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Let $\mathcal{H}iggs_{ss}$ be the stack of semistable rank two Higgs bundleson $\mathbf{P}^{1}$ with value in $O(n)$ where $n\geq 2$. In this paper we willconstruct the Poincar\'e sheaf $\mathcal{P}$ on$\mathcal{H}iggs_{ss}\times_{H}\mathcal{H}iggs_{ss}$ which is maximalCohen-Macaulay and flat over $\mathcal{H}iggs_{ss}$.

Construction of the Poincar\'e sheaf on the stack of rank two Higgs bundles of $\mathbf{P}^{1}$