Several definitions of projective varieties
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I am working on projective varieties and I am a little bit lost. Considering a local ring (in general a field) I think that $P^n(R)$ is the set of tuples $(x_0,...,x_n)$ with $x_i in R$ and some $x_j$ invertible modulo the equivalence relation:
$(x_0,....,x_n) equiv (y_0,...,y_n)$ means $exists , alpha in R$ invertible such that $x_i = alpha , y_i$ with $ 1 leq i leq n$. I am working on Algebraic Geometry $1$ realised by Ulrich Gortz and Torsten Wedhorn here a link of this book. I am trying to realise exercise $4.6$ (page $115$) whose aim is to prove this fact. For me it was a definition of a projective space over a local ring, so I am confused. Did I miss something? Is there another definition I don't know?
Thanks!
geometry algebraic-geometry
asked Nov 28 '18 at 22:01
user1123313131user1123313131
13
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$begingroup$
I am working on projective varieties and I am a little bit lost. Considering a local ring (in general a field) I think that $P^n(R)$ is the set of tuples $(x_0,...,x_n)$ with $x_i in R$ and some $x_j$ invertible modulo the equivalence relation:
$(x_0,....,x_n) equiv (y_0,...,y_n)$ means $exists , alpha in R$ invertible such that $x_i = alpha , y_i$ with $ 1 leq i leq n$. I am working on Algebraic Geometry $1$ realised by Ulrich Gortz and Torsten Wedhorn here a link of this book. I am trying to realise exercise $4.6$ (page $115$) whose aim is to prove this fact. For me it was a definition of a projective space over a local ring, so I am confused. Did I miss something? Is there another definition I don't know?
Thanks!
geometry algebraic-geometry
asked Nov 28 '18 at 22:01
user1123313131user1123313131
13
$endgroup$
add a comment |
$begingroup$
I am working on projective varieties and I am a little bit lost. Considering a local ring (in general a field) I think that $P^n(R)$ is the set of tuples $(x_0,...,x_n)$ with $x_i in R$ and some $x_j$ invertible modulo the equivalence relation:
$(x_0,....,x_n) equiv (y_0,...,y_n)$ means $exists , alpha in R$ invertible such that $x_i = alpha , y_i$ with $ 1 leq i leq n$. I am working on Algebraic Geometry $1$ realised by Ulrich Gortz and Torsten Wedhorn here a link of this book. I am trying to realise exercise $4.6$ (page $115$) whose aim is to prove this fact. For me it was a definition of a projective space over a local ring, so I am confused. Did I miss something? Is there another definition I don't know?
Thanks!
geometry algebraic-geometry
asked Nov 28 '18 at 22:01
user1123313131user1123313131
13
$endgroup$
I am working on projective varieties and I am a little bit lost. Considering a local ring (in general a field) I think that $P^n(R)$ is the set of tuples $(x_0,...,x_n)$ with $x_i in R$ and some $x_j$ invertible modulo the equivalence relation:
$(x_0,....,x_n) equiv (y_0,...,y_n)$ means $exists , alpha in R$ invertible such that $x_i = alpha , y_i$ with $ 1 leq i leq n$. I am working on Algebraic Geometry $1$ realised by Ulrich Gortz and Torsten Wedhorn here a link of this book. I am trying to realise exercise $4.6$ (page $115$) whose aim is to prove this fact. For me it was a definition of a projective space over a local ring, so I am confused. Did I miss something? Is there another definition I don't know?
Thanks!
geometry algebraic-geometry
geometry algebraic-geometry
asked Nov 28 '18 at 22:01
user1123313131user1123313131
13
asked Nov 28 '18 at 22:01
user1123313131user1123313131
13
edited Nov 28 '18 at 22:11
user1123313131
asked Nov 28 '18 at 22:01
user1123313131user1123313131
13
asked Nov 28 '18 at 22:01
user1123313131user1123313131
13
asked Nov 28 '18 at 22:01
user1123313131user1123313131
13
13
add a comment |
add a comment |
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