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What is the role of morphisms $A rightarrow mathbb{Z}$ in algebraic geometry?
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From Steve Awodey's Category Theory, p. 37:
Ring homomorphisms $A rightarrow mathbb{Z}$ into the initial ring $mathbb{Z}$ play an equally important role in algebraic geometry.
My question is thus, for someone who knows a reasonable amount of abstract algebra but zero algebraic geometry, what exactly is the role of such morphisms in algebraic geometry?
algebraic-geometry category-theory
asked Feb 2 at 8:46
xuq01xuq01
1084
$endgroup$
add a comment |
$begingroup$
From Steve Awodey's Category Theory, p. 37:
Ring homomorphisms $A rightarrow mathbb{Z}$ into the initial ring $mathbb{Z}$ play an equally important role in algebraic geometry.
My question is thus, for someone who knows a reasonable amount of abstract algebra but zero algebraic geometry, what exactly is the role of such morphisms in algebraic geometry?
algebraic-geometry category-theory
asked Feb 2 at 8:46
xuq01xuq01
1084
$endgroup$
$begingroup$
The category of affine schemes is equivalent (and may even be defined as) $mathbf{CRing}^{op}$. This means that a commutative ring homomorphism $Atomathbb Z$ is a global element (i.e. an arrow from a terminal object) of $mathbf{Aff}$.
$endgroup$
– Derek Elkins
Feb 2 at 8:52
add a comment |
$begingroup$
From Steve Awodey's Category Theory, p. 37:
Ring homomorphisms $A rightarrow mathbb{Z}$ into the initial ring $mathbb{Z}$ play an equally important role in algebraic geometry.
My question is thus, for someone who knows a reasonable amount of abstract algebra but zero algebraic geometry, what exactly is the role of such morphisms in algebraic geometry?
algebraic-geometry category-theory
asked Feb 2 at 8:46
xuq01xuq01
1084
$endgroup$
From Steve Awodey's Category Theory, p. 37:
Ring homomorphisms $A rightarrow mathbb{Z}$ into the initial ring $mathbb{Z}$ play an equally important role in algebraic geometry.
My question is thus, for someone who knows a reasonable amount of abstract algebra but zero algebraic geometry, what exactly is the role of such morphisms in algebraic geometry?
algebraic-geometry category-theory
algebraic-geometry category-theory
asked Feb 2 at 8:46
xuq01xuq01
1084
asked Feb 2 at 8:46
xuq01xuq01
1084
asked Feb 2 at 8:46
xuq01xuq01
1084
asked Feb 2 at 8:46
xuq01xuq01
1084
asked Feb 2 at 8:46
xuq01xuq01
1084
1084
$begingroup$
The category of affine schemes is equivalent (and may even be defined as) $mathbf{CRing}^{op}$. This means that a commutative ring homomorphism $Atomathbb Z$ is a global element (i.e. an arrow from a terminal object) of $mathbf{Aff}$.
$endgroup$
– Derek Elkins
Feb 2 at 8:52
add a comment |
$begingroup$
The category of affine schemes is equivalent (and may even be defined as) $mathbf{CRing}^{op}$. This means that a commutative ring homomorphism $Atomathbb Z$ is a global element (i.e. an arrow from a terminal object) of $mathbf{Aff}$.
$endgroup$
– Derek Elkins
Feb 2 at 8:52
$begingroup$
The category of affine schemes is equivalent (and may even be defined as) $mathbf{CRing}^{op}$. This means that a commutative ring homomorphism $Atomathbb Z$ is a global element (i.e. an arrow from a terminal object) of $mathbf{Aff}$.
$endgroup$
– Derek Elkins
Feb 2 at 8:52
$begingroup$
The category of affine schemes is equivalent (and may even be defined as) $mathbf{CRing}^{op}$. This means that a commutative ring homomorphism $Atomathbb Z$ is a global element (i.e. an arrow from a terminal object) of $mathbf{Aff}$.
$endgroup$
– Derek Elkins
Feb 2 at 8:52
add a comment |
1 Answer
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These correspond to scheme maps $text{Spec},Bbb Ztotext{Spec},A$,
so to points of $text{Spec},A$ defined over $Bbb Z$.
answered Feb 2 at 8:51
Lord Shark the UnknownLord Shark the Unknown
108k1162136
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add a comment |
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1 Answer
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1 Answer
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active
oldest
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$begingroup$
These correspond to scheme maps $text{Spec},Bbb Ztotext{Spec},A$,
so to points of $text{Spec},A$ defined over $Bbb Z$.
answered Feb 2 at 8:51
Lord Shark the UnknownLord Shark the Unknown
108k1162136
$endgroup$
add a comment |
$begingroup$
These correspond to scheme maps $text{Spec},Bbb Ztotext{Spec},A$,
so to points of $text{Spec},A$ defined over $Bbb Z$.
answered Feb 2 at 8:51
Lord Shark the UnknownLord Shark the Unknown
108k1162136
$endgroup$
add a comment |
$begingroup$
These correspond to scheme maps $text{Spec},Bbb Ztotext{Spec},A$,
so to points of $text{Spec},A$ defined over $Bbb Z$.
answered Feb 2 at 8:51
Lord Shark the UnknownLord Shark the Unknown
108k1162136
$endgroup$
These correspond to scheme maps $text{Spec},Bbb Ztotext{Spec},A$,
so to points of $text{Spec},A$ defined over $Bbb Z$.
answered Feb 2 at 8:51
Lord Shark the UnknownLord Shark the Unknown
108k1162136
answered Feb 2 at 8:51
Lord Shark the UnknownLord Shark the Unknown
108k1162136
answered Feb 2 at 8:51
Lord Shark the UnknownLord Shark the Unknown
108k1162136
answered Feb 2 at 8:51
Lord Shark the UnknownLord Shark the Unknown
108k1162136
108k1162136
add a comment |
add a comment |
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$begingroup$
The category of affine schemes is equivalent (and may even be defined as) $mathbf{CRing}^{op}$. This means that a commutative ring homomorphism $Atomathbb Z$ is a global element (i.e. an arrow from a terminal object) of $mathbf{Aff}$.
$endgroup$
– Derek Elkins
Feb 2 at 8:52