Interpolating Between Hilbert-Samuel and Hilbert-Kunz Multiplicity - - PowerPoint PPT Presentation

Interpolating Between Hilbert-Samuel and Hilbert-Kunz Multiplicity

William D. Taylor

University of Arkansas

KUMUNUjr, April 8, 2017

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 1 / 12

Setting and Notation

(R, m) is a local ring of characteristic p > 0 and dimension d I ⊆ R is an m-primary ideal of R λ(M) denotes the length of the R-module M

Definition

The Hilbert-Samuel multiplicity of I is defined to be e(I) = lim

n→∞

d! · λ(R/I n) nd . The Hilbert-Kunz multiplicity of I is defined to be eHK(I) = lim

e→∞

λ

• R/I [pe]

ped .

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 2 / 12

Properties of e(I) and eHK(I)

1 If I = J, then e(I) = e(J) 2 If I ∗ = J∗, then eHK(I) = eHK(J) 3 If R is regular then e(m) = eHK(m) = 1 4 There is an Associativity Formula relating each multiplicity to the

multiplicity after quotienting by the set of primes of maximal dimension.

Theorem (Rees ‘61, Hochster-Huneke ‘90)

If R is a complete domain and I ⊆ J, then the converses of 1 and 2 hold.

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 3 / 12

Definition of s-multiplicity

Remark

Let s be a positive real number. If s is small, then for e ≫ 0, I ⌈spe⌉ ⊇ I [pe],

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 4 / 12

Definition of s-multiplicity

Remark

Let s be a positive real number. If s is small, then for e ≫ 0, I ⌈spe⌉ ⊇ I [pe], hence I ⌈spe⌉ + I [pe] = I ⌈spe⌉,

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 4 / 12

Definition of s-multiplicity

Remark

Let s be a positive real number. If s is small, then for e ≫ 0, I ⌈spe⌉ ⊇ I [pe], hence I ⌈spe⌉ + I [pe] = I ⌈spe⌉, hence lim

e→∞

λ

• R/(I ⌈spe⌉ + I [pe])
• ped

= λ

• R/I ⌈spe⌉

ped = sd d!e(I).

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 4 / 12

Definition of s-multiplicity

Remark

Let s be a positive real number. If s is small, then for e ≫ 0, I ⌈spe⌉ ⊇ I [pe], hence I ⌈spe⌉ + I [pe] = I ⌈spe⌉, hence lim

e→∞

λ

• R/(I ⌈spe⌉ + I [pe])
• ped

= λ

• R/I ⌈spe⌉

ped = sd d!e(I). Similarly, If s is large, then for e ≫ 0, I ⌈spe⌉ ⊆ I [pe], hence I ⌈spe⌉ + I [pe] = I [pe], hence lim

e→∞

λ

• R/(I ⌈spe⌉ + I [pe])
• ped

= lim

e→∞

λ

• R/I [pe]

ped = eHK(I).

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 4 / 12

Definition of s-multiplicity

Definition (-)

For a positive real number s, the s-multiplicity of I is es(I) = lim

e→∞

λ

• R/(I ⌈spe⌉ + I [pe])
• pedHs(d)

where Hs(d) =

⌊s⌋

• i=0

(−1)i d! d i

• (s − i)d.

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 5 / 12

Properties of s-multiplicity

Proposition (-)

The function es(I) has the following properties:

1 es(I) is a continuous function of s. 2 If s is sufficiently small, then es(I) = e(I). 3 If s is sufficiently large, then es(I) = eHK(I). 4 If R is regular, then es(m) = 1 for all s. William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 6 / 12

Properties of s-multiplicity

Proposition (-)

The function es(I) has the following properties:

1 es(I) is a continuous function of s. 2 If s ≤ 1, then es(I) = e(I). 3 If s ≥ d, then es(I) = eHK(I). 4 If R is regular, then es(m) = 1 for all s. William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 7 / 12

The Associativity Formula

Theorem (-)

If s is any positive real number, then es(I) =

• p∈Assh R

eR/p

s

(I(R/p)) λRp(Rp) where Assh R = {p ∈ Spec R | dim R/p = dim R}. This theorem generalizes the Associativity Formulae for the Hilbert-Samuel and Hilbert-Kunz multiplicities.

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 8 / 12

Closures

Recall: If I = J, then e(I) = e(J) If I ∗ = J∗, then eHK(I) = eHK(J) If I ⊆ J and R is “nice”, then the converse holds.

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 9 / 12

Closures

Recall: If I = J, then e(I) = e(J) If I ∗ = J∗, then eHK(I) = eHK(J) If I ⊆ J and R is “nice”, then the converse holds. Question: Are there closures that similarly relate to s-multiplicity?

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 9 / 12

Closures

Recall: If I = J, then e(I) = e(J) If I ∗ = J∗, then eHK(I) = eHK(J) If I ⊆ J and R is “nice”, then the converse holds. Question: Are there closures that similarly relate to s-multiplicity? Answer: Yes, we call them s-closures.

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 9 / 12

s-closures

Definition (-)

Let s ≥ 1. We say x ∈ I cls, the s-closure of I, if there exists c not in any minimal prime of R such that for all e ≫ 0, cxpe ∈ I ⌈spe⌉ + I [pe].

Remark

When s = 1, s-closure is integral closure. When s ≥ d, s-closure is tight

• closure. As s increases, the s-closures get smaller.

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 10 / 12

s-closures

Question: Do any new closures actually appear?

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 11 / 12

s-closures

Question: Do any new closures actually appear? Short Answer: Yes!

Example

Let R = k[[x, y]], and let I = (x3, y3). Then I cls =      (x, y)3 = I if s = 1 (x3, x2y2, y3) if 1 < s ≤ 4

3

(x3, y3) = I ∗ if s > 4

3.

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 11 / 12

s-closures

Question: Do any new closures actually appear? Short Answer: Yes!

Example

Let R = k[[x, y]], and let I = (x3, y3). Then I cls =      (x, y)3 = I if s = 1 (x3, x2y2, y3) if 1 < s ≤ 4

3

(x3, y3) = I ∗ if s > 4

3.

Long Answer: In many cases, there are uncountably many distinct s-closures on a fixed ring R.

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 11 / 12

s-closures

Theorem (-)

Let I and J be m-primary ideals of R. If I cls = Jcls, then es(I) = es(J). If R is an F-finite complete domain and I ⊆ J, then the converse holds.

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 12 / 12

s-closures

Theorem (-)

Let I and J be m-primary ideals of R. If I cls = Jcls, then es(I) = es(J). If R is an F-finite complete domain and I ⊆ J, then the converse holds.

Remark

The proof of this theorem uses a nice result of Polstra and Tucker on limits related to positive characteristic rings.

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 12 / 12

s-closures

Theorem (-)

Let I and J be m-primary ideals of R. If I cls = Jcls, then es(I) = es(J). If R is an F-finite complete domain and I ⊆ J, then the converse holds.

Remark

The proof of this theorem uses a nice result of Polstra and Tucker on limits related to positive characteristic rings.

Remark

Extending the converse result to the non-domain case seems difficult.

William D. Taylor (U. Arkansas) Interpolating Between e(I) and eHK (I) KUMUNUjr 2017 12 / 12