Mixing
Writing a projective scheme as a union of irreducible subschemes.
$begingroup$
Suppose $i: X hookrightarrow mathbb{P}^n_k$ is a projective scheme, over some field $k$ (I'm not sure if algebraically closed is necessary here). So $X$ comes together with an ideal sheaf and a short exact sequence
$$ 0 rightarrow mathcal{I}_X rightarrow mathcal{O}_{mathbb{P}^n} rightarrow i_* mathcal{O}_X rightarrow 0.$$
Topologically $X$ consists of finitely many irreducible components. I would like to know if it is always possible to write $X$ as a scheme-theoretic union of irreducible subschemes $X_i hookrightarrow X$, i.e. $mathcal{I}_X = bigcap_i mathcal{I}_{X_i}$.
I think this works if $X$ is reduced, then one can just take the irreducible components together with the reduced structure as $X_i$, and $mathcal{I}_X = bigcap_i mathcal{I}_{X_i}$, because the intersection of radical ideals is again radical, and two radical ideals with the same vanishing locus are equal, by Hilbert's Nullstellensatz.
Does this have anything to do with primary decomposition? Wikipedia states that there is some scheme theoretic interpretation of primary decomposition, but suppose $mathcal{I}_X$ is the intersection of non-primary ideals, then why should it be possible to write it as the intersection of primary ideals?
algebraic-geometry primary-decomposition
asked Jan 29 at 15:53
red_trumpetred_trumpet
1,025319
$endgroup$
|
show 4 more comments
$begingroup$
Suppose $i: X hookrightarrow mathbb{P}^n_k$ is a projective scheme, over some field $k$ (I'm not sure if algebraically closed is necessary here). So $X$ comes together with an ideal sheaf and a short exact sequence
$$ 0 rightarrow mathcal{I}_X rightarrow mathcal{O}_{mathbb{P}^n} rightarrow i_* mathcal{O}_X rightarrow 0.$$
Topologically $X$ consists of finitely many irreducible components. I would like to know if it is always possible to write $X$ as a scheme-theoretic union of irreducible subschemes $X_i hookrightarrow X$, i.e. $mathcal{I}_X = bigcap_i mathcal{I}_{X_i}$.
I think this works if $X$ is reduced, then one can just take the irreducible components together with the reduced structure as $X_i$, and $mathcal{I}_X = bigcap_i mathcal{I}_{X_i}$, because the intersection of radical ideals is again radical, and two radical ideals with the same vanishing locus are equal, by Hilbert's Nullstellensatz.
Does this have anything to do with primary decomposition? Wikipedia states that there is some scheme theoretic interpretation of primary decomposition, but suppose $mathcal{I}_X$ is the intersection of non-primary ideals, then why should it be possible to write it as the intersection of primary ideals?
algebraic-geometry primary-decomposition
asked Jan 29 at 15:53
red_trumpetred_trumpet
1,025319
$endgroup$
$begingroup$
It's a theorem that in a noetherian ring every ideal has a primary decomposition (c.f., Atiyah-MacDonald, Thm 7.13), which deals with the case of affine schemes. I think there is a similar result for graded rings in Eisenbud, which would mean the same is true for projective schemes.
$endgroup$
– André 3000
Jan 29 at 17:54
1
$begingroup$
Yeah I'm still trying to figure out primary ideals. Is it true that an ideal $mathfrak{a} subset A$ is primary if and only if $text{Spec}(A/mathfrak{a})$ is irreducible?
$endgroup$
– red_trumpet
Jan 29 at 17:57
$begingroup$
Yes, I think that follows from this question: math.stackexchange.com/q/541943
$endgroup$
– André 3000
Jan 29 at 18:46
$begingroup$
I don't think this is true anymore :D See the discussion here, and the counterexample here.
$endgroup$
– red_trumpet
Jan 29 at 18:49
$begingroup$
But...you're not using the definition of primary, just a strictly weaker property.
