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I'm reading the Wiki article about the Grothendieck group.

What's the reason we define $[A] - [B] + [C] = 0 $ rather than $[A] + [B] - [C] = 0 $ (or something else) for every exact sequence $0 \to A \to B \to C \to 0$? What is the property we obtain if we define it this way? I suppose it has something to do with exactness at $B$ but what?

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    Consider the short exact sequences $0 \to A \to A \oplus C \to C \to 0$...2012-08-05

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To get a feel for this kind of relation, consider a short exact sequence

$0 \rightarrow V_1 \rightarrow V_2 \rightarrow V_3 \rightarrow 0$

of finite-dimensional vector spaces over a field. What is the relation between their dimensions?

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    @PeteL.Clark I didn't realise the maps were linear, I thought first we're talking about arbitrary maps on sets. Of course it makes more sense that it's morphisms. So then we have $\mathrm{dim}V_2 = \mathrm{dim}V_1 + \mathrm{dim}V_3$? And hence you have answered my question : ) (more or less, I'm not sure how to define dimension for an arbitrary (non-free) $R$-module.2012-08-06