Decomposition theorem over more general base schemes

Question

The BBDG decomposition theorem says that if $f\colon X \to Y$ is a projective morphism of finite type $\mathbf{C}$-schemes and $X$ is smooth of (pure) dimension $d$ then $\mathbf{R}f_*\mathbf{Q}_\ell[d]\cong \oplus {}^p\mathrm{R}^i f _*(\mathbf{Q}_\ell[d])$ is a direct sum of its perverse cohomology sheaves, and each ${}^p\mathrm{R}^i f _*(\mathbf{Q}_\ell[d])$ is semisimple.

Is it expected that the decomposition theorem holds over more general base schemes? Are there any positive results in this direction? (see the update below)

One issue with this possible generalization is that the semisimplicity part of the decomposition theorem holds only over an algebraically closed field. So, in general, one needs to impose some analogous condition on the base.

I think the statement is already non-trivial (and interesting) for $Y=\mathrm{Spec} \ \mathbf{C}[[T]]$ and $X$ a regular, projective $Y$-scheme.

It seems somewhat plausible that the decomposition theorem is correct for the example above since there is a version of the decomposition theorem for projective morphisms of analytic spaces due to Saito.

Update: As mentioned in the comments, ${}^p\mathrm{R}^i f _*(\mathbf{Q}_\ell[d])$ cannot be always semi-simple over $\mathrm{Spec} \ \mathbf{C}[[T]]$. But it still makes sense to ask if each ${}^p\mathrm{R}^i f _*(\mathbf{Q}_\ell[d])$ is a direct sum of IC-sheaves $\mathrm{IC}_{Z_i}(\mathcal{L}_i)$ for closed subscheme $Z_i\subset Y$ and local systems $\mathcal{L}_i$ on open subschemes $U_i \subset Z_i$.

Answer 1

If $X$ is any excellent scheme with equicodimensional irreducible components, there is a perverse t-structure on $D^b_c(X,\mathbf{Q}_\ell)$ with all expected properties ($\ell$ any prime invertible on $X$). This follows from some results of Gabber, see e.g. sections 2.1-2.2 of this paper.

For $f: Y \to X$ any projective morphism of such schemes, I conjecture that the decomposition theorem holds except without requiring semisimplicity. In other words, one should have $Rf_{\ast} IC_Y \simeq \oplus IC_{Z_i}(L_i)[n_i]$ where the $Z_i \subset X$ are some closed subschemes and $L_i$ is some (possibly non-semisimple) lisse $\mathbf{Q}_\ell$-sheaf on $Z_i^{\mathrm{reg}}$.

This seems to be a very difficult problem. One especially interesting case is when $X=\mathrm{Spec}A$ with $A$ a $K$-affinoid ring in the sense of rigid geometry. In this case, the conjecture amounts to an algebraic variant of Conjecture 4.17 here.