Let $(X,\mathcal O_X)$ be a ringed space and $\mathcal I$ a sheaf of ideals of $\mathcal O_X$, is the presheaf defined by $U\mapsto (\mathcal I(U))^2$ a sheaf of ideals of $\mathcal O_X$?
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See https://math.stackexchange.com/questions/2679258/product-of-sheaf-of-ideals – Watson Aug 19 '19 at 06:44
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See as well https://mathoverflow.net/questions/422670/is-the-ideal-product-presheaf-a-sheaf-do-we-have-any-reasons-to-believe-it-will – Elías Guisado Villalgordo May 18 '22 at 16:10