$endgroup$
– André 3000
Jan 29 at 19:02
|
show 4 more comments
$begingroup$
Suppose $i: X hookrightarrow mathbb{P}^n_k$ is a projective scheme, over some field $k$ (I'm not sure if algebraically closed is necessary here). So $X$ comes together with an ideal sheaf and a short exact sequence
$$ 0 rightarrow mathcal{I}_X rightarrow mathcal{O}_{mathbb{P}^n} rightarrow i_* mathcal{O}_X rightarrow 0.$$
Topologically $X$ consists of finitely many irreducible components. I would like to know if it is always possible to write $X$ as a scheme-theoretic union of irreducible subschemes $X_i hookrightarrow X$, i.e. $mathcal{I}_X = bigcap_i mathcal{I}_{X_i}$.
I think this works if $X$ is reduced, then one can just take the irreducible components together with the reduced structure as $X_i$, and $mathcal{I}_X = bigcap_i mathcal{I}_{X_i}$, because the intersection of radical ideals is again radical, and two radical ideals with the same vanishing locus are equal, by Hilbert's Nullstellensatz.
Does this have anything to do with primary decomposition? Wikipedia states that there is some scheme theoretic interpretation of primary decomposition, but suppose $mathcal{I}_X$ is the intersection of non-primary ideals, then why should it be possible to write it as the intersection of primary ideals?
algebraic-geometry primary-decomposition
asked Jan 29 at 15:53
red_trumpetred_trumpet
1,025319
$endgroup$
Suppose $i: X hookrightarrow mathbb{P}^n_k$ is a projective scheme, over some field $k$ (I'm not sure if algebraically closed is necessary here). So $X$ comes together with an ideal sheaf and a short exact sequence
$$ 0 rightarrow mathcal{I}_X rightarrow mathcal{O}_{mathbb{P}^n} rightarrow i_* mathcal{O}_X rightarrow 0.$$
Topologically $X$ consists of finitely many irreducible components. I would like to know if it is always possible to write $X$ as a scheme-theoretic union of irreducible subschemes $X_i hookrightarrow X$, i.e. $mathcal{I}_X = bigcap_i mathcal{I}_{X_i}$.
I think this works if $X$ is reduced, then one can just take the irreducible components together with the reduced structure as $X_i$, and $mathcal{I}_X = bigcap_i mathcal{I}_{X_i}$, because the intersection of radical ideals is again radical, and two radical ideals with the same vanishing locus are equal, by Hilbert's Nullstellensatz.
Does this have anything to do with primary decomposition? Wikipedia states that there is some scheme theoretic interpretation of primary decomposition, but suppose $mathcal{I}_X$ is the intersection of non-primary ideals, then why should it be possible to write it as the intersection of primary ideals?
algebraic-geometry primary-decomposition
algebraic-geometry primary-decomposition
asked Jan 29 at 15:53
red_trumpetred_trumpet
1,025319
asked Jan 29 at 15:53
red_trumpetred_trumpet
1,025319
asked Jan 29 at 15:53
red_trumpetred_trumpet
1,025319
asked Jan 29 at 15:53
red_trumpetred_trumpet
1,025319
asked Jan 29 at 15:53
red_trumpetred_trumpet
1,025319
1,025319
$begingroup$
It's a theorem that in a noetherian ring every ideal has a primary decomposition (c.f., Atiyah-MacDonald, Thm 7.13), which deals with the case of affine schemes. I think there is a similar result for graded rings in Eisenbud, which would mean the same is true for projective schemes.
$endgroup$
– André 3000
Jan 29 at 17:54
1
$begingroup$
Yeah I'm still trying to figure out primary ideals. Is it true that an ideal $mathfrak{a} subset A$ is primary if and only if $text{Spec}(A/mathfrak{a})$ is irreducible?
$endgroup$
– red_trumpet
Jan 29 at 17:57
$begingroup$
Yes, I think that follows from this question: math.stackexchange.com/q/541943
$endgroup$
– André 3000
Jan 29 at 18:46
$begingroup$
I don't think this is true anymore :D See the discussion here, and the counterexample here.
$endgroup$
– red_trumpet
Jan 29 at 18:49
$begingroup$
But...you're not using the definition of primary, just a strictly weaker property.
$endgroup$
– André 3000
Jan 29 at 19:02
|
show 4 more comments
$begingroup$
It's a theorem that in a noetherian ring every ideal has a primary decomposition (c.f., Atiyah-MacDonald, Thm 7.13), which deals with the case of affine schemes. I think there is a similar result for graded rings in Eisenbud, which would mean the same is true for projective schemes.
