Question on lemma 3.5 of Beilinson's paper, Notes on Absolute Hodge Cohomology

Question

I am reading Beilinson's paper, Notes on Absolute Hodge Cohomology (unfortunately I could find an link to this paper ), and I don't understand lemma 3.5.

For $A$ a Noetherian subring of $\mathbb{R}$ such that $A \otimes \mathbb{Q}$ is a field, let $D^*_H$ be the derived category of an $A$-Hodge complexes whose definition and construction could be found in section 3 of this paper, which admits a functor $\underline{H}:D^*_H \rightarrow \text{MHS}_A$. Then Lemma 3.5 claims that $\underline{H}$ is the cohomological functor of a certain (unique) non-degenerate $t$-structure on $D^*_H$ and the natural inclusion $\text{MHS}_A \hookrightarrow D^*_H$ is equivalence with the heart of $D^*_H$. I don't understand the two claims, could anyone provide a careful explanation?

I apologize for not giving the construction of $D^*_H$ and $\underline{H}$ here, also for not provide a online link. I only find the paper in library.

Answer 1

Basically, the author wants to say that $\underline{H}$ is like the functor of taking 0-th cohomology of a complex. For instance, if $C(A)$ is the derived category of an abelian category $A$ then $C(A)$ inherits a t-structure so that the core is $A$, then taking 0-th cohomology is a functor from $C(A)$ to $A$. Taking $i$-th cohomology can be achieved by first shifting in the derived category and then applying $H^0$. If you need a more precise statement you should read the definition of t-structure.