Under an assumption on the existence of $p$ -adic Galois representations, we carry out Taylor–Wiles patching (in the derived category) for the completed homology of the locally symmetric spaces associated with $\operatorname{GL}_{n}$ over a number field. We use our construction, and some new results in non-commutative algebra, to show that standard conjectures on completed homology imply ‘big $R=\text{big}~\mathbb{T}$ ’ theorems in situations where one cannot hope to appeal to the Zariski density of classical points (in contrast to all previous results of this kind). In the case where $n=2$ and  $p$ splits completely in the number field, we relate our construction to the $p$ -adic local Langlands correspondence for  $\operatorname{GL}_{2}(\mathbb{Q}_{p})$ .

在假设存在$p$ -adic Galois 表示的情况下，我们对与$\operatorname{GL}_{n}$相关的局部对称空间的完全同调进行 Taylor-Wiles 修补（在派生类别中）在一个数字字段上。我们使用我们的构造和非交换代数中的一些新结果来证明完全同调的标准猜想在人们无法希望的情况下暗示“大$R=\text{big}~\mathbb{T}$ ”定理诉诸经典点的 Zariski 密度（与以前的所有此类结果相反）。在$n=2$和 $p$在数字字段中完全分裂的情况下，我们将我们的构造与$p$联系起来 - $\operatorname{GL}_{2}(\mathbb{Q}_{p})$的 adic local Langlands 对应关系 。