A survey on Stanley decompositions

A survey on Stanley decompositions

J ¨urgen Herzog Universit ¨at Duisburg-Essen

Ellwangen, March 2011

Outline

The conjecture

Known cases

How to compute the Stanley depth

Upper and lower bounds

Stanley depth for syzygies

Outline

The conjecture

Known cases

How to compute the Stanley depth

Upper and lower bounds

Stanley depth for syzygies

Outline

The conjecture

Known cases

How to compute the Stanley depth

Upper and lower bounds

Stanley depth for syzygies

Outline

The conjecture

Known cases

How to compute the Stanley depth

Upper and lower bounds

Stanley depth for syzygies

Outline

The conjecture

Known cases

How to compute the Stanley depth

Upper and lower bounds

Stanley depth for syzygies

The conjecture

Richard Stanley in his article “Linear Diophantine equations and local cohomology”, Invent. Math. 68 (1982) made a striking conjecture concerning the depth of multigraded modules.

The conjecture

Richard Stanley in his article “Linear Diophantine equations and local cohomology”, Invent. Math. 68 (1982) made a striking conjecture concerning the depth of multigraded modules.

Here we concentrate on the case that M is a finitely generated Zⁿ-graded S-module, whereS =K[x₁, . . . ,x_n]is the polynomial ring.

The conjecture

Richard Stanley in his article “Linear Diophantine equations and local cohomology”, Invent. Math. 68 (1982) made a striking conjecture concerning the depth of multigraded modules.

Here we concentrate on the case that M is a finitely generated Zⁿ-graded S-module, whereS =K[x₁, . . . ,x_n]is the polynomial ring.

An important special case for aZⁿ-graded S-module is M=I/J where J ⊂I⊂S are monomial ideals.

AStanley decompositionDof M is direct sum ofZⁿ-graded K -vector spaces

D:M=

r

M

j=1

m_jK[Z_j],

where each m_j ∈M is homogeneous, Z_j ⊂X ={x₁, . . . ,x_n} and each m_jK[Z_j]is a free K[Z_j]-module.

AStanley decompositionDof M is direct sum ofZⁿ-graded K -vector spaces

D:M=

r

M

j=1

m_jK[Z_j],

where each m_j ∈M is homogeneous, Z_j ⊂X ={x₁, . . . ,x_n} and each m_jK[Z_j]is a free K[Z_j]-module.

We set sdepth(D) =min{|Z_j| j =1, . . . ,r}, and

sdepth M =max{sdepth(D) :Dis a Stanley decomposition of M}. is called theStanley depthof M.

AStanley decompositionDof M is direct sum ofZⁿ-graded K -vector spaces

D:M=

r

M

j=1

m_jK[Z_j],

where each m_j ∈M is homogeneous, Z_j ⊂X ={x₁, . . . ,x_n} and each m_jK[Z_j]is a free K[Z_j]-module.

We set sdepth(D) =min{|Z_j| j =1, . . . ,r}, and

sdepth M =max{sdepth(D) :Dis a Stanley decomposition of M}. is called theStanley depthof M.

Conjecture (Stanley)sdepth M ≥depth M.

Example: I = (x1x₂³,x₁³x₂)

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The figure displays Stanley decompositions of

I=x₁x₂³K[x₁,x₂]⊕x₁³x₂²K[x₁]⊕x₁³x₂K[x₁], and

S/I=K[x2]⊕x₁K[x1]⊕x₁x₂K ⊕x₁x₂²K ⊕x₁²x₂K ⊕x₁²x₂²K.

Known cases

The Stanley depth for modules of the form I/J where J ⊂I⊂S_K[x1, . . . ,x_n]are monomial ideals is apure

combinatorial invariant, in particular, it does not depend on the field K , while the depth ishomological invariantand in case of squarefree monomial ideal, a topological invariant of the attached simplicial complex, and may very well depend on the field K .

Known cases

The Stanley depth for modules of the form I/J where J ⊂I⊂S_K[x1, . . . ,x_n]are monomial ideals is apure

combinatorial invariant, in particular, it does not depend on the field K , while the depth ishomological invariantand in case of squarefree monomial ideal, a topological invariant of the attached simplicial complex, and may very well depend on the field K .

