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\newtheorem{teo}[subsection]{Theorem}
\newtheorem{prop}[subsection]{Proposition}
\newtheorem{cor}[subsection]{Corollary}
\newtheorem{lem}[subsection]{Lemma}
\newtheorem{ex}[subsection]{Example}

\begin{document}

\section{Preliminaries}

\subsection{Symmetric powers \label{sympow}} Let $\mathcal{C}$ be a $\mathbb{Q}$-linear Karoubian additive symmetric monoidal category. Let $n$ be a nonegative integer, and let $S_n$ be the group of permutations of the set of integers from $1$ to $n$. For any object $M$ of $\mathcal{C}$, the group $S_n$ acts on the left on $M^{\otimes n}$. We define $\mathrm{Sym}^{n}(M)$ as the image of the idempotent $p_n = \frac{1}{n!} \sum_{\sigma \in S_n} \sigma$ of $M^{\otimes n}$. We have
$$
\mathrm{dim}(\mathrm{Sym}^n(M)) = \mathrm{Tr}(p_n) =  \frac{1}{n!} \sum_{\sigma \in S_n} \mathrm{Tr}(\sigma | M^{\otimes n} ) = \frac{1}{n!} \prod_{0 \leq j <n} (\mathrm{dim(M)}+ j).
$$

%Let $\mathcal{C}$ For any object $M$ of $\mathcal{C}$, the graded commutative ring 
%$$
%\mathrm{Sym}^{\bullet}(M) = \bigoplus_{n \in \mathbb{N}} \mathrm{Sym}^{n}(M),
%$$
%is defined as largest commutative quotient of the tensor algebra $T(M) = \bigoplus_{n \in \mathbb{N}} M^{\otimes n}$.

\subsection{Exterior powers \label{extpow}} Let $\mathcal{C}$ and $n$ be as in \ref{sympow}. For any object $M$ of $\mathcal{C}$, we define $\Lambda^{n}(M)$ as the image of the idempotent $a_n = \frac{1}{n!} \sum_{\sigma \in S_n} \varepsilon(\sigma) \sigma$ of $M^{\otimes n}$, where $\varepsilon : S_n \rightarrow \{ \pm 1 \}$ is the signature homomorphism. We have
$$
\mathrm{dim}(\Lambda^n(M)) = \mathrm{Tr}(a_n) =  \frac{1}{n!} \sum_{\sigma \in S_n} \varepsilon(\sigma) \mathrm{Tr}(\sigma | M^{\otimes n} ) = \frac{1}{n!} \prod_{0 \leq j <n} (\mathrm{dim(M)} - j).
$$

\section{Embedding result}

\subsection{Deligne's condition \label{delcond}} An additive symmetric monoidal category $\mathcal{C}$, in which any object is dualizable, is said to satisfy \textbf{Deligne's condition} if the following assertions hold:
\begin{enumerate}
\item for any object $M$ of $\mathcal{C}$ with $\mathrm{dim}(M) = 0$ then $M = 0$;
\item for any object $M$ of $\mathcal{C}$ and any positive integer $j$, the endomorphism $\mathrm{dim}(M) + j$ is invertible;
\item for any object $M$ of $\mathcal{C}$, there exists an integer $n$ such that $\prod_{0 \leq j <n} (\mathrm{dim(M)} - j) = 0$.
\end{enumerate}

\subsection{Rigid exact symmetric monoidal categories \label{exactcat}} A \textbf{rigid exact} symmetric monoidal category is an additive symmetric monoidal category $\mathcal{C}$, in which any object is dualizable, together with a class $\mathrm{Seq}$ of sequences $M_1 \rightarrow M_2 \rightarrow M_3$ of homomorphisms of $\mathcal{C}$ such that:
\begin{enumerate}
\item the couple $(\mathcal{C}, \mathrm{Seq})$ is an exact category;
\item for any object $M$ of $\mathcal{C}$, the functor $M \otimes -$ sends $\mathrm{Seq}$ to itself;
\item the duality functor from $\mathcal{C}$ to itself sends $\mathrm{Seq}$ to itself.
\end{enumerate}

In particular, the dual of an admissible monomorphism is an admissible epimorphism, and conversely.

\begin{ex} For any ring $R$, the category $\mathrm{Vect}_R$ of finite projective $R$-modules is naturally a rigid exact symmetric monoidal category. Moreover, it satisfies Deligne's condition \ref{delcond} if and only if $R$ is a $\mathbb{Q}$-algebra.
\end{ex}

\begin{teo} \label{teo1} Let $\mathcal{C}$ be a rigid exact symmetric monoidal category which is $\mathbb{Q}$-linear, Karoubian and essentially small. Then the following are equivalent:
\begin{itemize}
\item[(i)] Deligne's condition \ref{delcond} holds for $\mathcal{C}$;
\item[(ii)] there exists a $\mathbb{Q}$-algebra $R$ and a faithful exact monoidal functor from $\mathcal{C}$ to $\mathrm{Vect}_R$.
\end{itemize}
\end{teo}


\section{Splitting lemmas}

Let $\mathcal{C}$ be as in Theorem \ref{teo1}.

