Algebraic K-theory with commutative semirings? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T07:19:52Z http://mathoverflow.net/feeds/question/104796 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104796/algebraic-k-theory-with-commutative-semirings Algebraic K-theory with commutative semirings? Sebastian 2012-08-15T21:47:23Z 2012-08-16T12:30:41Z <p>My question is basically given in the title: Are there any references for a generalization of algebraic K-theory to the scenario where the domain of the functors consists of commutative semirings (provided this has been done at all)?</p> http://mathoverflow.net/questions/104796/algebraic-k-theory-with-commutative-semirings/104820#104820 Answer by Jeffrey Giansiracusa for Algebraic K-theory with commutative semirings? Jeffrey Giansiracusa 2012-08-16T08:37:23Z 2012-08-16T12:30:41Z <p>There are a few papers out there dealing with a slightly different focus - algebraic K-theory over the "field with one element" \$\mathbb{F}_1\$. Some of the frameworks for \$\mathbb{F}_1\$ algebra include semirings, so these papers might contain material that covers what you are interested in as well.</p> <p>The thesis of Nicolai Durov <a href="http://arxiv.org/abs/0704.2030" rel="nofollow">arXiv:0704.2030</a> describes a setting for algebraic geometry over a class of objects more general than rings. These objects are commutative algebraic theories. Commutative rings, commutative semirings and commutative monoids all form full subcategories of commutative algebraic theories. Among the many things Durov does in his thesis, he includes some discussion of algebraic K-theory in the final chapter.</p> <p>The "blueprints" of Oliver Lorscheid (<a href="http://arxiv.org/abs/1103.1745" rel="nofollow">arxiv.org/1103.1745</a> and its sequels) also contain semirings as a full subcategory. According to the abstract, he will eventually get to K-theory of blueprints, which will contain K-theory of semirings as a special case.</p>