What is know for I/J?

◮ (Jahan, Zheng, H) Stanley’s conjecture holds for all algebrasS/I,Ia monomial ideal, if it holds for all such Cohen–Macaulay algebras.

◮ (Jahan, Zheng, H) Stanley’s conjecture holds for all algebrasS/I,Ia monomial ideal, if it holds for all such Cohen–Macaulay algebras.

◮ (H, Popescu) If∆is a simplicial complex. Then the Stanley Reisner ringK[∆]of∆satisfies Stanley’s conjecture, if∆ is shellable.

◮ (Jahan, Zheng, H) Stanley’s conjecture holds for all algebrasS/I,Ia monomial ideal, if it holds for all such Cohen–Macaulay algebras.

◮ (H, Popescu) If∆is a simplicial complex. Then the Stanley Reisner ringK[∆]of∆satisfies Stanley’s conjecture, if∆ is shellable.

◮ (H, Jahan, Yassemi) Stanley’s conjecture holds forS/I whenIis Cohen-Macaulay ideal of codimension 2 or Gorenstein of codimension 3.

◮ (Jahan, Zheng, H) Stanley’s conjecture holds for all algebrasS/I,Ia monomial ideal, if it holds for all such Cohen–Macaulay algebras.

◮ (H, Popescu) If∆is a simplicial complex. Then the Stanley Reisner ringK[∆]of∆satisfies Stanley’s conjecture, if∆ is shellable.

◮ (H, Jahan, Yassemi) Stanley’s conjecture holds forS/I whenIis Cohen-Macaulay ideal of codimension 2 or Gorenstein of codimension 3.

◮ (Popescu) IfSis a polynomial ring in at most 5 variables, then Stanley’s conjecture holds forS/I.

◮ (Jahan, Zheng, H) Stanley’s conjecture holds for all algebrasS/I,Ia monomial ideal, if it holds for all such Cohen–Macaulay algebras.

◮ (H, Popescu) If∆is a simplicial complex. Then the Stanley Reisner ringK[∆]of∆satisfies Stanley’s conjecture, if∆ is shellable.

◮ (H, Jahan, Yassemi) Stanley’s conjecture holds forS/I whenIis Cohen-Macaulay ideal of codimension 2 or Gorenstein of codimension 3.

◮ (Popescu) IfSis a polynomial ring in at most 5 variables, then Stanley’s conjecture holds forS/I.

◮ (Cimpoeas) IfI ⊂S=K[x1, . . . ,x_n]is generated by at most 2n−1monomials, then Stanley’s conjecture holds forI

◮ (Jahan, Zheng, H) Stanley’s conjecture holds for all algebrasS/I,Ia monomial ideal, if it holds for all such Cohen–Macaulay algebras.

◮ (H, Popescu) If∆is a simplicial complex. Then the Stanley Reisner ringK[∆]of∆satisfies Stanley’s conjecture, if∆ is shellable.

◮ (H, Jahan, Yassemi) Stanley’s conjecture holds forS/I whenIis Cohen-Macaulay ideal of codimension 2 or Gorenstein of codimension 3.

◮ (Popescu) IfSis a polynomial ring in at most 5 variables, then Stanley’s conjecture holds forS/I.

◮ (Cimpoeas) IfI ⊂S=K[x1, . . . ,x_n]is generated by at most 2n−1monomials, then Stanley’s conjecture holds forI

◮ (Apel, Okazaki, Yanagawa) IfIis a cogeneric monomial ideal, then Stanley’s conjecture holds forS/I.

How to compute the Stanley depth

We return to the case M =I/J where J⊂I ⊂S=K[x₁, . . . ,xn] are monomial ideals.

How to compute the Stanley depth

We return to the case M =I/J where J⊂I ⊂S=K[x₁, . . . ,xn] are monomial ideals.

We choose g ∈Nⁿsuch that g≥a for all generators x^aof I and J, and consider the finite poset

P_I/J^g ={a∈Nⁿ x^a∈I\J, a≤g}.

We call it thecharacteristic posetof I/J with respect to g.

