Timeline for Relation between the Spec and the Proj of a ring

4 events

when toggle format	what		by	license	comment
Jun 18, 2019 at 13:11	comment	added	user130124		You're welcome, yes there is a grading on $R$ in the case when $R=k[x]$ but when you take Proj R[z] the convention is that you forget the grading that $R$ may or may not have. By definition the grading on $R[z]$ is defined such that all elements in $R\subset R[z]$ are defined to be in degree zero, i.e $R\subset R[z]_0$. The element $z$ is defined to have degree $1$. Thus in the case $R=k[x]$ any element $x^i$ has degree $0$ and $x^i z^j$ has degree $j$.
Jun 18, 2019 at 11:15	vote	accept	User43029
Jun 18, 2019 at 11:13	comment	added	User43029		Thank you! I think I can prove that $Proj~R[z] = Spec~R \otimes_{Spec~k} Proj~k[x] $. My only problem is to see that $x^iz^j \in R[z]_j$, once the grading says that I have $R_i \subset R[z]_i$. Doesn't this mean that $x^i \in R[z]_i$?
Jun 18, 2019 at 10:57	history	answered	user130124	CC BY-SA 4.0