# Motives of commutative groups schemes and relative 1-motives

Lehalleur, Simon Pepin. Motives of commutative groups schemes and relative 1-motives. 2015, University of Zurich, Faculty of Science.

## Abstract

In this dissertation, we present several new results in the theory of mixed motives. More precisely, we study the case of relative 1-motives and motives of commutative group schemes in the context of the triangulated categories of mixed motives of Voevodsky. Given a scheme $\mathit{S}$, Voevodsky introduced a triangulated category $\mathbf{DA}(\mathit{S})$ of mixed motives over $\mathit{S}$. This category is built and studied with the hope that it is a suitable approximation to the derived category of a conjectural abelian category$\mathbf{MM}(\mathit{S})$ of mixed motives over S. The expectation is that $\mathbf{DA}(\mathit{S})$ carries a motivic t-structure whose heart would be this conjectural abelian cate- gory. One can construct unconditionally a \six operations formalism" [Ayo07a] [Ayo07b] for the system of categories $\mathbf{DA}(\mathit{-})$ and realisation functors to derived categories of systems of coefficients (constructible sheaves, l-adic sheaves) [Ayo10b] [Ayo14a] which are compatible with the classical theory of the six operations in the Betti and ℓ-adic contexts, so that $\mathbf{DA}(\mathit{S})$ already behaves to a large extend like a \derived category of motivic sheaves" . Important examples of motives in $\mathbf{DA}(\mathit{S})$ are constructed out of commutative group schemes over S, and almost all our results are concerned with these. Let $\mathit{G/S}$ be such a smooth commutative group scheme. Then we have two natural motives built from $\mathit{G}$: first, $\Sigma^{\infty}\mathit{G}_{\mathbb{Q}}$ $\in$ $\mathbf{DA}(\mathit{S})$, which is $\mathit{G}$ seen as a sheaf of $\mathbb{Q}$-vector spaces", and $\mathit{M_{S}(G}$, that is, the homological motive of $\mathit{G}$ as an $\mathit{S}$-scheme. In Chapter 2, written in collaboration with Giuseppe Ancona and Annette Huber, we compare the two, and we prove the following canonical \Küunneth decomposition" of the motive $\mathit{M_{S}(G}$In this dissertation, we present several new results in the theory of mixed motives. More precisely, we study the case of relative 1-motives and motives of commutative group schemes in the context of the triangulated categories of mixed motives of Voevodsky. Given a scheme $\mathit{S}$, Voevodsky introduced a triangulated category $\mathbf{DA}(\mathit{S})$ of mixed motives over $\mathit{S}$. This category is built and studied with the hope that it is a suitable approximation to the derived category of a conjectural abelian category$\mathbf{MM}(\mathit{S})$ of mixed motives over S. The expectation is that $\mathbf{DA}(\mathit{S})$ carries a motivic t-structure whose heart would be this conjectural abelian cate- gory. One can construct unconditionally a \six operations formalism" [Ayo07a] [Ayo07b] for the system of categories $\mathbf{DA}(\mathit{-})$ and realisation functors to derived categories of systems of coefficients (constructible sheaves, l-adic sheaves) [Ayo10b] [Ayo14a] which are compatible with the classical theory of the six operations in the Betti and ℓ-adic contexts, so that $\mathbf{DA}(\mathit{S})$ already behaves to a large extend like a \derived category of motivic sheaves" . Important examples of motives in $\mathbf{DA}(\mathit{S})$ are constructed out of commutative group schemes over S, and almost all our results are concerned with these. Let $\mathit{G/S}$ be such a smooth commutative group scheme. Then we have two natural motives built from $\mathit{G}$: first, $\Sigma^{\infty}\mathit{G}_{\mathbb{Q}}$ $\in$ $\mathbf{DA}(\mathit{S})$, which is $\mathit{G}$ seen as a sheaf of $\mathbb{Q}$-vector spaces", and $\mathit{M_{S}(G}$, that is, the homological motive of $\mathit{G}$ as an $\mathit{S}$-scheme. In Chapter 2, written in collaboration with Giuseppe Ancona and Annette Huber, we compare the two, and we prove the following canonical "Künneth decomposition" of the motive $\mathit{M_{S}(G)}$.

