The spectrum of a well‐generated tensor‐triangulated category.

For a tensor‐triangulated category and any regular cardinal α$\alpha$, we study the frame of α$\alpha$‐localizing tensor ideals and its associated space of points. For a well‐generated category and its frame of localizing tensor ideals, we provide conditions such that the associated space is obtained by refining the topology of the corresponding space for the triangulated subcategory of α$\alpha$‐compact objects. This is illustrated by several known examples for α=ℵ0$\alpha =\aleph _0$, and new spaces arise for α>ℵ0$\alpha >\aleph _0$. [ABSTRACT FROM AUTHOR]