Linear independence in torsion modules

Stream: maths

[Eric Wieser]

Consider the int-module zmod 4 × int. Does mathlib consider the vectors a=(1, 1) and b=(0, 1) docs#linear_independent?

[Adam Topaz]

It shouldn't.

[Eric Wieser]

Obviously 4*a=4*b, so my impression is that it doesn't

[Adam Topaz]

They generate, so if they were independent you would have a free module

[Eric Wieser]

Are "free module" and "torsion-free module" synonyms?

[Adam Topaz]

No, but free implies torsion free

[Adam Topaz]

There's a weaker notion of independence. I don't know what it's called, but these satisfy that.

[Eric Wieser]

That weaker notion is what I'm seeking here, I think

[Adam Topaz]

I've called it quasi-independent in one of my old papers.

[Adam Topaz]

But again there might be a standard name

[Eric Wieser]

And this came up because the tfae statement made by wikipedia about docs#alternating_map appears to be untrue in this module

[Adam Topaz]

What's the statement?

[Eric Wieser]

https://en.m.wikipedia.org/wiki/Alternating_multilinear_map#Definition

[Adam Topaz]

Right. So in this case the wedge product of the module you mentioned with itself is isomorphic to zmod 4.

[Eric Wieser]

As far as I can tell, condition 2 does not imply 3 on that page

[Adam Topaz]

So two vectors a,b are quasi-independent if a \wedge b generates that module

[Eric Wieser]

I didn't understand your previous message but I understand at least the gist of that one

[Adam Topaz]

You can formulate quasi-independence in some way analogous to usual independence, but it's a bit tedious to write down (and I'm on mobile, so I'll have to do it later, if you're still interested)

[Eric Wieser]

I'm also on mobile, later is fine!

[Kevin Buzzard]

Free only implies torsion-free for integral domains, maybe? Or maybe there's a cleverer definition of torsion.

[Kevin Buzzard]

An example of a torsion-free module that isn't free is the rationals as a Z-module. It's not finitely-generated but any two elements are dependent!

[Kevin Buzzard]

Another example is the ideal generated by X and Y in k[X,Y] where k is a field.

[Adam Topaz]

Oh yeah, I was thinking explicitly of Z :)

[Adam Topaz]

I mean with the usual def of torsion free, Z/4 is not torsion free over itself. The definition of torsion free is kind of stupid imo

[Adam Topaz]

@Eric Wieser on second thought the wedge product thing is probably not good enough... E.g. (1,0) and (0,3) would satisfy that condition.