Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
added references.
Any book that develops a bit of algebraic geometry of non-unital commutative rings or one that discusses what would be hte major things that break?
$CRng$ is equivalent to $CRing_{/ \mathbb{Z}}$ via the augmentation functor that adjoins a unit to a rng, with quasi-inverse given by the augmentation ideal, so $CRng^{op}$ is equivalent to the category of affine schemes under $\operatorname{Spec} \mathbb{Z}$. I don’t know how helpful that is, though.
That is a good point, thanks. I have added that remark to rng.
Also,I discovered this article and added a pointer to it:
CRings over $\mathbb{Z}$ make one think of $E_\infty$-rings over $\mathbb{S}$, which in turn might make one think of $\infty$-groups over $\mathbb{S}$, of which Sagave showed (here) that they accomodate all $\infty$-groups of units of $E_\infty$-rings and of which Kapranov essentially suggested (here) that they are the source and generalization of all supergeometry.
On the other hand of course rings over $(\mathbb{Z},+,\cdot)$ are freely $(\mathbb{Z},+)$-graded as abelian groups (which was the point above) and so its not clear if there is indeed a useful relation here.
Maybe something more interesting happens for $E_\infty$-rings over $\mathbb{S}$? Need to think about it.
As I was hanging in my student years more around ring theorists than category theorists, I vote for the page title nonunital ring rather than rng. Among my colleagues much more widely spread name, often called just ring.
Re #5: I think that’s a good idea. Then rng can redirect to that.
Very well, I am all in favor of that. I have edited the entry accordingly and in particular I have rewritten the idea-section and its comments on terminology.
Now maybe Toby will be unhappy, though? If so, we can still roll back or maybe find another compromise.
added (here) pointers to
Daniel Quillen, $K_0$ for nonunital rings and Morita invariance, J. Reine Angew. Math., 472:197-217, 1996.
Snigdhayan Mahanta, Higher nonunital Quillen K’-theory, KK-dualities and applications to topological T-dualities, J. Geom. Phys., 61 (5), 875-889, 2011. (pdf)
added to nonunital ring a pointer to prop. 5.2.3.14 in Higher Algebra, which is vastly general but should subsume in particular the equivalence that I was after above:
$E_{\infty}Ring^{nonunital} \simeq E_\infty Ring_{/\mathbb{S}} \,.$I am not unhappy. But some duplicated discussion on terminology was left at nonunital ring, which I have fixed.
1 to 10 of 10