Artin-Nagata properties, minimal multiplicities, and
depth of fiber cones
Jonathan Montaño
Purdue University
AMS Sectional Meeting
Dalhousie University
October 19, 2014
Jonathan Montaño (Purdue University)
Minimal multiplicities & depth of fiber cones
October 19, 2014
1 / 14
Setting and Notation
I
(R, m, k) Cohen-Macaulay (CM), |k| = ∞, dim(R) = d > 0.
I
I an R-ideal.
M
R(I ) =
I n , the Rees algebra of I .
I
n>0
I
G(I ) =
M
I n /I n+1 , the associated graded algebra of I .
n>0
I
F(I ) =
M
I n /mI n , the fiber cone of I .
n>0
R(I ), G(I ), and F(I ) are called the blowup algebras of I .
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Minimal multiplicities & depth of fiber cones
October 19, 2014
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Seeting and Notation
Let M = m + R(I )+ = m ⊕ I ⊕ I 2 ⊕ · · · and S any of the blowup algebras.
depth S := depth M S .
S is CM if depth S = dim S.
Recall,
I
dim R(I ) = d + 1, provided ht I > 0.
I
dim G(I ) = d.
I
`(I ) := dim F(I ), the analytic spread of I .
J ⊆ I is a minimal reduction of I if µ(J) = `(I ) and I n+1 = JI n for some n ∈ N.
r (I ) = min{n | I n+1 = JI n , for some minimal reduction J}, the reduction number
of I .
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Minimal multiplicities & depth of fiber cones
October 19, 2014
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Depths of blowup algebras
Question: How do the depths of the blowup algebras relate to each other?
Assume ht I > 1.
I
R(I ) is CM ⇒ G(I ) is CM. (Huneke, ’82)
√
“⇐” if I = m and r (I ) < d. (Goto-Shimoda, ’82)
I
“⇐” if a(G(I )) < 0. (Ikeda-Trung, ’89)
I
“⇐” if R is pseudo-rational (e.g. regular) (Lipman, ’94)
I
G(I ) is not CM ⇒ depth R(I ) = depth G(I ) + 1 (Huckaba-Marly, ’94)
I
Jonathan Montaño (Purdue University)
Minimal multiplicities & depth of fiber cones
October 19, 2014
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Depths of blowup algebras
Question: How do the depths of the blowup algebras relate to each other?
Assume ht I > 1.
I
R(I ) is CM ⇒ G(I ) is CM. (Huneke, ’82)
√
“⇐” if I = m and r (I ) < d. (Goto-Shimoda, ’82)
I
“⇐” if a(G(I )) < 0. (Ikeda-Trung, ’89)
I
“⇐” if R is pseudo-rational (e.g. regular) (Lipman, ’94)
I
G(I ) is not CM ⇒ depth R(I ) = depth G(I ) + 1 (Huckaba-Marly, ’94)
I
However, in general:
I
F(I ) is CM 6⇒ G(I ) is CM.
I
F(I ) is CM 6⇐ R(I ) is CM.
Jonathan Montaño (Purdue University)
Minimal multiplicities & depth of fiber cones
October 19, 2014
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Minimal multiplicities
If I is m-primary, the Hilbert-Samuel multiplicity of I is defined as
λ(I n /I n+1 )
.
n→∞
nd−1
e(I ) = (d − 1)! lim
Jonathan Montaño (Purdue University)
Minimal multiplicities & depth of fiber cones
October 19, 2014
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Minimal multiplicities
If I is m-primary, the Hilbert-Samuel multiplicity of I is defined as
λ(I n /I n+1 )
.
n→∞
nd−1
e(I ) = (d − 1)! lim
Notions of minimal multiplicity provide conditions for strong relations among the
depths of blowup algebras. They originated in Abhyankar’s inequality
e(m) > µ(m) − d + 1.
I
R has minimal multiplicity ⇒ G(m) is CM (Sally, 77’)
Sally’s conjecture:
I
R has almost minimal multiplicity ⇒ depth G(m) > d − 1. (Rossi-Valla, 96’,
Wang, 97’)
Jonathan Montaño (Purdue University)
Minimal multiplicities & depth of fiber cones
October 19, 2014
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Minimal multiplicities
√
I = m.
Let J ⊆ I be a minimal reduction. Recall e(I ) = e(J) = λ(R/J).
