There exists a rich theory for inner product spaces (i.e. vector spaces with a symmetric nondegenerate bilinear form) over fields, and it can be discussed in the context of local rings and free modules. Furthermore, there exists a very nice result using cohomology in a paper of Jardine

https://www.uwo.ca/math/faculty/jardine/preprints/preprint-coh-inv.pdf

that talks about these objects up to isomorphisms and relates it to the absolute Galois group of a field. This has made me wonder if there exists a “criterion” or existence theorem for such structures on a module over a commutative Unitas ring. For instance, if one has a Noetherian algebra over a field of characteristic prime to two and the module of Kahler differentials is finitely generated projective, then when does it have a symmetric nondegenerate bilinear form? Such conditions ensure that its isomorphic to its dual, so when one works with say a smooth variety, its canonical bundle has 2-torsion, does there exists geometric conditions that ensure there exists such a form? Ideally, it would be nice to have a statement that says under certain assumptions that there exists a nondegenerate symmetric bilinear form on a finitely generated module iff *some condition*. This might be too general, but even something in a particular context would be interesting as it seems this theory seems to become difficult to study when one steps away from certain finiteness conditions like free modules.

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