Zulip Chat Archive

Stream: maths

Topic: Linear independence in torsion modules


view this post on Zulip Eric Wieser (Jan 02 2021 at 23:02):

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

view this post on Zulip Adam Topaz (Jan 02 2021 at 23:04):

It shouldn't.

view this post on Zulip Eric Wieser (Jan 02 2021 at 23:05):

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

view this post on Zulip Adam Topaz (Jan 02 2021 at 23:06):

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

view this post on Zulip Eric Wieser (Jan 02 2021 at 23:07):

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

view this post on Zulip Adam Topaz (Jan 02 2021 at 23:08):

No, but free implies torsion free

view this post on Zulip Adam Topaz (Jan 02 2021 at 23:09):

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

view this post on Zulip Eric Wieser (Jan 02 2021 at 23:09):

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

view this post on Zulip Adam Topaz (Jan 02 2021 at 23:10):

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

view this post on Zulip Adam Topaz (Jan 02 2021 at 23:10):

But again there might be a standard name

view this post on Zulip Eric Wieser (Jan 02 2021 at 23:11):

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

view this post on Zulip Adam Topaz (Jan 02 2021 at 23:12):

What's the statement?

view this post on Zulip Eric Wieser (Jan 02 2021 at 23:13):

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

view this post on Zulip Adam Topaz (Jan 02 2021 at 23:15):

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

view this post on Zulip Eric Wieser (Jan 02 2021 at 23:16):

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

view this post on Zulip Adam Topaz (Jan 02 2021 at 23:16):

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

view this post on Zulip Eric Wieser (Jan 02 2021 at 23:17):

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

view this post on Zulip Adam Topaz (Jan 02 2021 at 23:18):

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)

view this post on Zulip Eric Wieser (Jan 02 2021 at 23:19):

I'm also on mobile, later is fine!

view this post on Zulip Kevin Buzzard (Jan 02 2021 at 23:34):

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

view this post on Zulip Kevin Buzzard (Jan 02 2021 at 23:35):

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!

view this post on Zulip Kevin Buzzard (Jan 02 2021 at 23:36):

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

view this post on Zulip Adam Topaz (Jan 02 2021 at 23:46):

Oh yeah, I was thinking explicitly of Z :)

view this post on Zulip Adam Topaz (Jan 02 2021 at 23:48):

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

view this post on Zulip Adam Topaz (Jan 03 2021 at 00:13):

@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.


Last updated: May 12 2021 at 07:17 UTC