Description of admissibility Definition J is I -admissible if J v ( - - PowerPoint PPT Presentation

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Description of admissibility Definition J is I -admissible if J v ( I ). Lemma This is equivalent to IO Y = E I , with J = J and I an ideal. Indeed, on Y the center J becomes E , in particular principal. Dan

SLIDE 1

Description of admissibility

Definition

J is I-admissible if J ≤ v(I).

Lemma

This is equivalent to IOY ′ = E ℓI′, with J = ¯ Jℓ and I′ an ideal. Indeed, on Y ′ the center J becomes E ℓ, in particular principal.

Dan Abramovich Admissibility and coefficient ideals New York, July 27, 2020 1 / 2

SLIDE 2

Description of admissibility

Definition

J is I-admissible if J ≤ v(I).

Lemma

This is equivalent to IOY ′ = E ℓI′, with J = ¯ Jℓ and I′ an ideal. Indeed, on Y ′ the center J becomes E ℓ, in particular principal. So on Y ′, we have J ≤ v(I) ⇔ E ℓ ⊇ IOY ′.♠ This is more subtle in Quek’s theorem!

Dan Abramovich Admissibility and coefficient ideals New York, July 27, 2020 1 / 2

SLIDE 3

Description of admissibility

Definition

J is I-admissible if J ≤ v(I).

Lemma

This is equivalent to IOY ′ = E ℓI′, with J = ¯ Jℓ and I′ an ideal. Indeed, on Y ′ the center J becomes E ℓ, in particular principal. So on Y ′, we have J ≤ v(I) ⇔ E ℓ ⊇ IOY ′.♠ This is more subtle in Quek’s theorem! Write J = (xa1

1 , . . . , xak k ) and I = (f1, . . . , fm).

Expand fi = cαxα1

1 · · · xα1 n .

J < v(I) ⇔ vJ(fi) ≥ 1 for all i

Dan Abramovich Admissibility and coefficient ideals New York, July 27, 2020 1 / 2

SLIDE 4

Description of admissibility

Definition

J is I-admissible if J ≤ v(I).

Lemma

This is equivalent to IOY ′ = E ℓI′, with J = ¯ Jℓ and I′ an ideal. Indeed, on Y ′ the center J becomes E ℓ, in particular principal. So on Y ′, we have J ≤ v(I) ⇔ E ℓ ⊇ IOY ′.♠ This is more subtle in Quek’s theorem! Write J = (xa1

1 , . . . , xak k ) and I = (f1, . . . , fm).

Expand fi = cαxα1

1 · · · xα1 n .

J < v(I) ⇔ vJ(fi) ≥ 1 for all i ⇔ αj

aj ≥ 1 for all i and α such that cα = 0.

Dan Abramovich Admissibility and coefficient ideals New York, July 27, 2020 1 / 2

SLIDE 5

Consequences

J is I1, I2-admissible ⇒ J is I1 + I2-admissible. J is I-admissible ⇒ Ja is Ia-admissible. J is I-admissible ⇒ J1− 1

a1 is D(I)-admissible. Dan Abramovich Admissibility and coefficient ideals New York, July 27, 2020 2 / 2

SLIDE 6

Consequences

J is I1, I2-admissible ⇒ J is I1 + I2-admissible. J is I-admissible ⇒ Ja is Ia-admissible. J is I-admissible ⇒ J1− 1

a1 is D(I)-admissible.

Proof.

vJ

∂xα

∂xj

= αi

ai − 1 aj ≥ 1 − 1 a1 .

Dan Abramovich Admissibility and coefficient ideals New York, July 27, 2020 2 / 2

SLIDE 7

Consequences

J is I1, I2-admissible ⇒ J is I1 + I2-admissible. J is I-admissible ⇒ Ja is Ia-admissible. J is I-admissible ⇒ J1− 1

a1 is D(I)-admissible.

Proof.

vJ

∂xα

∂xj

= αi

ai − 1 aj ≥ 1 − 1 a1 .

♠ Combining:

Proposition

A center J is I-admissible if and only if J(a1−1)! is C(I, a1)-admissible.

Dan Abramovich Admissibility and coefficient ideals New York, July 27, 2020 2 / 2