R
I
J
e(I ) > λ(I /I 2 ) − (d − 1)λ(R/I ) with equality iff r (I ) 6 1.
I
I2
I is of minimal multiplicity ⇒ G(I ) is CM. (Valla, ’78)
I is of minimal multiplicity ⇒ F(I ) is CM. (Huneke-Sally, ’88)
JI
I
I is of almost
minimal multiplicity ⇒ depth G(I ) > d − 1. (Rossi, ’00)
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Minimal multiplicities & depth of fiber cones
October 19, 2014
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Minimal multiplicities
√
I = m.
Let J ⊆ I be a minimal reduction. Recall e(I ) = e(J) = λ(R/J).
R
I
e(I ) > λ(I /I 2 ) − (d − 1)λ(R/I ) with equality iff r (I ) 6 1.
J
I
I2
I is of minimal multiplicity ⇒ G(I ) is CM. (Valla, ’78)
I is of minimal multiplicity ⇒ F(I ) is CM. (Huneke-Sally, ’88)
JI
I
R
I
e(I ) > µ(I ) − d + λ(R/I ) with equality iff I m = Jm.
J
I
I is of Goto-minimal multiplicity:
R(I ) is CM ⇔ G(I ) is CM ⇔ r (I ) 6 1. (Goto, ’00)
I
I is of almost Goto-minimal multiplicity:
depth G(I ) > d − 2 ⇒ depth F(I ) > d − 1.
(Jayanthan-Verma, ’05)
Im
Jm
I is of almost
minimal multiplicity ⇒ depth G(I ) > d − 1. (Rossi, ’00)
Jonathan Montaño (Purdue University)
Minimal multiplicities & depth of fiber cones
October 19, 2014
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j-multiplicity
If I is any ideal, the j-multiplicity of I is defined as
λ H0m (I n /I n+1 )
j(I ) = (d − 1)! lim
.
n→∞
nd−1
(Achilles-Manaresi, ’93)
I
j(I ) is a non-negative integer.
I
j(I ) 6= 0 ⇔ `(I ) = d.
√
I = m ⇒ e(I ) = j(I ).
I
Jonathan Montaño (Purdue University)
Minimal multiplicities & depth of fiber cones
October 19, 2014
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j-multiplicity
If I is any ideal, the j-multiplicity of I is defined as
λ H0m (I n /I n+1 )
j(I ) = (d − 1)! lim
.
n→∞
nd−1
(Achilles-Manaresi, ’93)
I
j(I ) is a non-negative integer.
I
j(I ) 6= 0 ⇔ `(I ) = d.
√
I = m ⇒ e(I ) = j(I ).
I
Theorem (Achilles-Manaresi, Xie)
Let x1 , . . . , xd−1 be d − 1 general elements in I and R := R/(x1 , . . . , xd−1 ) : I ∞ .
1
R is CM and dim(R) 6 1.
2
The ideal I := I R is m-primary.
3
j(I ) = e(I ).
Jonathan Montaño (Purdue University)
Minimal multiplicities & depth of fiber cones
October 19, 2014
7 / 14
Artin-Nagata properties
Goal: Use the j-multiplicity to extend properties of minimal multiplicity to
non-m-primary ideals.
Jonathan Montaño (Purdue University)
Minimal multiplicities & depth of fiber cones
October 19, 2014
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Artin-Nagata properties
Goal: Use the j-multiplicity to extend properties of minimal multiplicity to
non-m-primary ideals.
I satisfies Gs if µ(Ip ) 6 ht p for every p ∈ V (I ) such that ht p < s.
−
Classes of ideals satisfying G`(I ) and the Artin-Nagata property AN`(I
)−2 :
I
Equimultiple ideals.
I
Ideals of analytic deviation one that are generically complete intersection.
I
Strongly Cohen-Macaulay ideals satisfying G`(I ) ; e.g., in a Gorenstein ring:
I
I
I
I
Cohen-Macaulay ideals generated by ht I + 2 elements.
Perfect ideals of height two
Perfect Gorenstein ideals of height three.
Ideals in the linkage class of a complete intersection (licci ideals).
Jonathan Montaño (Purdue University)
Minimal multiplicities & depth of fiber cones
October 19, 2014
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Minimal j-multiplicity
Polini and Xie defined minimal j-multiplcity and extended the results of Rossi,
Valla, and Wang to ideals that satisfy the residual assumptions `(I ) = d, Gd , and
−
ANd−2
.