$endgroup$
– André 3000
Jan 29 at 17:54
1
$begingroup$
Yeah I'm still trying to figure out primary ideals. Is it true that an ideal $mathfrak{a} subset A$ is primary if and only if $text{Spec}(A/mathfrak{a})$ is irreducible?
$endgroup$
– red_trumpet
Jan 29 at 17:57
$begingroup$
Yes, I think that follows from this question: math.stackexchange.com/q/541943
$endgroup$
– André 3000
Jan 29 at 18:46
$begingroup$
I don't think this is true anymore :D See the discussion here, and the counterexample here.
$endgroup$
– red_trumpet
Jan 29 at 18:49
$begingroup$
But...you're not using the definition of primary, just a strictly weaker property.
$endgroup$
– André 3000
Jan 29 at 19:02
$begingroup$
It's a theorem that in a noetherian ring every ideal has a primary decomposition (c.f., Atiyah-MacDonald, Thm 7.13), which deals with the case of affine schemes. I think there is a similar result for graded rings in Eisenbud, which would mean the same is true for projective schemes.
$endgroup$
– André 3000
Jan 29 at 17:54
$begingroup$
It's a theorem that in a noetherian ring every ideal has a primary decomposition (c.f., Atiyah-MacDonald, Thm 7.13), which deals with the case of affine schemes. I think there is a similar result for graded rings in Eisenbud, which would mean the same is true for projective schemes.
$endgroup$
– André 3000
Jan 29 at 17:54
1
1
$begingroup$
Yeah I'm still trying to figure out primary ideals. Is it true that an ideal $mathfrak{a} subset A$ is primary if and only if $text{Spec}(A/mathfrak{a})$ is irreducible?
$endgroup$
– red_trumpet
Jan 29 at 17:57
$begingroup$
Yeah I'm still trying to figure out primary ideals. Is it true that an ideal $mathfrak{a} subset A$ is primary if and only if $text{Spec}(A/mathfrak{a})$ is irreducible?
$endgroup$
– red_trumpet
Jan 29 at 17:57
$begingroup$
Yes, I think that follows from this question: math.stackexchange.com/q/541943
$endgroup$
– André 3000
Jan 29 at 18:46
$begingroup$
Yes, I think that follows from this question: math.stackexchange.com/q/541943
$endgroup$
– André 3000
Jan 29 at 18:46
$begingroup$
I don't think this is true anymore :D See the discussion here, and the counterexample here.
$endgroup$
– red_trumpet
Jan 29 at 18:49
$begingroup$
I don't think this is true anymore :D See the discussion here, and the counterexample here.
$endgroup$
– red_trumpet
Jan 29 at 18:49
$begingroup$
But...you're not using the definition of primary, just a strictly weaker property.
$endgroup$
– André 3000
Jan 29 at 19:02
$begingroup$
But...you're not using the definition of primary, just a strictly weaker property.
$endgroup$
– André 3000
Jan 29 at 19:02
|
show 4 more comments
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$begingroup$
It's a theorem that in a noetherian ring every ideal has a primary decomposition (c.f., Atiyah-MacDonald, Thm 7.13), which deals with the case of affine schemes. I think there is a similar result for graded rings in Eisenbud, which would mean the same is true for projective schemes.
$endgroup$
– André 3000
Jan 29 at 17:54
1
$begingroup$
Yeah I'm still trying to figure out primary ideals. Is it true that an ideal $mathfrak{a} subset A$ is primary if and only if $text{Spec}(A/mathfrak{a})$ is irreducible?
$endgroup$
– red_trumpet
Jan 29 at 17:57
$begingroup$
Yes, I think that follows from this question: math.stackexchange.com/q/541943
$endgroup$
– André 3000
Jan 29 at 18:46
$begingroup$
I don't think this is true anymore :D See the discussion here, and the counterexample here.
$endgroup$
– red_trumpet
Jan 29 at 18:49
$begingroup$
But...you're not using the definition of primary, just a strictly weaker property.
$endgroup$
– André 3000
Jan 29 at 19:02