How to compute the Stanley depth

We return to the case M =I/J where J⊂I ⊂S=K[x₁, . . . ,xn] are monomial ideals.

We choose g ∈Nⁿsuch that g≥a for all generators x^aof I and J, and consider the finite poset

P_I/J^g ={a∈Nⁿ x^a∈I\J, a≤g}.

We call it thecharacteristic posetof I/J with respect to g.

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How to compute the Stanley depth

We return to the case M =I/J where J⊂I ⊂S=K[x₁, . . . ,xn] are monomial ideals.

We choose g ∈Nⁿsuch that g≥a for all generators x^aof I and J, and consider the finite poset

P_I/J^g ={a∈Nⁿ x^a∈I\J, a≤g}.

We call it thecharacteristic posetof I/J with respect to g.

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If∆is a simplicial complex andg = (1, . . . ,1), thenP_K[∆]^g can be identified with the face poset of∆.

The characteristic poset ofm= (x1,x₂,x₃)with respect to g = (1,1,1)is given by

◦

◦ ◦ ◦

◦ ◦ ◦

1 2 3

12 23 13

123

Given any poset P and a,b ∈P. Then

[a,b] ={c∈P : a≤c ≤b}is called aninterval.

Given any poset P and a,b ∈P. Then

[a,b] ={c∈P : a≤c ≤b}is called aninterval.

Suppose P is a finite poset. Apartitionof P is a disjoint union

P : P=

r

[

i=1

[a_i,b_i]

of intervals.

Given any poset P and a,b ∈P. Then

[a,b] ={c∈P : a≤c ≤b}is called aninterval.

Suppose P is a finite poset. Apartitionof P is a disjoint union

P : P=

r

[

i=1

[a_i,b_i]

of intervals.

P : Pm^g = [1,12]∪[2,23]∪[3,13]∪[123,123].

is a partition of Pm^g.

◦

◦ ◦ ◦

◦ ◦ ◦

1 2 3

12 23 13

123

Each partition of P_I/J^g gives rise to a Stanley decomposition of I/J.

Each partition of P_I/J^g gives rise to a Stanley decomposition of I/J.

In order to describe the Stanley decomposition of I/J coming from a partition of P_I/J^g we shall need the following notation: for each b ∈P_I/J^g , we set

Z_b={x_j : b(j) =g(j)},

Each partition of P_I/J^g gives rise to a Stanley decomposition of I/J.

In order to describe the Stanley decomposition of I/J coming from a partition of P_I/J^g we shall need the following notation: for each b ∈P_I/J^g , we set

Z_b={x_j : b(j) =g(j)}, and define

ρ P_I/J^g →Z_≥0, b7→ρ(b) =|Z_b|.

Theorem (a) LetP : P_I/J^g =Sr

i=1[c_i,d_i]be a partition ofP_I/J^g . Then

D(P) : I/J =

r

M

i=1

(M

c

x^cK[Z_di])

is a Stanley decomposition of I/J, where the inner direct sum is taken over allc∈[ci,d_i]for which c(j) =c_i(j)for all j with x_j ∈Z_di. Moreover,sdepthD(P) =min{ρ(d_i) i=1, . . . ,r}.

Theorem (a) LetP : P_I/J^g =Sr

i=1[c_i,d_i]be a partition ofP_I/J^g . Then

D(P) : I/J =

r

M

i=1

(M

c

x^cK[Z_di])

is a Stanley decomposition of I/J, where the inner direct sum is taken over allc∈[ci,d_i]for which c(j) =c_i(j)for all j with x_j ∈Z_di. Moreover,sdepthD(P) =min{ρ(d_i) i=1, . . . ,r}. (b) LetDbe a Stanley decomposition of I/J. Then there exists a partitionP of P_I/J^g such that

sdepthD(P)≥sdepthD.

In particular, sdepth I/J can be computed as the maximum of the numbers sdepthD(P), whereP runs over the (finitely many) partitions of P_I/J^g .

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This theorem has been used to compute or to estimate the Stanley depth in several cases:

◮ (C. Biro, D. Howard, M. Keller, W. Trotter, S. Young) sdepth(x₁, . . . ,xn) =⌈n/2⌉.