Jonathan Montaño (Purdue University)
Minimal multiplicities & depth of fiber cones
October 19, 2014
9 / 14
Minimal j-multiplicity
Polini and Xie defined minimal j-multiplcity and extended the results of Rossi,
Valla, and Wang to ideals that satisfy the residual assumptions `(I ) = d, Gd , and
−
ANd−2
.
R = R/(x1 , . . . , xd−1 ) : I ∞ for x1 , . . . , xd−1 general elements in I and I = I R.
If `(I ) = d then, I is of Goto-minimal j-multiplicity if I is of Goto-minimal
multiplicity. This property is well-defined. (M)
Proposition (M)
−
Assume I satisfies Gd and ANd−2
, then
I is of Goto-minimal j-multiplicity ⇔ I m = Jm for one (hence every) minimal
reduction J of I .
Jonathan Montaño (Purdue University)
Minimal multiplicities & depth of fiber cones
October 19, 2014
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Theorem 1
−
From now on, we will assume assume I satisfies G`(I ) and AN`(I
)−2 .
Let g = ht (I ) and s = `(I ). Let J be a minimal reduction of I .
Theorem (M)
Assume J ∩ I n m = JI n−1 m for every 2 6 n 6 r (I ), then the following are
equivalent:
i) F(I ) is CM.
ii) depth G(I ) > s − 1.
iii) depth R(I ) > s.
Jonathan Montaño (Purdue University)
Minimal multiplicities & depth of fiber cones
October 19, 2014
10 / 14
Theorem 1
−
From now on, we will assume assume I satisfies G`(I ) and AN`(I
)−2 .
Let g = ht (I ) and s = `(I ). Let J be a minimal reduction of I .
Theorem (M)
Assume J ∩ I n m = JI n−1 m for every 2 6 n 6 r (I ), then the following are
equivalent:
i) F(I ) is CM.
ii) depth G(I ) > s − 1.
iii) depth R(I ) > s.
We obtain as corollary
r (I ) 6 1 ⇒ F(I ) is CM
in the following classical cases:
I
I
I is equimultiple. (Shah, ’91)
I has analytic deviation one and is generically a complete intersection.
(Cortadellas-Zarzuela, ’97)
Jonathan Montaño (Purdue University)
Minimal multiplicities & depth of fiber cones
October 19, 2014
10 / 14
Theorem 2
Theorem (M)
Assume I m = Jm, consider the following statements:
i) R(I ) is CM (when g > 2),
ii) G(I ) is CM,
iii) F(I ) is CM and a(F(I )) 6 −g + 1,
iv) r (I ) 6 s − g + 1.
Then (i) ⇔ (ii) ⇒ (iii) ⇒ (iv).
If in addition depth R/I j > d − g − j + 1 for every 1 6 j 6 s − g + 1, then all the
statements are equivalent.
Jonathan Montaño (Purdue University)
Minimal multiplicities & depth of fiber cones
October 19, 2014
11 / 14
Examples
1
The monomial ideals
I = (x12 , x1 x2 , . . . , x1 xd , x22 , x2 x3 , . . . , x2 xn )
in the ring k[[x1 , . . . , xd ]] are strongly stable ideals, have height 2, and satisfy
−
Gd and ANd−2
. I is of Goto-minimal j-multiplicity and the algebras R(I ),
G(I ), and F(I ) are CM.
2
Let R = k[[x, y , z, w ]] and
M=
x
w
y
x
z
y
w
z
.
The ideal I = I2 (M) has height 3 and satisfies G4 and AN2− . I is of
Goto-minimal j-multiplicity and the algebras R(I ), G(I ), and F(I ) are CM.
Jonathan Montaño (Purdue University)
Minimal multiplicities & depth of fiber cones
October 19, 2014
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Theorem 3
The following theorem generalizes Jayanthan-Verma’s theorem.
Theorem (M)
Assume s = d, depth R/I j > min{d − g − j + 1, 1} for j = 1 and 2, and I is of
almost Goto-minimal j-multiplicity, then
depth G(I ) > d − 2 ⇒ depth F(I ) > d − 1.
Jonathan Montaño (Purdue University)
Minimal multiplicities & depth of fiber cones
October 19, 2014
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Thank you!
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Minimal multiplicities & depth of fiber cones
October 19, 2014
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