Prove that 𝕀2 is not a flat ℤ-module.

Name: Problem 27, Chapter 3: An Introduction to Homological Algebra
Uploaded: 2022-01-02T07:34:51-08:00
Duration: 3 min 59 s
Channel: Aidan Mcnabb
Description: Prove that 𝕀2 is not a flat ℤ-module.

Key Concepts

Flat Module

A flat module over a ring is one for which the functor of tensoring with that module is exact; that is, it preserves the exactness of short exact sequences. This is an important concept because it ensures that no “unexpected torsion” or other obstructions are introduced in various algebraic constructions, making the module behave nicely with respect to limits and colimits in homological algebra.

Torsion and Torsion-Free Modules

In the special case of modules over a principal ideal domain like the ring of integers ?, flat modules are characterized as those modules that are torsion-free. A module has torsion if there exists a nonzero element of the ring that annihilates a nonzero element of the module. The presence of torsion implies that the module will fail to satisfy the flatness condition because multiplication by a nonzero scalar can kill non-zero elements, distorting the exact sequence structure.

Exactness in Tensor Products

The notion of exactness in tensor products is central to the definition of flat modules. When you tensor an exact sequence of modules by a flat module, the resulting sequence remains exact. This property is used to identify flat modules in practice. In the context of abelian groups viewed as ?-modules, any group that contains torsion elements cannot preserve the exactness of all short exact sequences, and thus is not flat.

Transcript

00:01 Hello and welcome to problem 2 .1 .9.

00:04 Here we're asked to should the set of non -singular 2x2 matrices is not a vector space.

00:10 And we're also asked should the set of singular 2x2 matrices is not a vector space.

00:17 So let's go to the definition of what these terms mean.

00:22 A singular matrix is one with a zero determinants and a non -singular matrix is one with a non -singular matrix is one with a non -signular matrix.

00:30 Zero determinant.

00:32 So we can write this out more formally as a non -singular matrix.

00:36 So a is singular if and only if a equals whoops, the a is some matrix, a, b, c, and d, where a, d minus b, c, and 0 and a, b, c, d are elements of the real numbers.

01:13 Great, so that is a nice formal definition of what we're finding.

01:17 And to make this problem simpler, let's give an example of what a singular matrix is.

01:26 The most simple one is if both of these terms are equal to zero, then this must be singular.

01:32 So let's take the matrix, 0, 0, 2, 3.

01:39 Seems like an easy one.

01:41 This obviously has a zero determinant, because 3 times 0 is 0 minus 2 times 0 is 0.

01:48 All right.

01:51 Now, to show that the set of non -singular matrices is not a subspace, we have to show that it is not closed under addition.

02:02 And one way that we could show that it's not closed is if the sum of two non -singular matrices equals a singular matrix...

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