This theorem has been used to compute or to estimate the Stanley depth in several cases:

◮ (C. Biro, D. Howard, M. Keller, W. Trotter, S. Young) sdepth(x₁, . . . ,xn) =⌈n/2⌉.

◮ (Shen) LetI⊂S=K[x1, . . . ,xn]be a complete intersection monomial ideal minimally generated by m elements. Thensdepth(I) =n− ⌊m/2⌋.

This theorem has been used to compute or to estimate the Stanley depth in several cases:

◮ (C. Biro, D. Howard, M. Keller, W. Trotter, S. Young) sdepth(x₁, . . . ,xn) =⌈n/2⌉.

◮ (Shen) LetI⊂S=K[x1, . . . ,xn]be a complete intersection monomial ideal minimally generated by m elements. Thensdepth(I) =n− ⌊m/2⌋.

◮ (Floystad, H) Letsbe the largest integer such that n+1≥(2s+1)(s+1).Then the Stanley depth of any squarefree monomial ideal in n variables is greater or equal to 2s+1. Explicitly this lower bound is

2

√2n+2.25+0.5 2

−1.

Upper and lower bounds

Let M be aZⁿ-graded S=K[x1, . . . ,x_n]-module. Then there exists

F : 0=M₀⊂M₁⊂ · · · ⊂Mm=M a chain ofZⁿ-graded submodules of M such that

M_i/M_i−1≃(S/P_i)(−a_i)where a_i ∈Zⁿand where each P_i is a monomial prime ideal.

Upper and lower bounds

Let M be aZⁿ-graded S=K[x1, . . . ,x_n]-module. Then there exists

F : 0=M₀⊂M₁⊂ · · · ⊂Mm=M a chain ofZⁿ-graded submodules of M such that

M_i/M_i−1≃(S/P_i)(−a_i)where a_i ∈Zⁿand where each P_i is a monomial prime ideal.

One has

Min(M)⊂Ass(M)⊂ {P1, . . . ,Pr} ⊂Supp(M), and

Upper and lower bounds

Let M be aZⁿ-graded S=K[x1, . . . ,x_n]-module. Then there exists

F : 0=M₀⊂M₁⊂ · · · ⊂Mm=M a chain ofZⁿ-graded submodules of M such that

M_i/M_i−1≃(S/P_i)(−a_i)where a_i ∈Zⁿand where each P_i is a monomial prime ideal.

One has

Min(M)⊂Ass(M)⊂ {P1, . . . ,Pr} ⊂Supp(M), and

min{dim S/P1, . . . ,S/Pr} ≤ depth M,sdepth M

≤ min{dim S/P: P ∈Ass(M)}. The upper inequality has been proved by Apel.

Let M be a finitely generatedZⁿ-graded

S =K[x1, . . . ,x_n]-module. It is alsoZ-graded,i.e.,

M =M

i∈Z

M_i, with M_i = M

a∈Zn

|a|=i

M_a.

Let M be a finitely generatedZⁿ-graded

S =K[x1, . . . ,x_n]-module. It is alsoZ-graded,i.e.,

M =M

i∈Z

M_i, with M_i = M

a∈Zn

|a|=i

M_a.

H_M(t) =X

i∈Z

dim_KM_itⁱ

Let M be a finitely generatedZⁿ-graded

S =K[x1, . . . ,x_n]-module. It is alsoZ-graded,i.e.,

M =M

i∈Z

M_i, with M_i = M

a∈Zn

|a|=i

M_a.

H_M(t) =X

i∈Z

dim_KM_itⁱ = Q(t)

(1−t)^d, Q(1)6=0 is the Hilbert series of M.

Let M be a finitely generatedZⁿ-graded

S =K[x1, . . . ,x_n]-module. It is alsoZ-graded,i.e.,

M =M

i∈Z

M_i, with M_i = M

a∈Zn

|a|=i

M_a.

H_M(t) =X

i∈Z

dim_KM_itⁱ = Q(t)

(1−t)^d, Q(1)6=0 is the Hilbert series of M.

Example: H_S(t) = _(1−t¹₎n forS=K[x₁, . . . ,xn].

