LOCAL COHOMOLOGY, CONNECTEDNESS AND INITIAL IDEALS

LOCAL COHOMOLOGY, CONNECTEDNESS AND INITIAL IDEALS

Matteo Varbaro

Dipartimento di Matematica Universit´ a di Genova

arXiv:0802.1800

We start with a simple question

Given J = (ua, za, ya, xa, uv , zv , yv , xv , xyu, xyz, xzu, yzu) in S := k[x , y , z, u, v , w , a] is it possible to find:

(a) a term order ≺ on S

(b) I ⊆ S homogeneus such that S/I is Cohen-Macaulay such that:

pLT _≺ (I ) = J ?

We start with a simple question

Given J = (ua, za, ya, xa, uv , zv , yv , xv , xyu, xyz, xzu, yzu) in S := k[x , y , z, u, v , w , a] is it possible to find:

(a) a term order ≺ on S

(b) I ⊆ S homogeneus such that S/I is Cohen-Macaulay such that:

pLT _≺ (I ) = J ?

We start with a simple question

Given J = (ua, za, ya, xa, uv , zv , yv , xv , xyu, xyz, xzu, yzu) in S := k[x , y , z, u, v , w , a] is it possible to find:

(a) a term order ≺ on S

(b) I ⊆ S homogeneus such that S/I is Cohen-Macaulay such that:

pLT _≺ (I ) = J ?

We start with a simple question

Given J = (ua, za, ya, xa, uv , zv , yv , xv , xyu, xyz, xzu, yzu) in S := k[x , y , z, u, v , w , a] is it possible to find:

(a) a term order ≺ on S

(b) I ⊆ S homogeneus such that S/I is Cohen-Macaulay

such that:

pLT _≺ (I ) = J ?

We start with a simple question

Given J = (ua, za, ya, xa, uv , zv , yv , xv , xyu, xyz, xzu, yzu) in S := k[x , y , z, u, v , w , a] is it possible to find:

(a) a term order ≺ on S

(b) I ⊆ S homogeneus such that S/I is Cohen-Macaulay such that:

pLT _≺ (I ) = J ?

We start with a simple question

Given J = (ua, za, ya, xa, uv , zv , yv , xv , xyu, xyz, xzu, yzu) in S := k[x , y , z, u, v , w , a] is it possible to find:

(a) a term order ≺ on S

(b) I ⊆ S homogeneus such that S/I is Cohen-Macaulay such that:

pLT ≺ (I ) = J ?

MAIN GOAL OF THE TALK

We’ll generalize the following theorem of Kalkbrener and Sturmfels: k = ¯ k , I ⊆ S := k[x ₁ , . . . , x _n ] and ω ∈ (Z + ) ⁿ .

If I is a prime ideal, then S / in _ω (I ) is connected in codimension 1. So, in particular, they proved that for every term order ≺ on S

I prime ⇒ S /LT ≺ (I ) connected in codimension 1

MAIN GOAL OF THE TALK

We’ll generalize the following theorem of Kalkbrener and Sturmfels:

k = ¯ k , I ⊆ S := k[x ₁ , . . . , x _n ] and ω ∈ (Z + ) ⁿ .

If I is a prime ideal, then S / in _ω (I ) is connected in codimension 1. So, in particular, they proved that for every term order ≺ on S

I prime ⇒ S /LT ≺ (I ) connected in codimension 1

MAIN GOAL OF THE TALK

We’ll generalize the following theorem of Kalkbrener and Sturmfels:

k = ¯ k , I ⊆ S := k[x ₁ , . . . , x _n ] and ω ∈ (Z + ) ⁿ .

If I is a prime ideal, then S / in ω (I ) is connected in codimension 1.

So, in particular, they proved that for every term order ≺ on S

I prime ⇒ S /LT ≺ (I ) connected in codimension 1

MAIN GOAL OF THE TALK

We’ll generalize the following theorem of Kalkbrener and Sturmfels:

k = ¯ k , I ⊆ S := k[x ₁ , . . . , x _n ] and ω ∈ (Z + ) ⁿ .

If I is a prime ideal, then S / in ω (I ) is connected in codimension 1.

So, in particular, they proved that for every term order ≺ on S

I prime ⇒ S /LT ≺ (I ) connected in codimension 1

MAIN GOAL OF THE TALK

We’ll generalize the following theorem of Kalkbrener and Sturmfels:

k = ¯ k , I ⊆ S := k[x ₁ , . . . , x _n ] and ω ∈ (Z + ) ⁿ .

If I is a prime ideal, then S / in ω (I ) is connected in codimension 1.

So, in particular, they proved that for every term order ≺ on S

I prime ⇒ S /LT ≺ (I ) connected in codimension 1

Structure of the talk

Two principal chapters:

- Local cohomology and connectedness

- Connectedness of an ideal versus connectedness of its initial ideals

Structure of the talk

Two principal chapters:

- Local cohomology and connectedness

- Connectedness of an ideal versus connectedness of its initial ideals

Structure of the talk

Two principal chapters:

- Local cohomology and connectedness

- Connectedness of an ideal versus connectedness of its initial ideals

Structure of the talk

Two principal chapters:

- Local cohomology and connectedness

- Connectedness of an ideal versus connectedness of its initial ideals

PART 1

LOCAL COHOMOLOGY AND CONNECTEDNESS

Notation, definitions and ”basic” results

Notation, definitions and ”basic” results

- R noetherian ring, commutative with 1, a ⊆ R ideal.

Notation, definitions and ”basic” results

- R noetherian ring, commutative with 1, a ⊆ R ideal.

- cd(R; a) is the infimum d ∈ N s.t. H a ⁱ (M) = 0 for every R-module

M and i ≥ d .

Notation, definitions and ”basic” results

- R noetherian ring, commutative with 1, a ⊆ R ideal.

