Mixing
Why is there a correspondence between $ |text{Proj};S| - V(f) $ and $ text{Spec}; (S_{f})_{0}$?
$begingroup$
Forgive me for the length of this post, but I feel this question is deserving of some detail.
Suppose $ S $ is a positively graded $ A$-algebra, where $ A $ is a ring. That is $ S = bigoplus_{i=0}^{infty} S_{i}, $ where $ S_{i} cdot S_{j} subset S_{i+j}. $
$f $ is a homogeneous element of $ S $ of degree 1, that is, $ f in S_{1},$ and $ V(f) $ denotes the set of elements of $ text{Proj};S $ which do not contain $ f $.
I am reading Eisenbud & Harris' "Geometry of Schemes". In Exercise III-6(a), the authors make the following statement:
The intersection $ (I cdot S[f^{-1}]) cap S[f^{-1}]_{0}) $ is generated by elements obtained by choosing a set of homogeneous generators of $ I $ and multiplying them by the appropriate negative powers of $ f.$
I think I understand this statement intuitively. But they go on to infer the correspondence from here, and proceed give the following hint concerning the manner of constructing the maps:
The correspondence is given by taking a prime $ mathfrak{p} $ of $ S[f^{-1}] $ to $ mathfrak{p} = mathfrak{p} cap S[f^{-1}]_{0} $ and taking the prime $ mathfrak{q} $ of $ S[f^{-1}] _{0} $ to $ mathfrak{q}S[f^{-1}].$
My difficulty boils down to not knowing how to use this hint to establish the result explicitly/mechanically.
algebraic-geometry schemes projective-schemes
asked Jan 8 at 15:59
Overwhelmed AG ApprenticeOverwhelmed AG Apprentice
458
$endgroup$
|
show 4 more comments
$begingroup$
Forgive me for the length of this post, but I feel this question is deserving of some detail.
Suppose $ S $ is a positively graded $ A$-algebra, where $ A $ is a ring. That is $ S = bigoplus_{i=0}^{infty} S_{i}, $ where $ S_{i} cdot S_{j} subset S_{i+j}. $
$f $ is a homogeneous element of $ S $ of degree 1, that is, $ f in S_{1},$ and $ V(f) $ denotes the set of elements of $ text{Proj};S $ which do not contain $ f $.
I am reading Eisenbud & Harris' "Geometry of Schemes". In Exercise III-6(a), the authors make the following statement:
The intersection $ (I cdot S[f^{-1}]) cap S[f^{-1}]_{0}) $ is generated by elements obtained by choosing a set of homogeneous generators of $ I $ and multiplying them by the appropriate negative powers of $ f.$
I think I understand this statement intuitively. But they go on to infer the correspondence from here, and proceed give the following hint concerning the manner of constructing the maps:
The correspondence is given by taking a prime $ mathfrak{p} $ of $ S[f^{-1}] $ to $ mathfrak{p} = mathfrak{p} cap S[f^{-1}]_{0} $ and taking the prime $ mathfrak{q} $ of $ S[f^{-1}] _{0} $ to $ mathfrak{q}S[f^{-1}].$
My difficulty boils down to not knowing how to use this hint to establish the result explicitly/mechanically.
algebraic-geometry schemes projective-schemes
asked Jan 8 at 15:59
Overwhelmed AG ApprenticeOverwhelmed AG Apprentice
458
$endgroup$
1
$begingroup$
I will refer you to Lemma 10.56.2 of the Stacks project.
$endgroup$
– jgon
Jan 8 at 17:29
$begingroup$
@jgon Thanks for the heads-up, especially since this lemma is a generalisation of my question. I am following the argument. But I have hit a snag. I'm not sure why the following is true: "If $ mathfrak{a}mathfrak{b} in mathfrak{p}_{0}S, $ with $ mathfrak{a} $ and $ mathfrak{b} $ homogeneous, then $ a^{d}b^{d}/f^{text{deg}(mathfrak{a}) + text{deg}(mathfrak{b})} in mathfrak{p}_{0}.$" What am I missing? Any help would be very much appreciated.
$endgroup$
– Overwhelmed AG Apprentice
Jan 9 at 4:10
1
$begingroup$
unfortunately I'm about to sleep, but if your difficulty is unresolved when I wake up in the morning, I will answer the question then
$endgroup$
– jgon
Jan 9 at 4:19
1
$begingroup$
Sorry I couldn't read that last night, since I was on the app, which doesn't tex comments. I'm not sure why you're using fraktur $a$ and $b$, since they're not in fraktur in the lemma, which is important because they aren't ideals, they are homogeneous elements of the ring. Then notice that $a^db^d/f^{deg(a)+deg(b)}$ has total degree $0$, and is a multiple of $ab$, so it is in $mathfrak{p}_0Scap S_0$, which by the first sentence is $mathfrak{p}_0$.