$\mathit{M}_{\mathit{S}}(\mathit{G})$ $\tilde{\longrightarrow}$ $\left( \bigoplus_{\mathit{n}\ge0}^{kd(\mathit{G/S})} Sym^{\mathit{n}}\Sigma^{\infty}\mathit{G_{\mathbb{Q}}} \right)$ $\otimes$ $\mathit{M}_{\mathit{S}}(\mathrm{\pi}_0(\mathit{G/S}))$.

From this result, we can expect that the motive $\Sigma^{\infty}\mathit{G}_{\mathbb{Q}}$ should be a prime example of a relative homological 1-motive, that is, a motive in the triangulated subcategory generated by the relative homology of curves over $\mathit{S}$. This motivates the systematic study of this subcategory. In Chapter 3.1, we define more generally for every $\mathit{n}\in \mathbb{N}$ the category in $\mathbf{DA_{n}}(\mathit{S})$of homological n- motives (resp. $\mathbf{DA^{n}}(\mathit{S})$ of cohomological n-motives) as the full triangulated subcategory of $\mathbf{DA}(\mathit{S})$which is generated by homological motives of smooth (resp. cohomological motives of projective) S-schemes of relative dimension less than n. We study their general properties and their behaviour under the six operations. Then we show in Chapter 3.2 that $\Sigma^{\infty}\mathit{G}_{\mathbb{Q}}$ lies indeed in$\mathbf{DA}(\mathit{S})$. in Chapter 3.3, we de_ne for any cohomological motive$\mathit{S}$ its \1-motivic approximation" $\mathit{\omega}^{1}$ $\in$ $\mathbf{DA^{1}}(\mathit{S})$. The functor !1 is called the motivic Picard functor. We then compute !1 in an important special case.
In Chapter 3.4, we de_ne a $\mathit{t}$-structure $\mathit{t_{MM,1}}$ on $\mathbf{DA}(\mathit{S})$. We study some properties of $\mathit{t_{MM,1}}$, which together suggest, that $\mathit{t_{MM,1}}$ is the restriction to $\mathbf{DA}(\mathit{S})$ of the conjectural motivic $\mathit{t}$- structure on $\mathbf{DA}(\mathit{S})$. We then study further the heart $\mathbf{MM}_{1}(\mathit{S})$ and we connect it to an existing classical theory of 1-motives, namely, Deligne's theory of 1-motives from [Del74].

## Abstract

In this dissertation, we present several new results in the theory of mixed motives. More precisely, we study the case of relative 1-motives and motives of commutative group schemes in the context of the triangulated categories of mixed motives of Voevodsky. Given a scheme $\mathit{S}$, Voevodsky introduced a triangulated category $\mathbf{DA}(\mathit{S})$ of mixed motives over $\mathit{S}$. This category is built and studied with the hope that it is a suitable approximation to the derived category of a conjectural abelian category$\mathbf{MM}(\mathit{S})$ of mixed motives over S. The expectation is that $\mathbf{DA}(\mathit{S})$ carries a motivic t-structure whose heart would be this conjectural abelian cate- gory. One can construct unconditionally a \six operations formalism" [Ayo07a] [Ayo07b] for the system of categories $\mathbf{DA}(\mathit{-})$ and realisation functors to derived categories of systems of coefficients (constructible sheaves, l-adic sheaves) [Ayo10b] [Ayo14a] which are compatible with the classical theory of the six operations in the Betti and ℓ-adic contexts, so that $\mathbf{DA}(\mathit{S})$ already behaves to a large extend like a \derived category of motivic sheaves" . Important examples of motives in $\mathbf{DA}(\mathit{S})$ are constructed out of commutative group schemes over S, and almost all our results are concerned with these. Let $\mathit{G/S}$ be such a smooth commutative group scheme. Then we have two natural motives built from $\mathit{G}$: first, $\Sigma^{\infty}\mathit{G}_{\mathbb{Q}}$ $\in$ $\mathbf{DA}(\mathit{S})$, which is $\mathit{G}$ seen as a sheaf of $\mathbb{Q}$-vector spaces", and $\mathit{M_{S}(G}$, that is, the homological motive of $\mathit{G}$ as an $\mathit{S}$-scheme. In Chapter 2, written in collaboration with Giuseppe Ancona and Annette Huber, we compare the two, and we prove the following canonical \Küunneth decomposition" of the motive $\mathit{M_{S}(G}$In this dissertation, we present several new results in the theory of mixed motives. More precisely, we study the case of relative 1-motives and motives of commutative group schemes in the context of the triangulated categories of mixed motives of Voevodsky. Given a scheme $\mathit{S}$, Voevodsky introduced a triangulated category $\mathbf{DA}(\mathit{S})$ of mixed motives over $\mathit{S}$. This category is built and studied with the hope that it is a suitable approximation to the derived category of a conjectural abelian category$\mathbf{MM}(\mathit{S})$ of mixed motives over S. The expectation is that $\mathbf{DA}(\mathit{S})$ carries a motivic t-structure whose heart would be this conjectural abelian cate- gory. One can construct unconditionally a \six operations formalism" [Ayo07a] [Ayo07b] for the system of categories $\mathbf{DA}(\mathit{-})$ and realisation functors to derived categories of systems of coefficients (constructible sheaves, l-adic sheaves) [Ayo10b] [Ayo14a] which are compatible with the classical theory of the six operations in the Betti and ℓ-adic contexts, so that $\mathbf{DA}(\mathit{S})$ already behaves to a large extend like a \derived category of motivic sheaves" . Important examples of motives in $\mathbf{DA}(\mathit{S})$ are constructed out of commutative group schemes over S, and almost all our results are concerned with these. Let $\mathit{G/S}$ be such a smooth commutative group scheme. Then we have two natural motives built from $\mathit{G}$: first, $\Sigma^{\infty}\mathit{G}_{\mathbb{Q}}$ $\in$ $\mathbf{DA}(\mathit{S})$, which is $\mathit{G}$ seen as a sheaf of $\mathbb{Q}$-vector spaces", and $\mathit{M_{S}(G}$, that is, the homological motive of $\mathit{G}$ as an $\mathit{S}$-scheme. In Chapter 2, written in collaboration with Giuseppe Ancona and Annette Huber, we compare the two, and we prove the following canonical "Künneth decomposition" of the motive $\mathit{M_{S}(G)}$.