- cd(R; a) is the infimum d ∈ N s.t. H a ⁱ (R) = 0 for every i ≥ d .

Notation, definitions and ”basic” results

- R noetherian ring, commutative with 1, a ⊆ R ideal.

- cd(R; a) is the infimum d ∈ N s.t. H a ⁱ (R) = 0 for every i ≥ d .

ht(a) ≤ cd(R; a) ≤ dim R.

Connectivity dimension

Let T be a noetherian topological space.

c(T ) is the infimum of dim Z such that Z ⊆ T is closed and

T \ Z is disconnected, with ∅ disconnected of dimension −1.

Connectivity dimension

Let T be a noetherian topological space.

c(T ) is the infimum of dim Z such that Z ⊆ T is closed and

T \ Z is disconnected, with ∅ disconnected of dimension −1.

Connectivity dimension

Let T be a noetherian topological space.

c(T ) is the infimum of dim Z such that Z ⊆ T is closed and

T \ Z is disconnected, with ∅ disconnected of dimension −1.

Connectivity dimension

Let T be a noetherian topological space.

c(T ) is the infimum of dim Z such that Z ⊆ T is closed and T \ Z is disconnected, with ∅ disconnected of dimension −1.

EXAMPLES

Connectivity dimension

Let T be a noetherian topological space.

c(T ) is the infimum of dim Z such that Z ⊆ T is closed and T \ Z is disconnected, with ∅ disconnected of dimension −1.

EXAMPLES

T is connected if and only if c(T ) ≥ 0.

Connectivity dimension

Let T be a noetherian topological space.

c(T ) is the infimum of dim Z such that Z ⊆ T is closed and T \ Z is disconnected, with ∅ disconnected of dimension −1.

EXAMPLES

If T is irreducible, then c(T ) = dim T .

Connectivity dimension

Let T be a noetherian topological space.

c(T ) is the infimum of dim Z such that Z ⊆ T is closed and T \ Z is disconnected, with ∅ disconnected of dimension −1.

EXAMPLES

a = (xz, xw , yz, yw ) ⊆ C[x, y , z, w ], T = Z(a) ⊆ A ⁴ . T = Z(x , y ) ∪ Z(z, w ), and Z(x , y ) ∩ Z(z, w ) = {(0, 0, 0, 0)},

so dim T = 2 and c(T ) = 0.

Connectivity dimension

Let T be a noetherian topological space.

c(T ) is the infimum of dim Z such that Z ⊆ T is closed and T \ Z is disconnected, with ∅ disconnected of dimension −1.

If T = Spec R, then c(R) := c(T ) is the infimum of dim R/a

with a ideal such that Spec R \ V(a) is disconnected.

A characterization of connectivity dimension

T.F.A.E.: (1) c(R) ≥ d ;

(2) ∀℘ ⁰ , ℘ ⁰⁰ ∈ Min R, are d-connected , i.e ∃ ℘ ⁰ = ℘ ₀ , ℘ ₁ ,

. . . , ℘ _r = ℘ ⁰⁰ , ℘ _i ∈ Min R and dim R/(℘ _j + ℘ _{j −1} ) ≥ d .

We will say that R is connected in codimension dim R − d .

A characterization of connectivity dimension

T.F.A.E.:

(1) c(R) ≥ d ;

(2) ∀℘ ⁰ , ℘ ⁰⁰ ∈ Min R, are d-connected , i.e ∃ ℘ ⁰ = ℘ ₀ , ℘ ₁ ,

. . . , ℘ _r = ℘ ⁰⁰ , ℘ _i ∈ Min R and dim R/(℘ _j + ℘ _{j −1} ) ≥ d .

We will say that R is connected in codimension dim R − d .

A characterization of connectivity dimension

T.F.A.E.:

(1) c(R) ≥ d ;

(2) ∀℘ ⁰ , ℘ ⁰⁰ ∈ Min R, are d-connected , i.e ∃ ℘ ⁰ = ℘ ₀ , ℘ ₁ ,

. . . , ℘ _r = ℘ ⁰⁰ , ℘ _i ∈ Min R and dim R/(℘ _j + ℘ _{j −1} ) ≥ d .

We will say that R is connected in codimension dim R − d .

A characterization of connectivity dimension

T.F.A.E.:

(1) c(R) ≥ d ;

(2) ∀℘ ⁰ , ℘ ⁰⁰ ∈ Min R, are d-connected

, i.e ∃ ℘ ⁰ = ℘ ₀ , ℘ ₁ ,

. . . , ℘ _r = ℘ ⁰⁰ , ℘ _i ∈ Min R and dim R/(℘ _j + ℘ _{j −1} ) ≥ d .

We will say that R is connected in codimension dim R − d .

A characterization of connectivity dimension

T.F.A.E.:

(1) c(R) ≥ d ;

(2) ∀℘ ⁰ , ℘ ⁰⁰ ∈ Min R, are d-connected , i.e ∃ ℘ ⁰ = ℘ ₀ , ℘ ₁ , . . . , ℘ _r = ℘ ⁰⁰ , ℘ _i ∈ Min R and dim R/(℘ _j + ℘ _{j −1} ) ≥ d .

We will say that R is connected in codimension dim R − d .

A characterization of connectivity dimension

T.F.A.E.:

(1) c(R) ≥ d ;

(2) ∀℘ ⁰ , ℘ ⁰⁰ ∈ Min R, are d-connected , i.e ∃ ℘ ⁰ = ℘ ₀ , ℘ ₁ ,

. . . , ℘ _r = ℘ ⁰⁰ , ℘ _i ∈ Min R and dim R/(℘ _j + ℘ _{j −1} ) ≥ d .

We will say that R is connected in codimension dim R − d .

