View source for Projective spectrum of a ring

A [[Scheme|scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753601.png" /> associated with a graded ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753602.png" /> (cf. also [[Graded module|Graded module]]). As a set of points, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753603.png" /> is a set of homogeneous prime ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753604.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753605.png" /> does not contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753606.png" />. The topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753607.png" /> is defined by the following basis of open sets: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753608.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536010.png" />. The structure sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536011.png" /> of the locally ringed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536012.png" /> is defined on the basis open sets as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536013.png" />, that is, the subring of the elements of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536014.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536015.png" /> of fractions with respect to the multiplicative system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536016.png" />.

The most important example of a projective spectrum is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536017.png" />. The set of its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536018.png" />-valued points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536019.png" /> for any field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536020.png" /> is in natural correspondence with the set of points of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536021.png" />-dimensional projective space over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536022.png" />.

If all the rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536023.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536024.png" />-modules are spanned by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536025.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536026.png" /> terms), then an additional structure is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536027.png" />. Namely, the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536028.png" /> and the units <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536029.png" /> determine a Čech <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536030.png" />-cocycle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536031.png" /> to which an [[Invertible sheaf|invertible sheaf]], denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536032.png" />, corresponds. The symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536033.png" /> usually denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536034.png" />-th tensor power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536036.png" />. There exists a canonical homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536037.png" />, indicating the geometric meaning of the grading of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536038.png" /> (see [[#References|[1]]]). If, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536039.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536040.png" /> corresponds to a sheaf of hyperplane sections in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536041.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) {{MR|0209285}} {{ZBL|0187.42701}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck, "Eléments de géometrie algebrique" ''Publ. Math. IHES'' , '''1–4''' (1960–1967) {{MR|0238860}} {{MR|0217086}} {{MR|0199181}} {{MR|0173675}} {{MR|0163911}} {{MR|0217085}} {{MR|0217084}} {{MR|0163910}} {{MR|0163909}} {{MR|0217083}} {{MR|0163908}} {{ZBL|0203.23301}} {{ZBL|0144.19904}} {{ZBL|0135.39701}} {{ZBL|0136.15901}} {{ZBL|0118.36206}} </TD></TR></table>



====Comments====
See also [[Projective scheme|Projective scheme]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>