\subsection{Faithfulness.} A commutative algebra $A$ in $\mathrm{Ind}(\mathcal{C})$ is said to be \textbf{faithful} (or \textbf{faithfully flat}) if the unit morphism $1 \rightarrow A$ is an admissible monomorphism.

\begin{ex} If $\mathcal{C} = \mathrm{Vect}_R$ for some ring $R$ then $\mathrm{Ind}(\mathcal{C})$ is the symmetric monoidal category of flat $R$-modules by Lazard's theorem [058G]. Thus a commutative algebra in $\mathrm{Ind}(\mathcal{C})$ is simply a flat homomorphism $R \rightarrow A$ of rings. It is faithful as an object of $\mathrm{Ind}(\mathcal{C})$ if and only if $R \rightarrow A$ is injective with flat cokernel, or equivalently if $A$ is a faithfully flat $R$-algebra.
\end{ex}



\begin{prop}\label{lemme2.2} Let $M$ be an object of $\mathcal{C}$ and let $M \xrightarrow[]{\lambda}1$ be an admissible epimorphism. Let us consider the commutative algebra
$$
A = \mathrm{colim}_n \ \mathrm{Sym}^n(M^{\vee}),
$$
in $\mathrm{Ind}(\mathcal{C})$, with transition maps given by the multiplication by $1 \xrightarrow[]{\lambda^{\vee}}M^{\vee}$. Then we have the following assertions:
\begin{enumerate}
\item the epimorphism $A \otimes M \xrightarrow[]{\mathrm{id}_A \otimes v} A$ of $A$-modules admits an $A$-linear section;
\item the unit homomorphism $1 \rightarrow A$ is an admissible monomorphism, i.e. $A$ is faithful.
\end{enumerate}

\end{prop}
\begin{proof}
\begin{enumerate}
\item The canonical morphism $M^\vee \rightarrow A$ yields an $A$-linear morphism $A \rightarrow A \otimes M$, and its composition with $\lambda$ is the identity of $A$ by construction.
\item The composition of the morphism $\mathrm{Sym}^n(M) \rightarrow 1$ induced by $v$ with the projection $M^{\otimes n} \xrightarrow[]{p_n} \mathrm{Sym}^n(M)$ is the admissible epimorphism $\lambda^{\otimes n}$. Thus $\mathrm{Sym}^n(M) \rightarrow 1$ is an admissible epimorphism, and its dual $1 \rightarrow \mathrm{Sym}^n(M^\vee)$ is consequently an admissible monomorphism in $\mathcal{C}$. Hence the colimit $1 \rightarrow A$ is an admissible monomorphism in $\mathrm{Ind}(\mathcal{C})$.
\end{enumerate}
\end{proof}

\begin{cor}\label{lemme2.2bis} Let $M \rightarrow N$ be an admissible epimorphism in $\mathcal{C}$. Then there exists a faithful commutative algebra $A$ in $\mathrm{Ind}(\mathcal{C})$ such that the admissible epimorphism $A \otimes M \rightarrow A \otimes N$ splits.
\end{cor}

\begin{proof} For any commutative algebra $A$ in $\mathrm{Ind}(\mathcal{C})$, the epimorphism $A \otimes M \rightarrow A \otimes N$ splits if and only if the corresponding morphism $A \otimes M \otimes N^{\vee} \rightarrow A$ splits. The morphism $M \otimes N^\vee \rightarrow 1$ is the pullback along the coevaluation $1 \rightarrow N \otimes N^\vee$ of the admissible epimorphism $M \otimes N^\vee \rightarrow N \otimes N^\vee$, hence is itself admissible. The conclusion then follows from Lemma \ref{lemme2.2} applied to $M \otimes N^\vee \rightarrow 1$. 