A characterization of connectivity dimension

T.F.A.E.:

(1) c(R) ≥ d ;

(2) ∀℘ ⁰ , ℘ ⁰⁰ ∈ Min R, are d-connected , i.e ∃ ℘ ⁰ = ℘ ₀ , ℘ ₁ ,

. . . , ℘ _r = ℘ ⁰⁰ , ℘ _i ∈ Min R and dim R/(℘ _j + ℘ _{j −1} ) ≥ d .

We will say that R is connected in codimension dim R − d .

EXAMPLES

A characterization of connectivity dimension

T.F.A.E.:

(1) c(R) ≥ d ;

(2) ∀℘ ⁰ , ℘ ⁰⁰ ∈ Min R, are d-connected , i.e ∃ ℘ ⁰ = ℘ ₀ , ℘ ₁ , . . . , ℘ _r = ℘ ⁰⁰ , ℘ _i ∈ Min R and dim R/(℘ _j + ℘ _{j −1} ) ≥ d . We will say that R is connected in codimension dim R − d . EXAMPLES

S = k[x , y , z, v , w ], a = (xy , xv , xw , yv , yz, vz, wz), R = S /a.

R is connected in codimension 2, but not in codimension 1

A characterization of connectivity dimension

T.F.A.E.:

(1) c(R) ≥ d ;

(2) ∀℘ ⁰ , ℘ ⁰⁰ ∈ Min R, are d-connected , i.e ∃ ℘ ⁰ = ℘ ₀ , ℘ ₁ , . . . , ℘ _r = ℘ ⁰⁰ , ℘ _i ∈ Min R and dim R/(℘ _j + ℘ _{j −1} ) ≥ d . We will say that R is connected in codimension dim R − d . EXAMPLES

S = k[x , y , z, v , w ], a = (xy , xv , xw , yv , yz, vz, wz), R = S /a.

R is connected in codimension 2, but not in codimension 1, since

a = (x , y , z) ∩ (x , z, v ) ∩ (y , v , w ) = ℘ ₁ ∩ ℘ ₂ ∩ ℘ ₃

A characterization of connectivity dimension

T.F.A.E.:

(1) c(R) ≥ d ;

(2) ∀℘ ⁰ , ℘ ⁰⁰ ∈ Min R, are d-connected , i.e ∃ ℘ ⁰ = ℘ ₀ , ℘ ₁ , . . . , ℘ _r = ℘ ⁰⁰ , ℘ _i ∈ Min R and dim R/(℘ _j + ℘ _{j −1} ) ≥ d . We will say that R is connected in codimension dim R − d . EXAMPLES

S = k[x , y , z, v , w ], a = (xy , xv , xw , yv , yz, vz, wz), R = S /a.

R is connected in codimension 2, but not in codimension 1, since a = (x , y , z) ∩ (x , z, v ) ∩ (y , v , w ) = ℘ ₁ ∩ ℘ ₂ ∩ ℘ ₃

dim R = 2, dim S /(℘ ₁ + ℘ ₂ ) = 1,

dim S /(℘ ₁ + ℘ ₃ ) = 0, dim S /(℘ ₂ + ℘ ₃ ) = 0.

A characterization of connectivity dimension

T.F.A.E.:

(1) c(R) ≥ d ;

(2) ∀℘ ⁰ , ℘ ⁰⁰ ∈ Min R, are d-connected , i.e ∃ ℘ ⁰ = ℘ ₀ , ℘ ₁ , . . . , ℘ _r = ℘ ⁰⁰ , ℘ _i ∈ Min R and dim R/(℘ _j + ℘ _{j −1} ) ≥ d . We will say that R is connected in codimension dim R − d . EXAMPLES

S = k[x , y , z, v , w ], a = (zv , yv , xv , yw , yz, xz), R = S /a.

R is connected in codimension 1

A characterization of connectivity dimension

T.F.A.E.:

(1) c(R) ≥ d ;

(2) ∀℘ ⁰ , ℘ ⁰⁰ ∈ Min R, are d-connected , i.e ∃ ℘ ⁰ = ℘ ₀ , ℘ ₁ , . . . , ℘ _r = ℘ ⁰⁰ , ℘ _i ∈ Min R and dim R/(℘ _j + ℘ _{j −1} ) ≥ d . We will say that R is connected in codimension dim R − d . EXAMPLES

S = k[x , y , z, v , w ], a = (zv , yv , xv , yw , yz, xz), R = S /a.

R is connected in codimension 1, since

a = (x , y , z) ∩ (y , z, v ) ∩ (z, v , w ) ∩ (x , y , v ) = ℘ ₁ ∩ ℘ ₂ ∩ ℘ ₃ ∩ ℘ ₄

A characterization of connectivity dimension

T.F.A.E.:

(1) c(R) ≥ d ;

(2) ∀℘ ⁰ , ℘ ⁰⁰ ∈ Min R, are d-connected , i.e ∃ ℘ ⁰ = ℘ ₀ , ℘ ₁ , . . . , ℘ _r = ℘ ⁰⁰ , ℘ _i ∈ Min R and dim R/(℘ _j + ℘ _{j −1} ) ≥ d . We will say that R is connected in codimension dim R − d . EXAMPLES

S = k[x , y , z, v , w ], a = (zv , yv , xv , yw , yz, xz), R = S /a.

R is connected in codimension 1, since

a = (x , y , z) ∩ (y , z, v ) ∩ (z, v , w ) ∩ (x , y , v ) = ℘ ₁ ∩ ℘ ₂ ∩ ℘ ₃ ∩ ℘ ₄ dim R = 2, dim S /(℘ ₁ + ℘ ₂ ) = 1,

dim S /(℘ ₂ + ℘ ₃ ) = 1, dim S /(℘ ₃ + ℘ ₄ ) = 1.

The complete case

The complete case

Let (R, m) be local and complete.