$endgroup$
– jgon
Jan 9 at 15:10
1
$begingroup$
I see. We use the fact that $deg f = d$ and $deg alphabeta = deg alpha + deg beta$ (for homogeneous elements(!)). Then $deg (a^db^d/f^{deg a + deg b}) = ddeg a + ddeg b - (deg a+deg b)deg f = 0$.
$endgroup$
– jgon
Jan 9 at 15:17
|
show 4 more comments
$begingroup$
Forgive me for the length of this post, but I feel this question is deserving of some detail.
Suppose $ S $ is a positively graded $ A$-algebra, where $ A $ is a ring. That is $ S = bigoplus_{i=0}^{infty} S_{i}, $ where $ S_{i} cdot S_{j} subset S_{i+j}. $
$f $ is a homogeneous element of $ S $ of degree 1, that is, $ f in S_{1},$ and $ V(f) $ denotes the set of elements of $ text{Proj};S $ which do not contain $ f $.
I am reading Eisenbud & Harris' "Geometry of Schemes". In Exercise III-6(a), the authors make the following statement:
The intersection $ (I cdot S[f^{-1}]) cap S[f^{-1}]_{0}) $ is generated by elements obtained by choosing a set of homogeneous generators of $ I $ and multiplying them by the appropriate negative powers of $ f.$
I think I understand this statement intuitively. But they go on to infer the correspondence from here, and proceed give the following hint concerning the manner of constructing the maps:
The correspondence is given by taking a prime $ mathfrak{p} $ of $ S[f^{-1}] $ to $ mathfrak{p} = mathfrak{p} cap S[f^{-1}]_{0} $ and taking the prime $ mathfrak{q} $ of $ S[f^{-1}] _{0} $ to $ mathfrak{q}S[f^{-1}].$
My difficulty boils down to not knowing how to use this hint to establish the result explicitly/mechanically.
algebraic-geometry schemes projective-schemes
asked Jan 8 at 15:59
Overwhelmed AG ApprenticeOverwhelmed AG Apprentice
458
$endgroup$
Forgive me for the length of this post, but I feel this question is deserving of some detail.
Suppose $ S $ is a positively graded $ A$-algebra, where $ A $ is a ring. That is $ S = bigoplus_{i=0}^{infty} S_{i}, $ where $ S_{i} cdot S_{j} subset S_{i+j}. $
$f $ is a homogeneous element of $ S $ of degree 1, that is, $ f in S_{1},$ and $ V(f) $ denotes the set of elements of $ text{Proj};S $ which do not contain $ f $.
I am reading Eisenbud & Harris' "Geometry of Schemes". In Exercise III-6(a), the authors make the following statement:
The intersection $ (I cdot S[f^{-1}]) cap S[f^{-1}]_{0}) $ is generated by elements obtained by choosing a set of homogeneous generators of $ I $ and multiplying them by the appropriate negative powers of $ f.$
I think I understand this statement intuitively. But they go on to infer the correspondence from here, and proceed give the following hint concerning the manner of constructing the maps:
The correspondence is given by taking a prime $ mathfrak{p} $ of $ S[f^{-1}] $ to $ mathfrak{p} = mathfrak{p} cap S[f^{-1}]_{0} $ and taking the prime $ mathfrak{q} $ of $ S[f^{-1}] _{0} $ to $ mathfrak{q}S[f^{-1}].$
My difficulty boils down to not knowing how to use this hint to establish the result explicitly/mechanically.
algebraic-geometry schemes projective-schemes
algebraic-geometry schemes projective-schemes
asked Jan 8 at 15:59
Overwhelmed AG ApprenticeOverwhelmed AG Apprentice
458
asked Jan 8 at 15:59
Overwhelmed AG ApprenticeOverwhelmed AG Apprentice
458
edited Jan 8 at 16:14
Overwhelmed AG Apprentice
asked Jan 8 at 15:59
Overwhelmed AG ApprenticeOverwhelmed AG Apprentice
458
asked Jan 8 at 15:59
Overwhelmed AG ApprenticeOverwhelmed AG Apprentice
458
asked Jan 8 at 15:59
Overwhelmed AG ApprenticeOverwhelmed AG Apprentice
458
458
1
$begingroup$
I will refer you to Lemma 10.56.2 of the Stacks project.
