I think the good abstract way to think about the determinant is in terms of induced maps on the top exterior power. If you have an n dimensional vector space V and an endomorphism L:V→V, this induces a map ∧nV→∧nV, and since ∧nV is always one-dimensional this map must be of the form v→kv for some scalar k in the ground field. It’s this k that is the determinant of L.
This is indeed more fundamental than the concept of volume. We can interpret exterior powers as corresponding to volume if we’re working over a local field, for example, but actually the concept of exterior power generalizes far beyond this special case. This is why the determinant still preserves its nice properties even if we work over an arbitrary commtuative ring; since such rings still have exterior powers behaving in the usual way.
I didn’t present it like this in this post because it’s actually not too easy to introduce the concept of “exterior power” without the post becoming too abstract.
This is close to one thing I’ve been thinking about myself. The determinant is well defined for endomorphisms on finitely-generated projective modules over any ring. But the ‘top exterior power’ definition doesn’t work there because such things do not have a dimension. There are two ways I’ve seen for nevertheless defining the determinant.
View the module as a sheaf of modules over the spectrum of the ring. Then the dimension is constant on each connected component, so you can take the top exterior power on each and then glue them back together.
Use the fact that finitely-generated projective modules are precisely those which are the direct summands of a free module. So given an endomorphism A:V→V you can write V⊕V′=Rn and then define det(A)=det(A⊕idV′).
These both give the same answer. However, I don’t like the first definition because it feels very piecemeal and nonuniform, and I don’t like the second because it is effectively picking a basis. So I’ve been working on by own definition where instead of defining Λn(V) for natural numbers n we instead define ΛW(V) for finitely-generated projective modules W. Then the determinant is defined via ΛV(V).
I’m curious about this. I can see a reasonable way to define ΛW(V) in terms of sheaves of modules over Spec(R): Over each connected component, W has some constant dimension n, so we just let ΛW(V) be Λn(V) over that component. But it sounds like you might not like this definition, and I’d be interested to know if you had a better way of defining ΛW(V) (which will probably end up being equivalent to this). [Edit: Perhaps something in terms of generators and relations, with the generators being linear maps W→V?]
I’m curious about this. I can see a reasonable way to define ΛW(V) in terms of sheaves of modules over Spec(R): Over each connected component, W has some constant dimension n, so we just let ΛW(V) be Λn(V) over that component.
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.
But it sounds like you might not like this definition,
Right. I’m hoping for a simple definition that captures what the determinant ‘really means’ in the most general case. So it would be nice if it could be defined with just linear algebra without having to bring in the machinery of the spectrum.
and I’d be interested to know if you had a better way of defining ΛW(V) (which will probably end up being equivalent to this).
I’m still looking for a nice definition. Here’s what I’ve got so far.
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. Then it would apply equally well with Rn replaced with W, whereupon we can define ΛW(V) as the universal alternating map out of Hom(W,V).
[Edit: Perhaps something in terms of generators and relations, with the generators being linear maps W→V?]
Yeah exactly. That’s probably a simpler way to say what I was describing above. One embarrassing thing is that I don’t even know how to describe the simplest relations, i.e. what the0s should be.
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.
For what it’s worth, I strongly agree (1) that exterior algebra is the Right Way to think about determinants, conditional on there being other reasons in your life for knowing exterior algebra, and (2) that doing it that way in this post would have had too much overhead.
I think the good abstract way to think about the determinant is in terms of induced maps on the top exterior power. If you have an n dimensional vector space V and an endomorphism L:V→V, this induces a map ∧nV→∧nV, and since ∧nV is always one-dimensional this map must be of the form v→kv for some scalar k in the ground field. It’s this k that is the determinant of L.
This is indeed more fundamental than the concept of volume. We can interpret exterior powers as corresponding to volume if we’re working over a local field, for example, but actually the concept of exterior power generalizes far beyond this special case. This is why the determinant still preserves its nice properties even if we work over an arbitrary commtuative ring; since such rings still have exterior powers behaving in the usual way.
I didn’t present it like this in this post because it’s actually not too easy to introduce the concept of “exterior power” without the post becoming too abstract.
This is close to one thing I’ve been thinking about myself. The determinant is well defined for endomorphisms on finitely-generated projective modules over any ring. But the ‘top exterior power’ definition doesn’t work there because such things do not have a dimension. There are two ways I’ve seen for nevertheless defining the determinant.
View the module as a sheaf of modules over the spectrum of the ring. Then the dimension is constant on each connected component, so you can take the top exterior power on each and then glue them back together.
Use the fact that finitely-generated projective modules are precisely those which are the direct summands of a free module. So given an endomorphism A:V→V you can write V⊕V′=Rn and then define det(A)=det(A⊕idV′).
These both give the same answer. However, I don’t like the first definition because it feels very piecemeal and nonuniform, and I don’t like the second because it is effectively picking a basis. So I’ve been working on by own definition where instead of defining Λn(V) for natural numbers n we instead define ΛW(V) for finitely-generated projective modules W. Then the determinant is defined via ΛV(V).
I’m curious about this. I can see a reasonable way to define ΛW(V) in terms of sheaves of modules over Spec(R): Over each connected component, W has some constant dimension n, so we just let ΛW(V) be Λn(V) over that component. But it sounds like you might not like this definition, and I’d be interested to know if you had a better way of defining ΛW(V) (which will probably end up being equivalent to this). [Edit: Perhaps something in terms of generators and relations, with the generators being linear maps W→V?]
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.
Right. I’m hoping for a simple definition that captures what the determinant ‘really means’ in the most general case. So it would be nice if it could be defined with just linear algebra without having to bring in the machinery of the spectrum.
I’m still looking for a nice definition. Here’s what I’ve got so far.
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. Then it would apply equally well with Rn replaced with W, whereupon we can define ΛW(V) as the universal alternating map out of Hom(W,V).
Yeah exactly. That’s probably a simpler way to say what I was describing above. One embarrassing thing is that I don’t even know how to describe the simplest relations, i.e. what the 0s should be.
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.
For what it’s worth, I strongly agree (1) that exterior algebra is the Right Way to think about determinants, conditional on there being other reasons in your life for knowing exterior algebra, and (2) that doing it that way in this post would have had too much overhead.