Question about terminology regarding "Grothendieck Group," "Grothendieck Ring," perhaps "Grothendieck field"?

Question

Any commutative monoid $M$ has a "Grothendieck group" associated to it, which is universal in the sense that if some other group $G$ has $M$ embedded in it, it also has the Grothendieck group of $M$.

If $M$ is also a commutative semiring, we can build the Grothendieck group of the additive monoid, and then extend the semiring multiplication to it in a natural way to get a ring. Somewhere along the way I picked up the term "Grothendieck ring" for this, but this term seems much less ubiquitous than "Grothendieck group," and searching on here seems to give a lot of results about a "Grothendieck ring of varieties." From a terminology standpoint, are these somehow all the same type of thing? Or are there different meanings of Grothendieck ring?

Lastly, given this idea of a "Grothendieck ring" associated to a semiring, we can clearly go one step further and build the field of fractions to get a field associated with the original semiring, which I would guess (?) satisfies a similar universal property. Essentially we're just iterating the Grothendieck group method twice. Does this have a name, like "Grothendieck field" or something?

Answer 1

"Grothendieck group" also refers to a different but related construction which takes as input a category $C$ of some sort, typically abelian, and returns as output the free abelian group on isomorphism classes of objects $c \in C$ quotiented by some relations. If we quotient by the relation $[c \oplus d] \sim [c] + [d]$ then we get the Grothendieck group of the commutative monoid given by isomorphism classes of objects in $C$ under direct sum, but another common choice is to quotient by the relation that if $0 \to a \to b \to c \to 0$ is a short exact sequence then $[b] \sim [a] + [c]$; this is not a special case of the Grothendieck group of a monoid. This construction can be used, for example, to define certain flavors of K-theory, which is what Grothendieck used it to do and why it's named after him.

The Grothendieck ring of varieties refers to a related construction where $C$ is the category of varieties over some field and we quotient the free abelian group on isomorphism classes of varieties by the relation $[X \setminus Y] \sim [X] - [Y]$; this has a similar flavor to but is not a special case of the above construction for abelian categories, nor is it a special case of the Grothendieck group of a monoid. You can talk about a ring structure on the Grothendieck group if $C$ has a monoidal structure which distributes over whatever additive structure you're using to define the Grothendieck group; in the case of varieties this is the cartesian product.

I don't think "Grothendieck ring" is standard terminology for the Grothendieck group of (the underlying additive monoid of) a semiring. I would avoid it since the most common example of a Grothendieck ring is not a special case of this construction anyway.

Lastly, we don't have a field of fractions unless our commutative ring is an integral domain, and that will rarely be the case; for example Bjorn Poonen showed that the Grothendieck ring of varieties is not a domain. Nobody appears to use the term "Grothendieck field" at all.

Overall I would avoid the term "Grothendieck group" to refer to the monoid construction because I think it's too confusing, and I would avoid the term "Grothendieck ring" to refer to the semiring construction similarly. I might use "group completion" and "ring completion" instead.