$endgroup$
– jgon
Jan 8 at 17:29
$begingroup$
@jgon Thanks for the heads-up, especially since this lemma is a generalisation of my question. I am following the argument. But I have hit a snag. I'm not sure why the following is true: "If $ mathfrak{a}mathfrak{b} in mathfrak{p}_{0}S, $ with $ mathfrak{a} $ and $ mathfrak{b} $ homogeneous, then $ a^{d}b^{d}/f^{text{deg}(mathfrak{a}) + text{deg}(mathfrak{b})} in mathfrak{p}_{0}.$" What am I missing? Any help would be very much appreciated.
$endgroup$
– Overwhelmed AG Apprentice
Jan 9 at 4:10
1
$begingroup$
unfortunately I'm about to sleep, but if your difficulty is unresolved when I wake up in the morning, I will answer the question then
$endgroup$
– jgon
Jan 9 at 4:19
1
$begingroup$
Sorry I couldn't read that last night, since I was on the app, which doesn't tex comments. I'm not sure why you're using fraktur $a$ and $b$, since they're not in fraktur in the lemma, which is important because they aren't ideals, they are homogeneous elements of the ring. Then notice that $a^db^d/f^{deg(a)+deg(b)}$ has total degree $0$, and is a multiple of $ab$, so it is in $mathfrak{p}_0Scap S_0$, which by the first sentence is $mathfrak{p}_0$.
$endgroup$
– jgon
Jan 9 at 15:10
1
$begingroup$
I see. We use the fact that $deg f = d$ and $deg alphabeta = deg alpha + deg beta$ (for homogeneous elements(!)). Then $deg (a^db^d/f^{deg a + deg b}) = ddeg a + ddeg b - (deg a+deg b)deg f = 0$.
$endgroup$
– jgon
Jan 9 at 15:17
|
show 4 more comments
1
$begingroup$
I will refer you to Lemma 10.56.2 of the Stacks project.
$endgroup$
– jgon
Jan 8 at 17:29
$begingroup$
@jgon Thanks for the heads-up, especially since this lemma is a generalisation of my question. I am following the argument. But I have hit a snag. I'm not sure why the following is true: "If $ mathfrak{a}mathfrak{b} in mathfrak{p}_{0}S, $ with $ mathfrak{a} $ and $ mathfrak{b} $ homogeneous, then $ a^{d}b^{d}/f^{text{deg}(mathfrak{a}) + text{deg}(mathfrak{b})} in mathfrak{p}_{0}.$" What am I missing? Any help would be very much appreciated.
$endgroup$
– Overwhelmed AG Apprentice
Jan 9 at 4:10
1
$begingroup$
unfortunately I'm about to sleep, but if your difficulty is unresolved when I wake up in the morning, I will answer the question then
$endgroup$
– jgon
Jan 9 at 4:19
1
$begingroup$
Sorry I couldn't read that last night, since I was on the app, which doesn't tex comments. I'm not sure why you're using fraktur $a$ and $b$, since they're not in fraktur in the lemma, which is important because they aren't ideals, they are homogeneous elements of the ring. Then notice that $a^db^d/f^{deg(a)+deg(b)}$ has total degree $0$, and is a multiple of $ab$, so it is in $mathfrak{p}_0Scap S_0$, which by the first sentence is $mathfrak{p}_0$.
$endgroup$
– jgon
Jan 9 at 15:10
1
$begingroup$
I see. We use the fact that $deg f = d$ and $deg alphabeta = deg alpha + deg beta$ (for homogeneous elements(!)). Then $deg (a^db^d/f^{deg a + deg b}) = ddeg a + ddeg b - (deg a+deg b)deg f = 0$.
$endgroup$
– jgon
Jan 9 at 15:17
1
1
$begingroup$
I will refer you to Lemma 10.56.2 of the Stacks project.
$endgroup$
– jgon
Jan 8 at 17:29
$begingroup$
I will refer you to Lemma 10.56.2 of the Stacks project.
$endgroup$
– jgon
Jan 8 at 17:29
$begingroup$
@jgon Thanks for the heads-up, especially since this lemma is a generalisation of my question. I am following the argument. But I have hit a snag. I'm not sure why the following is true: "If $ mathfrak{a}mathfrak{b} in mathfrak{p}_{0}S, $ with $ mathfrak{a} $ and $ mathfrak{b} $ homogeneous, then $ a^{d}b^{d}/f^{text{deg}(mathfrak{a}) + text{deg}(mathfrak{b})} in mathfrak{p}_{0}.$" What am I missing? Any help would be very much appreciated.
