Questions tagged [grothendieck-construction]

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Does this construction define a ring and if so what properties does it have?

Let $a \oplus b := \gcd(a,b)$ for $a,b \in \mathbb{N_0}$. Starting from the observation that for each $a,b,c \in \mathbb{N_0}$ we have: $$a \gcd(b,c) = \gcd(ab,ac)$$ we might translate this to the ...
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Proof check: Calculating the Grothendieck group of $k[x]$ where $k$ is a field

I'd just like some verification of the following proof/computation, thanks in advance. Let $k$ be a field. Claim: $K_0(k[x])$ the Grothendieck group is isomorphic to $\mathbb{Z}$. By the Quillen-...
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Extension of Naturals via Grothendieck Group Construction

So there is a way of extending the set $N$ of natural numbers with 0, equipped with ordinary multiplication, to its Grothendieck group, the group of integers with respect to addition. This group, $Z$,...
In Lang, the example is stated as follows: Let M be a commutative monoid, written additively. There exists a commutative group K(M) and a monoid homomorphism. $$\gamma: M\rightarrow K(M)$$ having ...