We show that the cohomology ring of a quiver Grassmannian asssociated with a rigid quiver representation has property (S): there is no odd cohomology and the cycle map is an isomorphism; moreover, its Chow ring admits explicit generators defined over any field. From this we deduce the polynomial point count property. By restricting the quiver to finite or affine type, we are able to show a much stronger assertion: namely, that a quiver Grassmannian associated to an indecomposable (not necessarily rigid) representation admits a cellular decomposition. As a corollary, we establish a cellular decomposition for quiver Grassmannians associated with representations with rigid regular part. Finally, we study the geometry behind the cluster multiplication formula of Caldero and Keller, providing a new proof of a slightly more general result.

中文翻译：

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quiver Grassmannians 的单元分解和上同调代数性

我们证明了与刚性箭袋表示相关联的箭袋格拉斯曼的上同调环具有性质 (S)：不存在奇上同调且循环图是同构；此外，它的 Chow 环允许在任何字段上定义显式生成器。由此我们推导出多项式点计数属性。通过将 quiver 限制为有限或仿射类型，我们能够展示一个更强的断言：即，与不可分解（不一定是刚性）表示相关的 quiver Grassmannian 允许细胞分解。作为推论，我们为与刚性规则部分的表示相关的颤抖 Grassmannians 建立细胞分解。最后，我们研究了 Caldero 和 Keller 的簇乘法公式背后的几何结构，为稍微更一般的结果提供了新的证明。