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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 19th 2014

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?

• CommentRowNumber2.
• CommentAuthorZhen Lin
• CommentTimeAug 19th 2014

$CRng$ 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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeAug 20th 2014
• (edited Aug 20th 2014)

That is a good point, thanks. I have added that remark to rng.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeAug 20th 2014

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.

• CommentRowNumber5.
• CommentAuthorzskoda
• CommentTimeAug 20th 2014

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.

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeAug 20th 2014

Re #5: I think that’s a good idea. Then rng can redirect to that.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeAug 20th 2014

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.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeAug 21st 2014
• (edited Aug 21st 2014)

• 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)

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeAug 22nd 2014

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:

$E_{\infty}Ring^{nonunital} \simeq E_\infty Ring_{/\mathbb{S}} \,.$
• CommentRowNumber10.
• CommentAuthorTobyBartels
• CommentTimeAug 22nd 2014
• (edited Aug 22nd 2014)

I am not unhappy. But some duplicated discussion on terminology was left at nonunital ring, which I have fixed.