$\mathit{M}_{\mathit{S}}(\mathit{G})$ $\tilde{\longrightarrow}$ $\left( \bigoplus_{\mathit{n}\ge0}^{kd(\mathit{G/S})} Sym^{\mathit{n}}\Sigma^{\infty}\mathit{G_{\mathbb{Q}}} \right)$ $\otimes$ $\mathit{M}_{\mathit{S}}(\mathrm{\pi}_0(\mathit{G/S}))$.

From this result, we can expect that the motive $\Sigma^{\infty}\mathit{G}_{\mathbb{Q}}$ should be a prime example of a relative homological 1-motive, that is, a motive in the triangulated subcategory generated by the relative homology of curves over $\mathit{S}$. This motivates the systematic study of this subcategory. In Chapter 3.1, we define more generally for every $\mathit{n}\in \mathbb{N}$ the category in $\mathbf{DA_{n}}(\mathit{S})$of homological n- motives (resp. $\mathbf{DA^{n}}(\mathit{S})$ of cohomological n-motives) as the full triangulated subcategory of $\mathbf{DA}(\mathit{S})$which is generated by homological motives of smooth (resp. cohomological motives of projective) S-schemes of relative dimension less than n. We study their general properties and their behaviour under the six operations. Then we show in Chapter 3.2 that $\Sigma^{\infty}\mathit{G}_{\mathbb{Q}}$ lies indeed in$\mathbf{DA}(\mathit{S})$. in Chapter 3.3, we de_ne for any cohomological motive$\mathit{S}$ its \1-motivic approximation" $\mathit{\omega}^{1}$ $\in$ $\mathbf{DA^{1}}(\mathit{S})$. The functor !1 is called the motivic Picard functor. We then compute !1 in an important special case.
In Chapter 3.4, we de_ne a $\mathit{t}$-structure $\mathit{t_{MM,1}}$ on $\mathbf{DA}(\mathit{S})$. We study some properties of $\mathit{t_{MM,1}}$, which together suggest, that $\mathit{t_{MM,1}}$ is the restriction to $\mathbf{DA}(\mathit{S})$ of the conjectural motivic $\mathit{t}$- structure on $\mathbf{DA}(\mathit{S})$. We then study further the heart $\mathbf{MM}_{1}(\mathit{S})$ and we connect it to an existing classical theory of 1-motives, namely, Deligne's theory of 1-motives from [Del74].

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Item Type: Dissertation Ayoub Joseph, Kresch Andrew, Okonek Christian 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2015 03 Mar 2016 07:17 22 Sep 2016 11:45 100 http://www.recherche-portal.ch/ZAD:default_scope:ebi01_prod010592871