The complete case

Let (R, m) be local and complete. Then

c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a)

The complete case

Let (R, m) be local and complete. Then

c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a)

Corollary

The complete case

Let (R, m) be local and complete. Then

c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a)

Corollary (Grothendieck)

(R, m) complete and local. Then

c(R/a) ≥ min{c(R), dim R − 1} − ara(a).

The complete case

Let (R, m) be local and complete. Then

c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a)

Corollary (Grothendieck)

(R, m) complete and local. Then

c(R/a) ≥ min{c(R), dim R − 1} − ara(a).

Proof of Corollary

The complete case

Let (R, m) be local and complete. Then

c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a)

Corollary (Grothendieck)

(R, m) complete and local. Then

c(R/a) ≥ min{c(R), dim R − 1} − ara(a).

Proof of Corollary

ara(a) ≥ cd(R, a).

The complete case

Let (R, m) be local and complete. Then

c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a)

Corollary (Hochster and Huneke)

(R, m) complete, loc., equidimensional, H _m ^{dim R} (R) indecomposable.

If cd(R, a) ≤ dim R − 2 then Spec R/a \ {m} is connected.

The complete case

Let (R, m) be local and complete. Then

c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a)

Corollary (Hochster and Huneke)

(R, m) complete, loc., equidimensional, H _m ^{dim R} (R) indecomposable.

If cd(R, a) ≤ dim R − 2 then Spec R/a \ {m} is connected.

Proof of Corollary

The complete case

Let (R, m) be local and complete. Then

c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a)

Corollary (Hochster and Huneke)

(R, m) complete, loc., equidimensional, H _m ^{dim R} (R) indecomposable.

If cd(R, a) ≤ dim R − 2 then Spec R/a \ {m} is connected.

Proof of Corollary

H _m ^{dim R} (R) indecomposable iff R connected in codimension 1.

The complete case

Let (R, m) be local and complete. Then

c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a)

Corollary (Hochster and Huneke)

(R, m) complete, loc., equidimensional, H _m ^{dim R} (R) indecomposable.

If cd(R, a) ≤ dim R − 2 then Spec R/a \ {m} is connected.

Proof of Corollary

H _m ^{dim R} (R) indecomposable iff R connected in codimension 1.

c(Spec R/a \ m) = c(R/a) − 1.

The Cohen-Macaulay and graded cases

(R, m) local Cohen-Macaulay ring. Then c(R/a) ≥ dim R − cd(R, a) − 1 . Proof

R Cohen-Macaulay ⇔ b R Cohen-Macaulay Cohen-Macaulay ⇒ connected in codimension 1

c(R/a) ≥ c(b R/ab R) cd(R, a) = cd(b R, ab R)

R k-algebra finitely generated positively graded, a homogeneus.

Then c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a).

The Cohen-Macaulay and graded cases

(R, m) local Cohen-Macaulay ring.

Then c(R/a) ≥ dim R − cd(R, a) − 1 . Proof

R Cohen-Macaulay ⇔ b R Cohen-Macaulay Cohen-Macaulay ⇒ connected in codimension 1

c(R/a) ≥ c(b R/ab R) cd(R, a) = cd(b R, ab R)

R k-algebra finitely generated positively graded, a homogeneus.

Then c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a).

The Cohen-Macaulay and graded cases

(R, m) local Cohen-Macaulay ring.

Then c(R/a) ≥ dim R − cd(R, a) − 1 .

Proof

R Cohen-Macaulay ⇔ b R Cohen-Macaulay Cohen-Macaulay ⇒ connected in codimension 1

c(R/a) ≥ c(b R/ab R) cd(R, a) = cd(b R, ab R)

R k-algebra finitely generated positively graded, a homogeneus.

Then c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a).

The Cohen-Macaulay and graded cases

(R, m) local Cohen-Macaulay ring.

Then c(R/a) ≥ dim R − cd(R, a) − 1 . Proof

R Cohen-Macaulay ⇔ b R Cohen-Macaulay Cohen-Macaulay ⇒ connected in codimension 1

c(R/a) ≥ c(b R/ab R) cd(R, a) = cd(b R, ab R)

R k-algebra finitely generated positively graded, a homogeneus.

Then c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a).

The Cohen-Macaulay and graded cases

(R, m) local Cohen-Macaulay ring.

Then c(R/a) ≥ dim R − cd(R, a) − 1 . Proof

R Cohen-Macaulay ⇔ b R Cohen-Macaulay

Cohen-Macaulay ⇒ connected in codimension 1 c(R/a) ≥ c(b R/ab R)

cd(R, a) = cd(b R, ab R)

R k-algebra finitely generated positively graded, a homogeneus.

Then c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a).

The Cohen-Macaulay and graded cases

(R, m) local Cohen-Macaulay ring.

Then c(R/a) ≥ dim R − cd(R, a) − 1 . Proof

R Cohen-Macaulay ⇔ b R Cohen-Macaulay Cohen-Macaulay ⇒ connected in codimension 1

c(R/a) ≥ c(b R/ab R) cd(R, a) = cd(b R, ab R)

R k-algebra finitely generated positively graded, a homogeneus.

Then c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a).

The Cohen-Macaulay and graded cases

(R, m) local Cohen-Macaulay ring.

Then c(R/a) ≥ dim R − cd(R, a) − 1 . Proof

R Cohen-Macaulay ⇔ b R Cohen-Macaulay Cohen-Macaulay ⇒ connected in codimension 1

c(R/a) ≥ c(b R/ab R)

cd(R, a) = cd(b R, ab R)

R k-algebra finitely generated positively graded, a homogeneus.

Then c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a).

The Cohen-Macaulay and graded cases

(R, m) local Cohen-Macaulay ring.

Then c(R/a) ≥ dim R − cd(R, a) − 1 . Proof

R Cohen-Macaulay ⇔ b R Cohen-Macaulay Cohen-Macaulay ⇒ connected in codimension 1

c(R/a) ≥ c(b R/ab R) cd(R, a) = cd(b R, ab R)

R k-algebra finitely generated positively graded, a homogeneus.

