Content deleted Content added
uniform formatting of inline formulae |
m →Monoid schemes: link to Affine monoid |
||
Line 83:
The ''spectrum'' of a monoid ''A'', denoted {{nowrap|Spec ''A''}}, is the set of [[prime ideal]]s of ''A''. The spectrum of a monoid can be given a [[Zariski topology]], by defining [[basis (topology)|basic]] [[open set]]s
: <math>U_h = \{\mathfrak{p}\in\text{Spec}A:h\notin\mathfrak{p}\},</math>
for each ''h'' in ''A''. A ''monoidal space'' is a topological space along with a [[sheaf (mathematics)|sheaf]] of multiplicative monoids called the ''structure sheaf''. An ''[[affine monoid]] scheme'' is a monoidal space which is isomorphic to the spectrum of a monoid, and a '''monoid scheme''' is a sheaf of monoids which has an open cover by affine monoid schemes.
Monoid schemes can be turned into ring-theoretic schemes by means of a '''base extension''' [[functor]] <math>-\otimes_{\mathbf{F}_1}\mathbf{Z}</math> that sends the monoid ''A'' to the '''Z'''-module (i.e. ring) '''Z'''[''A'']/{{angle bracket|0<sub>''A''</sub>}}, and a monoid homomorphism {{nowrap|''f'' : ''A'' → ''B''}} extends to a ring homomorphism <math>f_{\mathbf{Z}}:A\otimes_{\mathbf{F}_1}\mathbf{Z}\to B\otimes_{\mathbf{F}_1}\mathbf{Z}</math> which is linear as a '''Z'''-module homomorphism. The base extension of an affine monoid scheme is defined via the formula
| |||