4 problems

Let $X$ be a regular $p$-adic formal scheme, and let $Y_\eta\to X_\eta$ be a proper smooth rigid space with analytically good reduction over $X_\eta$. For each $i\in\mathbb{N}$, an…

Let $K/\mathbf{Q}_p$ be a finite extension, and let $f:X\to Y$ be a projective map of rigid spaces over $K$ with $Y$ qcqs. Let $\mathrm{IC}_{X,\mathbf{Q}_\ell}$ be the intersection…

Let $K/\mathbf{Q}_p$ be a finite extension, let $G_K$ be its absolute Galois group, and let $X$ be a rigid space proper over $K$. Write $IH^i(X_C,\mathbf{Q}_p)$ for its $p$-adic in…

Let $p$ be a prime, let $n$ be a positive integer, let $K$ be a finite extension of $cQ_p$, and let $\overline\rho:G_K\to\operatorname{GL}_n(k_L)$ be a continuous representation. L…