Then c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a).

The Cohen-Macaulay and graded cases

(R, m) local Cohen-Macaulay ring.

Then c(R/a) ≥ dim R − cd(R, a) − 1 . Proof

R Cohen-Macaulay ⇔ b R Cohen-Macaulay Cohen-Macaulay ⇒ connected in codimension 1

c(R/a) ≥ c(b R/ab R) cd(R, a) = cd(b R, ab R)

R k-algebra finitely generated positively graded, a homogeneus.

Then c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a).

The Cohen-Macaulay and graded cases

(R, m) local Cohen-Macaulay ring.

Then c(R/a) ≥ dim R − cd(R, a) − 1 . Proof

R Cohen-Macaulay ⇔ b R Cohen-Macaulay Cohen-Macaulay ⇒ connected in codimension 1

c(R/a) ≥ c(b R/ab R) cd(R, a) = cd(b R, ab R)

R k-algebra finitely generated positively graded, a homogeneus.

Then c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a).

Vanishing of local cohomology and connectedness

(Hartshorne) (R, m) local ring. If H _m ⁱ (R) = 0 for every i ≤ k , then c(R) ≥ k.

Equivalently c(R) ≥ depth(R) − 1.

So, if R is a positively graded n-dimensional k-algebra connected in codimension 1 and m = ⊕ _{d >0} R _d ,

H _a ⁿ⁻ⁱ (R) = 0 ∀i ≤ k ⇒ c(R/a) ≥ k

H _m ⁱ (R/a) = 0 ∀i ≤ k ⇒ c(R/a) ≥ k

Vanishing of local cohomology and connectedness

(Hartshorne) (R, m) local ring. If H _m ⁱ (R) = 0 for every i ≤ k ,

then c(R) ≥ k.

Equivalently c(R) ≥ depth(R) − 1.

So, if R is a positively graded n-dimensional k-algebra connected in codimension 1 and m = ⊕ _{d >0} R _d ,

H _a ⁿ⁻ⁱ (R) = 0 ∀i ≤ k ⇒ c(R/a) ≥ k

H _m ⁱ (R/a) = 0 ∀i ≤ k ⇒ c(R/a) ≥ k

Vanishing of local cohomology and connectedness

(Hartshorne) (R, m) local ring. If H _m ⁱ (R) = 0 for every i ≤ k , then c(R) ≥ k.

Equivalently c(R) ≥ depth(R) − 1.

So, if R is a positively graded n-dimensional k-algebra connected in codimension 1 and m = ⊕ _{d >0} R _d ,

H _a ⁿ⁻ⁱ (R) = 0 ∀i ≤ k ⇒ c(R/a) ≥ k

H _m ⁱ (R/a) = 0 ∀i ≤ k ⇒ c(R/a) ≥ k

Vanishing of local cohomology and connectedness

(Hartshorne) (R, m) local ring. If H _m ⁱ (R) = 0 for every i ≤ k , then c(R) ≥ k.

Equivalently c(R) ≥ depth(R) − 1.

So, if R is a positively graded n-dimensional k-algebra connected in codimension 1 and m = ⊕ _{d >0} R _d ,

H _a ⁿ⁻ⁱ (R) = 0 ∀i ≤ k ⇒ c(R/a) ≥ k

H _m ⁱ (R/a) = 0 ∀i ≤ k ⇒ c(R/a) ≥ k

Vanishing of local cohomology and connectedness

(Hartshorne) (R, m) local ring. If H _m ⁱ (R) = 0 for every i ≤ k , then c(R) ≥ k.

Equivalently c(R) ≥ depth(R) − 1.

So, if R is a positively graded n-dimensional k-algebra connected in codimension 1 and m = ⊕ _{d >0} R _d ,

H _a ⁿ⁻ⁱ (R) = 0 ∀i ≤ k ⇒ c(R/a) ≥ k

H _m ⁱ (R/a) = 0 ∀i ≤ k ⇒ c(R/a) ≥ k

Vanishing of local cohomology and connectedness

(Hartshorne) (R, m) local ring. If H _m ⁱ (R) = 0 for every i ≤ k , then c(R) ≥ k.

Equivalently c(R) ≥ depth(R) − 1.

So, if R is a positively graded n-dimensional k-algebra connected in codimension 1 and m = ⊕ _{d >0} R _d ,

H _a ⁿ⁻ⁱ (R) = 0 ∀i ≤ k ⇒ c(R/a) ≥ k

H _m ⁱ (R/a) = 0 ∀i ≤ k ⇒ c(R/a) ≥ k

Vanishing of local cohomology and connectedness

(Hartshorne) (R, m) local ring. If H _m ⁱ (R) = 0 for every i ≤ k , then c(R) ≥ k.

Equivalently c(R) ≥ depth(R) − 1.

So, if R is a positively graded n-dimensional k-algebra connected in codimension 1 and m = ⊕ _{d >0} R _d ,

H _a ⁿ⁻ⁱ (R) = 0 ∀i ≤ k ⇒ c(R/a) ≥ k

H _m ⁱ (R/a) = 0 ∀i ≤ k ⇒ c(R/a) ≥ k

PART 2

APPLICATIONS TO INITIAL IDEALS

Notation

I S := k[x 1 , . . . , x n ];

I I ⊆ S ideal;

I we will say I Cohen-Macaulay for S /I Cohen-Macaulay;

I ≺ term order on S ;

I LT ≺ (I ) ⊆ S ideal of leading terms of I with respect to ≺;

I ω = (ω 1 , . . . , ω n ) ∈ (Z + ) ⁿ weight vector;

Notation

I S := k[x 1 , . . . , x n ];

I I ⊆ S ideal;

I we will say I Cohen-Macaulay for S /I Cohen-Macaulay;

