-groups, which is part of the conjectural story. For example, the group $$\mathcal{O}_F^\times \otimes \mathbb{Q}$$ should be $$\mathrm{Ext}^1_{\mathbb{Z}}(\mathbb{Q}(0), h^0(X)(1))$$ with $$X = \mathrm{Spec} F$$. Also the Mordell-Weil group of an elliptic curve $$E/\mathbb{Q}$$ tensored with $$\mathbb{Q}$$ is isomorphic to $$\mathrm{CH}^1(E)^0$$, hence should be $$\mathrm{Ext}^1_{\mathbb{Z}}(\mathbb{Q}(0), h^1(E)(1))$$.
For question 3., Ext-groups in $$\mathcal{MM}_\mathbb{Q}$$ are sometimes not (expected to be) finitely generated. For example, $$F^\times \otimes \mathbb{Q} \cong H^1_\mathcal{M}(\mathrm{Spec} F, \mathbb{Q}(1))$$ is clearly not finitely generated. Sometimes the Ext-group is expected to be finitely generated. An example is the motivic cohomology group $$H^2_\mathcal{M}(E,\mathbb{Q}(n))$$ for an elliptic curve $$E/\mathbb{Q}$$ and $$n \geq 2$$. Using the localisation sequence in motivic cohomology, one can understand the difference between the Ext-group over $$\mathbb{Z}$$ and the Ext-group over $$\mathbb{Q}$$. The difference is given in terms of $$K$$-groups of the bad fibers of a proper regular model of $$E$$ over $$\mathbb{Z}$$. It turns out that these $$K$$-groups are finite-dimensional (unconditionally), and in fact are 0 for $$n>2$$. I don't know the complete picture for more general varieties, but there are standard conjectures on $$K$$-groups of varieties over finite fields, which should imply similar (conditional) finite-generation statements.
• If you permit a small addendum to my question: do we expect finite generation of $\operatorname{Ext}^1_{\mathcal{O}_F}(\mathbb{1},M)$ when $M$ has positive weights? It seems to me that Beilinson's conjectures only deal with motives having negative weights... Dec 7, 2020 at 14:39
• @Stabilo There should be no extension between two objects when the weights do not permit it (I guess $\mathrm{Ext}(A,B)=0$ when all the weights of $A$ are strictly less than the weights of $B$). There should be a theory of weights in the category of mixed motives, see e.g. Bondarko's work. This is true at least in the different realisations. Dec 8, 2020 at 14:14