Kernel elements for the Grothendieck group map of a commutative monoid - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:53:23Z http://mathoverflow.net/feeds/question/119901 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119901/kernel-elements-for-the-grothendieck-group-map-of-a-commutative-monoid Kernel elements for the Grothendieck group map of a commutative monoid Tom LaGatta 2013-01-26T01:15:59Z 2013-01-27T04:18:53Z <p>This is just a nomenclature question. Let $T$ be a commutative monoid, and let <code>$T^*$</code> be its Grothendieck group. That is, $T^* \cong T \times T \ / \sim$, where $(s,s') \sim (t, t')$ if $s+t'+e = s'+t+e$ for some $e \in T$. </p> <p>Let $i : T \to T^*$ be the natural inclusion map, where $i(t) = [(t,0)]$, and let $K$ denote the kernel of this map. </p> <p>Does the kernel $K \subseteq T$ have a common name in the literature? What are elements $k \in K$ called?</p> http://mathoverflow.net/questions/119901/kernel-elements-for-the-grothendieck-group-map-of-a-commutative-monoid/119928#119928 Answer by Martin Brandenburg for Kernel elements for the Grothendieck group map of a commutative monoid Martin Brandenburg 2013-01-26T10:16:50Z 2013-01-26T10:16:50Z <p>I don't know a name for $K$ either, but here is a suggestion.</p> <p>The correct definition of $\sim$ is $(a,b) \sim (c,d) \Leftrightarrow \exists e \in T ~ (a+d+e=b+c+e)$. In particular $(a,b) \sim (0,0) \Leftrightarrow \exists e \in T : a+e=b+e$. I would suggest to call such elements $a,b$ <em>stably equal</em>. Remark that this coincides with the terminology <em>stably isomorphic</em> from topological $K$-theory (where $T$ is the commutative monoid of $\cong$-classes of vector bundles on a space $X$ and one usually takes w.l.o.g. $e$ to be the class of a trivial vector bundle when $X$ is paracompact). In particular, $a \in K \Leftrightarrow \exists e \in T : a+e=e$, in which case one may say that $a$ is <em>stably zero</em>.</p>