In Section 2 of Nevovar's paper "Beilinson's Conjectures", for a pure motive $M$ of the form $h^i(X)(n)$ where $X$ is a projective smooth variety over $\mathbb{Q}$ and $n$ is an integer such that the weight $w=i-2n$ is negative. Then Deligne's period map is defined to be \begin{equation} \alpha_M:M_B^+ \otimes \mathbb{R} \rightarrow (M_{dR}/F^0 M_{dR}) \otimes \mathbb{R} \end{equation} where $M_B$ is the Betti realisation of $M$, $M_B^+$ is the subspace of $M_B$ on which $\phi_{\infty}$ (induced from the action of conjugation on the complex-valued points $X(\mathbb{C})$ and on $\mathbb{Q}(n)=(2 \pi i)^n \mathbb{Q}$ ) acts as identity, $M_{dR}$ is the de Rham realisation of $M$ and $F^0$ is a subspace of $M_{dR}$ appears in the filtration. This homomorphism is induced from the standard comparison isomorphism $I_\infty$ followed by a quotient.
In the commutative diagram in section 2.2 (top of page 7), it claims that the following sequence is exact \begin{equation} 0 \rightarrow (F^0M_{dR} \oplus M_B^+)\otimes \mathbb{R} \rightarrow (M_B^-(-1) \oplus M_B^+ ) \otimes \mathbb{R} \rightarrow \text{Ker} (\alpha_{M^\vee(1)})^\vee \rightarrow 0 \end{equation} The weight of the pure motive $M^\vee(1)$ is $-w-2$, which might not be negative, but the period map is still defined to be \begin{equation} \alpha_{M^\vee(1)}:M^\vee(1)_B^+ \otimes \mathbb{R} \rightarrow (M^\vee(1)_{dR}/F^0M^\vee(1)_{dR}) \otimes \mathbb{R} \end{equation}
Question: How to prove the above short sequence is exact?
Two lines below the commuative diagram in page 7, it says that
"similarly, $\text{Coker}(\alpha_{M^\vee(1)}) \xrightarrow{\sim} \text{Ker} (\alpha_M)^\vee=0$"
which the author deduces from the commuative diagram. If assume $\text{Coker}(\alpha_{M^\vee(1)})=0$, I know how to show the above short sequence is exact. But without this assumption, I don't know how to prove the the short sequence is exact from the definition of $\alpha_{M^\vee(1)}$ directly!