I ≺ term order on S ;

I LT ≺ (I ) ⊆ S ideal of leading terms of I with respect to ≺;

I ω = (ω 1 , . . . , ω n ) ∈ (Z + ) ⁿ weight vector;

Notation

I S := k[x 1 , . . . , x n ];

I I ⊆ S ideal;

I we will say I Cohen-Macaulay for S /I Cohen-Macaulay;

I ≺ term order on S ;

I LT ≺ (I ) ⊆ S ideal of leading terms of I with respect to ≺;

I ω = (ω 1 , . . . , ω n ) ∈ (Z + ) ⁿ weight vector;

Notation

I S := k[x 1 , . . . , x n ];

I I ⊆ S ideal;

I we will say I Cohen-Macaulay for S /I Cohen-Macaulay;

I ≺ term order on S ;

I LT ≺ (I ) ⊆ S ideal of leading terms of I with respect to ≺;

I ω = (ω 1 , . . . , ω n ) ∈ (Z + ) ⁿ weight vector;

Notation

I S := k[x 1 , . . . , x n ];

I I ⊆ S ideal;

I we will say I Cohen-Macaulay for S /I Cohen-Macaulay;

I ≺ term order on S ;

I LT ≺ (I ) ⊆ S ideal of leading terms of I with respect to ≺;

I ω = (ω 1 , . . . , ω n ) ∈ (Z + ) ⁿ weight vector;

Notation

I S := k[x 1 , . . . , x n ];

I I ⊆ S ideal;

I we will say I Cohen-Macaulay for S /I Cohen-Macaulay;

I ≺ term order on S ;

I LT ≺ (I ) ⊆ S ideal of leading terms of I with respect to ≺;

I ω = (ω 1 , . . . , ω n ) ∈ (Z + ) ⁿ weight vector;

Notation

I S := k[x 1 , . . . , x n ];

I I ⊆ S ideal;

I we will say I Cohen-Macaulay for S /I Cohen-Macaulay;

I ≺ term order on S ;

I LT ≺ (I ) ⊆ S ideal of leading terms of I with respect to ≺;

I ω = (ω 1 , . . . , ω n ) ∈ (Z + ) ⁿ weight vector;

Initial ideals with respect to weight vectors

Initial ideals with respect to weight vectors

- f ∈ S , in _ω (f ) ∈ S is the leading coefficient of

f (x ₁ t ^ω

, . . . , x _n t ^ω

) ∈ S[t]

Initial ideals with respect to weight vectors

- f ∈ S , in _ω (f ) ∈ S is the leading coefficient of f (x ₁ t ^ω

, . . . , x _n t ^ω

) ∈ S[t]

E. g., f = 2x ₁ x ₂ ³ + x ₁ x ₃ ³ + 3x ₂ ⁴ x ₃ ∈ k[x ₁ , x ₂ , x ₃ ] and ω = (3, 2, 1);

Initial ideals with respect to weight vectors

- f ∈ S , in _ω (f ) ∈ S is the leading coefficient of f (x ₁ t ^ω

, . . . , x _n t ^ω

) ∈ S[t]

E. g., f = 2x ₁ x ₂ ³ + x ₁ x ₃ ³ + 3x ₂ ⁴ x ₃ ∈ k[x ₁ , x ₂ , x ₃ ] and ω = (3, 2, 1);

then f (x 1 t ³ , x 2 t ² , x 3 t) = 2x 1 x ₂ ³ t ⁹ + x 1 x ₃ ³ t ⁶ + 3x ₂ ⁴ x 3 t ⁹

Initial ideals with respect to weight vectors

- f ∈ S , in _ω (f ) ∈ S is the leading coefficient of f (x ₁ t ^ω

, . . . , x _n t ^ω

) ∈ S[t]

E. g., f = 2x ₁ x ₂ ³ + x ₁ x ₃ ³ + 3x ₂ ⁴ x ₃ ∈ k[x ₁ , x ₂ , x ₃ ] and ω = (3, 2, 1);

then f (x 1 t ³ , x 2 t ² , x 3 t) = 2x 1 x ₂ ³ t ⁹ + x 1 x ₃ ³ t ⁶ + 3x ₂ ⁴ x 3 t ⁹ , so

in ω (f ) = 2x 1 x ₂ ³ + 3x ₂ ⁴ x 3

Initial ideals with respect to weight vectors

- f ∈ S , in _ω (f ) ∈ S is the leading coefficient of f (x ₁ t ^ω

, . . . , x _n t ^ω

) ∈ S[t]

- in _ω (I ):= (in _ω (f ) : f ∈ I ) ⊆ S

Initial ideals with respect to weight vectors

- f ∈ S , in _ω (f ) ∈ S is the leading coefficient of f (x ₁ t ^ω

, . . . , x _n t ^ω

) ∈ S[t]

- in _ω (I ):= (in _ω (f ) : f ∈ I ) ⊆ S

for every I , ≺ there exists ω such that LT ≺ (I ) = in _ω (I )

Initial ideals with respect to weight vectors

- f ∈ S , in _ω (f ) ∈ S is the leading coefficient of f (x ₁ t ^ω

, . . . , x _n t ^ω

) ∈ S[t]

- in _ω (I ):= (in _ω (f ) : f ∈ I ) ⊆ S

for every I , ≺ there exists ω such that LT ≺ (I ) = in _ω (I )

- ^ω f (x 1 , . . . , x n , t) := f ( x 1

t ^ω

, . . . , x n

t ^ω

)t ^deg

^f ∈ S[t]

Initial ideals with respect to weight vectors

- f ∈ S , in _ω (f ) ∈ S is the leading coefficient of f (x ₁ t ^ω

, . . . , x _n t ^ω

) ∈ S[t]

