Tag Archives: projective module

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 […]