\end{proof}



\begin{prop}\label{lemme2.1} Let $M$ be an object of $\mathcal{C}$ and let us consider the commutative algebra
$$
A = \mathrm{colim}_n \ \mathrm{Sym}^n(M^{\vee} \oplus M),
$$
in $\mathrm{Ind}(\mathcal{C})$, with transition maps $\mathrm{Sym}^n(M^{\vee} \oplus M) \rightarrow \mathrm{Sym}^{n+2}(M^{\vee} \oplus M)$ given by the multiplication by the coevaluation $1  \xrightarrow[]{\mathrm{coev}} M^{\vee} \otimes M$. Then we have the following assertions:
\begin{enumerate}
\item the $A$-module $A \otimes M$ admits a direct summand which is a free $A$-module of rank $1$;
\item if $\mathrm{dim}(M) + j$ is invertible for any nonnegative integer $j$, then the unit homomorphism $1 \rightarrow A$ is a split monomorphism. In particular, $A$ is faithful.
\end{enumerate}

\end{prop}

\begin{proof}
\begin{enumerate}
\item The canonical morphisms $M^{\vee} \xrightarrow[]{u_0} A$ and $ M \xrightarrow[]{v_0} A$ induce $A$-module homomorphisms $A \xrightarrow[]{u} A \otimes M$ and $A \otimes M \xrightarrow[]{v} A$. The composition $v \circ u$ coincides with the $A$-linearization of the composition
$$
1 \xrightarrow[]{ \mathrm{coev}}  M^\vee \otimes M \xrightarrow[]{ u_0 \otimes v_0} A,
$$
which is simply the unit $1 \rightarrow A$ by construction of $A$. Thus $v \circ u$ is the identity of $A$, and consequently $u$ is a split monomorphism of $A$-modules.

\item The algebra $A$ admits the subalgebra
$$
B = \mathrm{colim}_n \ \mathrm{Sym}^n(M^{\vee}) \otimes \mathrm{Sym}^n( M),
$$
as a direct summand, where the transitions are again given by multiplying by the coevaluation $1  \xrightarrow[]{\mathrm{coev}} M^{\vee} \otimes M$. Let $e_n$ be the composition
$$
\mathrm{Sym}^n(M^{\vee}) \otimes \mathrm{Sym}^n( M) \rightarrow ( M^{\vee})^{\otimes n} \otimes M^{\otimes n} \xrightarrow[]{\mathrm{ev}^{\otimes n}} 1.
$$
Let $d_1$ be the dimension of $M$ and let $d_n = \frac{1}{n!} d_1 (d_1 + 1) \cdots (d_1 + n-1)$ be the dimension of $\mathrm{Sym}^n( M)$. By assumption, each $d_n$ is an invertible endomorphism of the unit $1$. The composition
$$
\mathrm{Sym}^n(M^{\vee}) \otimes \mathrm{Sym}^n( M) \xrightarrow[]{\mathrm{coev}} \mathrm{Sym}^{n+1}(M^{\vee}) \otimes \mathrm{Sym}^{n+1}( M) \xrightarrow[]{d_{n+1}^{-1} e_{n + 1}} 1,
$$
coincides with $d_n^{-1} e_n$, hence the morphisms $(d_n^{-1} e_n)_{n}$ provide a morphism $B \rightarrow 1$, which splits the unit of $B$.
\end{enumerate}
\end{proof}

\begin{cor}\label{lemme2.1bis} Let $M$ be an object of $\mathcal{C}$ and let $d$ be a nonnegative integer such that $\mathrm{dim}(M) + j$ is invertible for any integer $j > -d$. Then there exists a faithful commutative algebra $A$ in $\mathrm{Ind}(\mathcal{C})$ such that the $A$-module $A \otimes M$ admits a direct summand which is a free $A$-module of rank $d$.
\end{cor}

\begin{proof} For $d = 0$, we simply take $A = 1$, the unit object of $\mathcal{C}$. We prove the general case by induction. Let us assume that $d$ is positive and that the assertion is proved for $d - 1$. By Lemma \ref{lemme2.1}, we have a faithful commutative algebra $B$ in $\mathcal{C}$ such that the $B$-module $B \otimes M$ is isomorphic to $B \oplus N$ for some $B$-module $N$. Since $N$ is a direct summand of the dualizable $B$-module $B \otimes N$, it is a dualizable $B$-module as well. The rigid exact symmetric monoidal category of dualizable $B$-modules satisifies the assumptions of Corollary \ref{lemme2.1bis}, and its object $N$ is dualizable with dimension $\mathrm{dim}(N)=\mathrm{dim}(M) - 1$ such that $\mathrm{dim}(N) + j$ is invertible for any integer $j > -d + 1$. By induction there exists a faithful commutative $B$-algebra $A$ in $\mathrm{Ind}(\mathcal{C})$ such that the $A$-module $A \otimes_B N$ admits a free direct summand of rank $d-1$. Then $A$ satisfies the conclusion of Corollary \ref{lemme2.1bis} for $M$.
\end{proof}