- in _ω (I ):= (in _ω (f ) : f ∈ I ) ⊆ S

for every I , ≺ there exists ω such that LT ≺ (I ) = in _ω (I )

- deg _ω f := max{ P n

i =1 ω _i a _i : x ₁ ^a

· · · x _n ^a

term of f } - ^ω f (x ₁ , . . . , x _n , t) := f ( x 1

t ^ω

, . . . , x n

t ^ω

)t ^deg

^f ∈ S[t]

Initial ideals with respect to weight vectors

- f ∈ S , in _ω (f ) ∈ S is the leading coefficient of f (x ₁ t ^ω

, . . . , x _n t ^ω

) ∈ S[t]

- in _ω (I ):= (in _ω (f ) : f ∈ I ) ⊆ S

for every I , ≺ there exists ω such that LT ≺ (I ) = in _ω (I )

- deg _ω f := max{ P n

i =1 ω _i a _i : x ₁ ^a

· · · x _n ^a

term of f } - ^ω f (x ₁ , . . . , x _n , t) := f ( x 1

t ^ω

, . . . , x n

t ^ω

)t ^deg

^f ∈ S[t]

- ^ω I := ( ^ω f : f ∈ I ) ⊆ S [t]

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

Corollary

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

Corollary (Kalkbrener and Sturmfels).

For every ω and I , if I is prime, then

S / in _ω (I ) is connected in codimension 1.

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

Corollary (Kalkbrener and Sturmfels).

For every ω and I , if I is prime, then

S / in _ω (I ) is connected in codimension 1.

Proof of Corollary

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

Corollary (Kalkbrener and Sturmfels).

For every ω and I , if I is prime, then

S / in _ω (I ) is connected in codimension 1.

Proof of Corollary

I prime ⇒ c(S /I ) = dim S /I .

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

Corollary For every ≺ and I , then

c(S/LT ≺ (I )) ≥ min{c(S /I ), dim S /I − 1}.

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

Corollary For every ≺ and I , then

c(S/LT ≺ (I )) ≥ min{c(S /I ), dim S /I − 1}.

Proof of Corollary

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

Corollary For every ≺ and I , then

c(S/LT ≺ (I )) ≥ min{c(S /I ), dim S /I − 1}.

Proof of Corollary

Choose ω such that in ω (I ) = LT ≺ (I ).

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

Corollary For every ω and graded I , the following holds:

c(S/ in _ω (I )) ≥ depth(S /I ) − 1.

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

Corollary For every ω and graded I , the following holds:

c(S/ in _ω (I )) ≥ depth(S /I ) − 1.

So, I Cohen-Macaulay ⇒ S / in _ω (I ) connected in codimension 1.

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

Corollary For every ω and graded I , the following holds:

c(S/ in _ω (I )) ≥ depth(S /I ) − 1.

So, I Cohen-Macaulay ⇒ S / in _ω (I ) connected in codimension 1.

Proof of Corollary

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

Corollary For every ω and graded I , the following holds:

c(S/ in _ω (I )) ≥ depth(S /I ) − 1.

So, I Cohen-Macaulay ⇒ S / in _ω (I ) connected in codimension 1.

Proof of Corollary

Follows from the fact that c(S /I ) ≥ depth(S /I ) − 1.

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

Corollary For every ≺ and graded I , the following holds:

depth(S /I ) ≥ 2 ⇒ depth(S /pLT ≺ (I )) ≥ 2.

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

Corollary For every ≺ and graded I , the following holds:

depth(S /I ) ≥ 2 ⇒ depth(S /pLT ≺ (I )) ≥ 2.

Proof of Corollary

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

Corollary For every ≺ and graded I , the following holds:

depth(S /I ) ≥ 2 ⇒ depth(S /pLT ≺ (I )) ≥ 2.

Proof of Corollary

depth(S /I ) ≥ 2 ⇒ c(S /I ) ≥ 1 ⇒ c(S /LT ≺ (I )) ≥ 1

The main result

For every I and ω

c(S/ in _ω (I )) ≥ min{c(S /I ), dim S /I − 1}

Corollary For every ≺ and graded I , the following holds:

depth(S /I ) ≥ 2 ⇒ depth(S /pLT ≺ (I )) ≥ 2.

Proof of Corollary

depth(S /I ) ≥ 2 ⇒ c(S /I ) ≥ 1 ⇒ c(S /LT ≺ (I )) ≥ 1

⇒ depth(S/pLT ≺ (I )) ≥ 2.

Idea of the proof

Let be R := S [t]/ ^ω I and a := ( ^ω I + t)/ ^ω I ⊆ R.

S / in _ω (I ) ∼ = R/a

Give a positive graduation to R s.t. a is homogeneus as follows: deg ¯ x i = ω i , deg ¯ t = 1

Graded version of the main result of the first part let us conclude!

Idea of the proof

Let be R := S [t]/ ^ω I and a := ( ^ω I + t)/ ^ω I ⊆ R.

S / in _ω (I ) ∼ = R/a

Give a positive graduation to R s.t. a is homogeneus as follows: deg ¯ x i = ω i , deg ¯ t = 1

Graded version of the main result of the first part let us conclude!

Idea of the proof

Let be R := S [t]/ ^ω I and a := ( ^ω I + t)/ ^ω I ⊆ R.

S / in _ω (I ) ∼ = R/a

Give a positive graduation to R s.t. a is homogeneus as follows: deg ¯ x i = ω i , deg ¯ t = 1

Graded version of the main result of the first part let us conclude!

Idea of the proof

Let be R := S [t]/ ^ω I and a := ( ^ω I + t)/ ^ω I ⊆ R.

S / in _ω (I ) ∼ = R/a

Give a positive graduation to R s.t. a is homogeneus as follows:

deg ¯ x i = ω i , deg ¯ t = 1

Graded version of the main result of the first part let us conclude!

