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I see your point. I was searching for a geometrical picture for why the determinant is involved. Top exterior powers of finitely generated sub modules corresponds locally to tangent hyperplanes and the transformation between them induced by the orthogonal projection using the bilinear form is preciesly the determinant which is a very reasonable generalization since it corresponds to the volume of a projected unit box. Just as the inner product correspods to the length of a projected unit vector. Does this sound right to you?
My global constructions via maps between exterior powers was motivated by this precise definition. Since both of these agree locally (their localizations agree) Doesn't it mean that my geometric definition is the same as your "local" one?
@DenisNardin Thanks. I'll be satisfied by a construction of a pre-hodge star on a projective finitely presented module of finite rank. By pre-hodge I mean an twisted-hodge star (twisted by $\bigwedge^n M$) such that adding a volume form (in the case of a trivial top exterior power) makes it a hodge star. I'm not interested in the generality just for the sake of it, but rather becasue i'd like to understand it better and stripping away the unnecessary structure is one of the best ways i know.
Hi Qiachou, Lately I find myself reading a lot of your posts and find them really enlightening. What would you rcommend to a novice like myself who relates philosophically with the homotopical(-higher categorical) viewpoint but lacks a deep understanding of the foundations of homotopy theory? Is there a "Modern Homotopy theory demystified" book you can recommend?
Are there any consequences of this outside group theory? Analogous to how non-solvability of the symmetric groups of degree > 4 implies solution by radicals for equations of degree > 4 is impossible.
I must say I found do Carmo's notation to be very confusing. As someone who wants to obtain a deep and precise understanding of the objects at hand i spent a lot of time trying to decipher his short hand notations.. His notation for the "directional derivative" along paths, for example, had me totally lost untli I understood (after asking on MSE) that it's actually a pullback connection. I cannot recommend this book to anyone who seeks a precisely notated exposition.