Given a Stanley decomposition

D: M=

r

M

j=1

m_jK[Zj], deg m_j =a_j, b_j =|Z_j|.

Given a Stanley decomposition

D: M=

r

M

j=1

m_jK[Zj], deg m_j =a_j, b_j =|Z_j|.

One obtains

S : H_M(t) =

r

X

j=1

t^aj (1−t)^bj.

Given a Stanley decomposition

D: M=

r

M

j=1

m_jK[Zj], deg m_j =a_j, b_j =|Z_j|.

One obtains

S : H_M(t) =

r

X

j=1

t^aj (1−t)^bj.

For any such sum decompositionS of H_M(t)we set

hdepthS =min{b₁, . . . ,rr}

, and calls

hdepth M =max{hdepthS : S is a sum decomposition of H_M(t)}. theHilbert depthof M.

Given a Stanley decomposition

D: M=

r

M

j=1

m_jK[Zj], deg m_j =a_j, b_j =|Z_j|.

One obtains

S : H_M(t) =

r

X

j=1

t^aj (1−t)^bj.

For any such sum decompositionS of H_M(t)we set

hdepthS =min{b₁, . . . ,rr}

, and calls

hdepth M =max{hdepthS : S is a sum decomposition of H_M(t)}. theHilbert depthof M. Obviously,hdepth M ≥sdepth M.

The inequalityhdepth M≥depth M is unknown in general.

The inequalityhdepth M≥depth M is unknown in general.

The Hilbert depth has been computed is some interesting special cases

◮ (Bruns, Krattenthaler, Uliczka) LetM(n,k)be the k -syzygy module ofK =S/(x1, . . . ,x_n). Then

hdepth M(n,k) =n−1 for⌊n/2⌋ ≤k <n,

and

hdepth M(n,k)≥sdepth M(n,k) ≥ ⌊(n+k)/2⌋ for k <n/2.

The inequalityhdepth M≥depth M is unknown in general.

The Hilbert depth has been computed is some interesting special cases

◮ (Bruns, Krattenthaler, Uliczka) LetM(n,k)be the k -syzygy module ofK =S/(x1, . . . ,x_n). Then

hdepth M(n,k) =n−1 for⌊n/2⌋ ≤k <n,

and

hdepth M(n,k)≥sdepth M(n,k) ≥ ⌊(n+k)/2⌋

for k <n/2.

◮ (Bruns, Krattenthaler, Ulizcka) hdepth(x1, . . . ,x_n)^k =⌈n/(k +1)⌉

Stanley depth for syzygies

We have

hdepth M(n,k) =n−1≥depth M(n,k) =k for⌊n/2⌋ ≤k <n.

Stanley depth for syzygies

We have

hdepth M(n,k) =n−1≥depth M(n,k) =k

for⌊n/2⌋ ≤k <n.

In general, if M is a finitely generatedZⁿ-graded of depth t, and Z_k(M)is its k th syzygy module, then

depth Z_k(M) =t+k for k =1, . . . ,n−t.

In particular for I ⊂S one hasdepth I ≥depth S/I+1.

Stanley depth for syzygies

We have

hdepth M(n,k) =n−1≥depth M(n,k) =k

for⌊n/2⌋ ≤k <n.

In general, if M is a finitely generatedZⁿ-graded of depth t, and Z_k(M)is its k th syzygy module, then

depth Z_k(M) =t+k for k =1, . . . ,n−t.

In particular for I ⊂S one hasdepth I ≥depth S/I+1.

It is not known whethersdepth I =sdepth S/I+1.

Stanley depth for syzygies

We have

hdepth M(n,k) =n−1≥depth M(n,k) =k

for⌊n/2⌋ ≤k <n.

In general, if M is a finitely generatedZⁿ-graded of depth t, and Z_k(M)is its k th syzygy module, then

depth Z_k(M) =t+k for k =1, . . . ,n−t.

In particular for I ⊂S one hasdepth I ≥depth S/I+1.

It is not known whethersdepth I =sdepth S/I+1.

Theorem (Floystad, H)sdepth Z_k(M)≥k for all k .

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