\section{Proof of Theorem \ref{teo1}}

Clearly $(ii)$ implies $(i)$ in Theorem \ref{teo1}. Conversely, let $\mathcal{C}$ be as in Theorem \ref{teo1}, and let us assume that Deligne's condition holds for $\mathcal{C}$. Let $M$ be an object of $\mathcal{C}$. By assumption, there exists an integer $n$ and orthogonal idempotents $(e_d)_{d=0}^n$ of $\mathrm{End}(1)$ such that $\mathrm{dim}(M) e_d = d e_d$ and $\sum_d e_d = 1$. By Proposition \ref{lemme2.1bis}, there exists a faithful commutative algebra $A$ in $\mathrm{Ind}(\mathcal{C})$ such that for any integer $d$, the $e_d A$-module $e_d A \otimes e_dM$ is of the form $(e_d A)^{\oplus d} \oplus N_d$. The $e_d A$-module
$$
\Lambda^{d+1}_{e_d A}(e_d A \otimes e_d M) ) = e_d A \otimes \Lambda^{d+1}(e_dM),
$$
contains $\Lambda^d_{e_d A}(e_d A^{\oplus d}) \otimes N \simeq N$ as a direct summand, and vanishes since $\Lambda^{d+1}(e_dM)$ has vanishing dimension. Thus $N$ vanishes, and $e_d A \otimes e_dM$ is isomorphic to $(e_d A)^{\oplus d}$.

Since $\mathcal{C}$ is essentially small, and since the category of faithful algebras in $\mathrm{Ind}(\mathcal{C})$ is cocomplete, we can assume (and we do) that the same algebra $A$ has these properties for all objects $M$ of $\mathcal{C}$. Likewise by Proposition \ref{lemme2.2} there exists a faithful commutative $A$-algebra $B$ in $\mathrm{Ind}(\mathcal{C})$ such that $A \rightarrow B$ is an admissible monomorphism and such that any short exact sequence in $\mathcal{C}$ splits over $B$.

Let $R$ be the ring $\mathrm{Hom}_{\mathrm{Ind}(\mathcal{C})}(1, B )$ and let us consider the additive functor $F$ which sends an object $M$ of $\mathcal{C}$ to the $R$-module $\mathrm{Hom}_{\mathrm{Ind}(C)}(1, B \otimes M)$. Since any short exact sequence of $\mathcal{C}$ splits over $B$, the functor $F$ is exact. Moreover, for any object $M$ of $\mathcal{C}$, with $(e_d)_d$ as above, we have
$$
F(M) = F \left( \oplus_{d=0}^n e_d M \right) = \oplus_{d=0}^n F(e_d M) \simeq \oplus_{d=0}^n F(e_d)R^{\oplus d},
$$
hence $F(M)$ is a finite projective $R$-module. A similar computation shows that the natural transformation $F(M_1) \otimes_R F(M_2) \rightarrow F( M_1 \otimes M_2)$ is an isomorphism. Since we also have $F(1) = R$, the functor $F$ is monoidal.

It remains to prove that $F$ is faithful. Let $M_1,M_2$ be objects of $\mathcal{C}$. The fact that $1 \rightarrow B$ is an admissible monomorphism, together with the left exactness of the functor $\mathrm{Hom}_{\mathrm{Ind}(\mathcal{C})}(1, M_1^{\vee} \otimes M_2 \otimes -)$ on $\mathrm{Ind}(\mathcal{C})$, yields that 
$$
\mathrm{Hom}_{\mathcal{C}}(M_1, M_2) = \mathrm{Hom}_{\mathcal{C}}(1, M_1^{\vee} \otimes M_2) \subseteq \mathrm{Hom}_{\mathrm{Ind}(\mathcal{C})}(1, M_1^{\vee} \otimes M_2 \otimes B),
$$
and the last term, namely $F(M_1^{\vee} \otimes M_2)$, is canonically isomorphic to $\mathrm{Hom}_{\mathrm{Vect}_R}(F(M_1), F(M_2))$ since $F$ is monoidal.





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