The Clausen-Scholze theory of condensed mathematics offers an abelian category with enough projective objects that embraces the study of arbitrary locally compact (and Hausdorff) groups. The behaviour of the tensor product is managed by restricting to a subcategory of solid abelian groups within the category of condensed abelian groups and there is a solidification functor that is left adjoint to this inclusion. Putting my first toe in the water, I am restricting to F\_p vector spaces where p is prime: in other words, condensed abelian groups of prime exponent. The Clausen-Scholze theory already provides sufficient in this special case to be able to solve problems that were inaccessible with classical Galois cohomology. My question: is every condensed vector space over a finite field automatically solid? And if not, exactly what would be the advantage of solidification in this context?



Are condensed vector spaces over finite fields always solid?
Peter Scholze's comment gives a good answer to the main direct question and tells us that the condensed mod p vector space with basis a compact Hausdorff space S is solid if and only if S is finite. The advantages of solidification are going to become more apparent as we use the condensed maths more widely. Solidification is particularly important when using tensor product. One situation where I believe there is no need to invoke solidification is when tensoring a solid vector space (over F\_p) with a finite vector space: such a tensor is automatically solid from the get-go.