$endgroup$
– Overwhelmed AG Apprentice
Jan 9 at 4:10
$begingroup$
@jgon Thanks for the heads-up, especially since this lemma is a generalisation of my question. I am following the argument. But I have hit a snag. I'm not sure why the following is true: "If $ mathfrak{a}mathfrak{b} in mathfrak{p}_{0}S, $ with $ mathfrak{a} $ and $ mathfrak{b} $ homogeneous, then $ a^{d}b^{d}/f^{text{deg}(mathfrak{a}) + text{deg}(mathfrak{b})} in mathfrak{p}_{0}.$" What am I missing? Any help would be very much appreciated.
$endgroup$
– Overwhelmed AG Apprentice
Jan 9 at 4:10
1
1
$begingroup$
unfortunately I'm about to sleep, but if your difficulty is unresolved when I wake up in the morning, I will answer the question then
$endgroup$
– jgon
Jan 9 at 4:19
$begingroup$
unfortunately I'm about to sleep, but if your difficulty is unresolved when I wake up in the morning, I will answer the question then
$endgroup$
– jgon
Jan 9 at 4:19
1
1
$begingroup$
Sorry I couldn't read that last night, since I was on the app, which doesn't tex comments. I'm not sure why you're using fraktur $a$ and $b$, since they're not in fraktur in the lemma, which is important because they aren't ideals, they are homogeneous elements of the ring. Then notice that $a^db^d/f^{deg(a)+deg(b)}$ has total degree $0$, and is a multiple of $ab$, so it is in $mathfrak{p}_0Scap S_0$, which by the first sentence is $mathfrak{p}_0$.
$endgroup$
– jgon
Jan 9 at 15:10
$begingroup$
Sorry I couldn't read that last night, since I was on the app, which doesn't tex comments. I'm not sure why you're using fraktur $a$ and $b$, since they're not in fraktur in the lemma, which is important because they aren't ideals, they are homogeneous elements of the ring. Then notice that $a^db^d/f^{deg(a)+deg(b)}$ has total degree $0$, and is a multiple of $ab$, so it is in $mathfrak{p}_0Scap S_0$, which by the first sentence is $mathfrak{p}_0$.
$endgroup$
– jgon
Jan 9 at 15:10
1
1
$begingroup$
I see. We use the fact that $deg f = d$ and $deg alphabeta = deg alpha + deg beta$ (for homogeneous elements(!)). Then $deg (a^db^d/f^{deg a + deg b}) = ddeg a + ddeg b - (deg a+deg b)deg f = 0$.
$endgroup$
– jgon
Jan 9 at 15:17
$begingroup$
I see. We use the fact that $deg f = d$ and $deg alphabeta = deg alpha + deg beta$ (for homogeneous elements(!)). Then $deg (a^db^d/f^{deg a + deg b}) = ddeg a + ddeg b - (deg a+deg b)deg f = 0$.
$endgroup$
– jgon
Jan 9 at 15:17
|
show 4 more comments
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1
$begingroup$
I will refer you to Lemma 10.56.2 of the Stacks project.
$endgroup$
– jgon
Jan 8 at 17:29
$begingroup$
@jgon Thanks for the heads-up, especially since this lemma is a generalisation of my question. I am following the argument. But I have hit a snag. I'm not sure why the following is true: "If $ mathfrak{a}mathfrak{b} in mathfrak{p}_{0}S, $ with $ mathfrak{a} $ and $ mathfrak{b} $ homogeneous, then $ a^{d}b^{d}/f^{text{deg}(mathfrak{a}) + text{deg}(mathfrak{b})} in mathfrak{p}_{0}.$" What am I missing? Any help would be very much appreciated.
$endgroup$
– Overwhelmed AG Apprentice
Jan 9 at 4:10
1
$begingroup$
unfortunately I'm about to sleep, but if your difficulty is unresolved when I wake up in the morning, I will answer the question then
$endgroup$
– jgon
Jan 9 at 4:19
1
$begingroup$
Sorry I couldn't read that last night, since I was on the app, which doesn't tex comments. I'm not sure why you're using fraktur $a$ and $b$, since they're not in fraktur in the lemma, which is important because they aren't ideals, they are homogeneous elements of the ring. Then notice that $a^db^d/f^{deg(a)+deg(b)}$ has total degree $0$, and is a multiple of $ab$, so it is in $mathfrak{p}_0Scap S_0$, which by the first sentence is $mathfrak{p}_0$.
$endgroup$
– jgon
Jan 9 at 15:10
1
$begingroup$
I see. We use the fact that $deg f = d$ and $deg alphabeta = deg alpha + deg beta$ (for homogeneous elements(!)). Then $deg (a^db^d/f^{deg a + deg b}) = ddeg a + ddeg b - (deg a+deg b)deg f = 0$.
$endgroup$
– jgon
Jan 9 at 15:17