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Associated to a certain subquotient of the category of representations of the small quantum group at a root of unity is an invertible 4d TQFT known as Crane–Yetter. In fact, the non-semisimplified representation category is invertible in the Morita theory of braided tensor categories: under the cobordism hypothesis this defines a non-semisimple invertible TQFT. Such an invertible theory assigns to a closed 3-manifold a 1-dimensional vector space. In this talk, we define a relative TQFT which can be seen as varying the non-semisimple Crane-Yetter theory over the character stack: it assigns to a closed 3-manifold a line bundle on its G-character stack. We construct this theory by analysing invertibility of a 1-morphism in the Morita theory of symmetric tensor categories, coming from representations of Lusztig's quantum group at a root of unity regarded as a bimodule for Rep(G) using the quantum Frobenius map.