TY - JOUR
TI - Homological algebra for commutative monoids
DO - https://doi.org/doi:10.7282/T3N58P2X
PY - 2015
AB - We first study commutative, pointed monoids providing basic definitions and results in a manner similar commutative ring theory. Included are results on chain conditions, primary decomposition as well as normalization for a special class of monoids which lead to a study monoid schemes, divisors, Picard groups and class groups. It is shown that the normalization of a monoid need not be a monoid, but possibly a monoid scheme. After giving the definition of, and basic results for, A-sets, we classify projective A-sets and show they are completely determine by their rank. Subsequently, for a monoid A, we compute K_0 and K_1 and prove the Devissage Theorem for G_0 . With the definition of short exact sequence for A-sets in hand, we describe the set Ext(X,Y ) of extensions for A-sets X,Y and classify the set of square-zero extensions of a monoid A by an A-set X using the Hochschild cosimplicial set. We also examine the projective model structure on simplicial A-sets showcasing the difficulties involved in computing homotopy groups as well as determining the derived category for a monoid. The author defines the category Da(C) of double-arrow complexes for a class of non-abelian categories C and, in the case of A-sets, shows an adjunction with the category of simplicial A-sets.
KW - Mathematics
KW - Algebra, Homological
KW - Monoids
LA - eng
ER -