If we call this construction F(W,V) then the construction I’m thinking of is F(W,V)⊗F(W,W)∗. Note that F(W,W) is locally 1-dimensional, so my construction is locally isomorphic to yours but globally twisted. It depends on W via more than just its local dimension. Also note that with this definition we will get that ΛV(V) is always isomorphic to R
Oh right, I was picturing W being free on connected components when I suggested that. Silly me.
If we pick a basis of Rn then it induces a bijection between Hom(Rn,V) and V×⋯×V. So we could define a map Hom(Rn,V)→U to be ‘alternating’ if and only if the corresponding map V×⋯×V→U is alternating. The interesting thing I noticed about this definition is that it doesn’t depend on which basis you pick for Rn. So I have some hope that since this construction isn’t basis dependent, I might be able to write down a basis-independent definition of it.
F is alternating if F(f∘g)=det(g)F(f), right? So if we’re willing to accept kludgy definitions of determinant in the process of defining ΛW(V), then we’re all set, and if not, then we’ll essentially need another way to define determinant for projective modules because that’s equivalent to defining an alternating map?
if not, then we’ll essentially need another way to define determinant for projective modules because that’s equivalent to defining an alternating map?
There’s a lot of cases in mathematics where two notions can be stated in terms of each other, but it doesn’t tell us which order to define things in.
The only other thought I have is that I have to use the fact that W is projective and finitely generated. This is equivalent to W being dualisable. So the definition is likely to use W∗ somewhere.
Oh right, I was picturing W being free on connected components when I suggested that. Silly me.
F is alternating if F(f∘g)=det(g)F(f), right? So if we’re willing to accept kludgy definitions of determinant in the process of defining ΛW(V), then we’re all set, and if not, then we’ll essentially need another way to define determinant for projective modules because that’s equivalent to defining an alternating map?
There’s a lot of cases in mathematics where two notions can be stated in terms of each other, but it doesn’t tell us which order to define things in.
The only other thought I have is that I have to use the fact that W is projective and finitely generated. This is equivalent to W being dualisable. So the definition is likely to use W∗ somewhere.