# Tag Archives: projective module

## Stable Isomorphisms, Grothendieck Groups: Example

If $a$ and $b$ are two real numbers and $ax = bx$, then we can't conclude that $a = b$ because $x$ may be zero. The same is true for tensor products of modules: if $A$ and $B$ are two left $R$-modules and $X$ is a right $R$-module, then an isomorphism $X\otimes_R A\cong A\otimes_R B$ […]

## Yet Another non-Free Finitely Generated Projective

In the post Examples: Projective Modules that are Not Free, we saw nine examples of projective modules that are not free. On in particular was 'the' submodule $M = \oplus_{i=1}^\infty \mathbb{Z}$ of $\prod_{i=1}^\infty\mathbb{Z}$. Now, that's a cool example to be sure, but the way we showed that $M$ was not free was to cite that […]

## Being Noetherian Is Not Local…Or Is It?

A commutative ring $R$ can be non-Noetherian and have all of its localisations at prime ideals Noetherian, such as the infamous $\prod_{i=1}^\infty \mathbb{Z}/2$. So being Noetherian is not a local property. However, there is an interesting variant of 'local' that does work, which I learnt from Yves Lequain's paper [1]. It goes like this: Theorem. […]

## Wild Spectral Sequences Ep. 4: Schanuel's Lemma

It's time for another installment of Wild Spectral Sequences! We shall start our investigations with a classic theorem useful in many applications of homological algebra called Schanuel's lemma, named after Stephen Hoel Schanuel who first proved it. Consider for a ring $R$ the category of left $R$-modules, and let $A$ be any […]