Idea of the proof

Let be R := S [t]/ ^ω I and a := ( ^ω I + t)/ ^ω I ⊆ R.

S / in _ω (I ) ∼ = R/a

Give a positive graduation to R s.t. a is homogeneus as follows:

deg ¯ x i = ω i , deg ¯ t = 1

Graded version of the main result of the first part let us conclude!

Idea of the proof

Let be R := S [t]/ ^ω I and a := ( ^ω I + t)/ ^ω I ⊆ R.

S / in _ω (I ) ∼ = R/a

Give a positive graduation to R s.t. a is homogeneus as follows:

deg ¯ x i = ω i , deg ¯ t = 1

Graded version of the main result of the first part let us conclude!

The answer to the initial question

J = (ua, za, ya, xa, uv , zv , yv , xv , xyu, xyz, xzu, yzu)

The minimal primes of J ⊆ S := k[x , y , z, u, v , w , a] are:

℘ ₁ = (x , y , z, u) , ℘ ₂ = (x , y , v , a), ℘ ₃ = (x , z, v , a),

℘ 4 = (x , u, v , a), ℘ 5 = (y , z, v , a),

℘ 6 = (y , u, v , a), ℘ 7 = (z, u, v , a).

Note that dim S /(℘ ₁ + ℘ _i ) = 1 whereas dim S /J = 3.

So S /J is not connected in codimension 1, therefore

cannot exist I ⊆ S Cohen-Macaulay and ≺ s. t. pLT ≺ (I ) = J!

The answer to the initial question

J = (ua, za, ya, xa, uv , zv , yv , xv , xyu, xyz, xzu, yzu)

The minimal primes of J ⊆ S := k[x , y , z, u, v , w , a] are:

℘ ₁ = (x , y , z, u) , ℘ ₂ = (x , y , v , a), ℘ ₃ = (x , z, v , a),

℘ 4 = (x , u, v , a), ℘ 5 = (y , z, v , a),

℘ 6 = (y , u, v , a), ℘ 7 = (z, u, v , a).

Note that dim S /(℘ ₁ + ℘ _i ) = 1 whereas dim S /J = 3.

So S /J is not connected in codimension 1, therefore

cannot exist I ⊆ S Cohen-Macaulay and ≺ s. t. pLT ≺ (I ) = J!

The answer to the initial question

J = (ua, za, ya, xa, uv , zv , yv , xv , xyu, xyz, xzu, yzu)

The minimal primes of J ⊆ S := k[x , y , z, u, v , w , a] are:

℘ ₁ = (x , y , z, u) , ℘ ₂ = (x , y , v , a), ℘ ₃ = (x , z, v , a),

℘ ₄ = (x , u, v , a), ℘ ₅ = (y , z, v , a),

℘ 6 = (y , u, v , a), ℘ 7 = (z, u, v , a).

Note that dim S /(℘ ₁ + ℘ _i ) = 1 whereas dim S /J = 3.

So S /J is not connected in codimension 1, therefore

cannot exist I ⊆ S Cohen-Macaulay and ≺ s. t. pLT ≺ (I ) = J!

The answer to the initial question

J = (ua, za, ya, xa, uv , zv , yv , xv , xyu, xyz, xzu, yzu)

The minimal primes of J ⊆ S := k[x , y , z, u, v , w , a] are:

℘ ₁ = (x , y , z, u) , ℘ ₂ = (x , y , v , a), ℘ ₃ = (x , z, v , a),

℘ ₄ = (x , u, v , a), ℘ ₅ = (y , z, v , a),

℘ 6 = (y , u, v , a), ℘ 7 = (z, u, v , a).

Note that dim S /(℘ ₁ + ℘ _i ) = 1

whereas dim S /J = 3.

So S /J is not connected in codimension 1, therefore

cannot exist I ⊆ S Cohen-Macaulay and ≺ s. t. pLT ≺ (I ) = J!

The answer to the initial question

J = (ua, za, ya, xa, uv , zv , yv , xv , xyu, xyz, xzu, yzu)

The minimal primes of J ⊆ S := k[x , y , z, u, v , w , a] are:

℘ ₁ = (x , y , z, u) , ℘ ₂ = (x , y , v , a), ℘ ₃ = (x , z, v , a),

℘ ₄ = (x , u, v , a), ℘ ₅ = (y , z, v , a),

℘ 6 = (y , u, v , a), ℘ 7 = (z, u, v , a).

Note that dim S /(℘ ₁ + ℘ _i ) = 1 whereas dim S /J = 3.

So S /J is not connected in codimension 1, therefore

cannot exist I ⊆ S Cohen-Macaulay and ≺ s. t. pLT ≺ (I ) = J!

The answer to the initial question

J = (ua, za, ya, xa, uv , zv , yv , xv , xyu, xyz, xzu, yzu)

The minimal primes of J ⊆ S := k[x , y , z, u, v , w , a] are:

℘ ₁ = (x , y , z, u) , ℘ ₂ = (x , y , v , a), ℘ ₃ = (x , z, v , a),

℘ ₄ = (x , u, v , a), ℘ ₅ = (y , z, v , a),

℘ 6 = (y , u, v , a), ℘ 7 = (z, u, v , a).

Note that dim S /(℘ ₁ + ℘ _i ) = 1 whereas dim S /J = 3.

So S /J is not connected in codimension 1,

therefore

cannot exist I ⊆ S Cohen-Macaulay and ≺ s. t. pLT ≺ (I ) = J!

The answer to the initial question

J = (ua, za, ya, xa, uv , zv , yv , xv , xyu, xyz, xzu, yzu)

The minimal primes of J ⊆ S := k[x , y , z, u, v , w , a] are:

℘ ₁ = (x , y , z, u) , ℘ ₂ = (x , y , v , a), ℘ ₃ = (x , z, v , a),

℘ ₄ = (x , u, v , a), ℘ ₅ = (y , z, v , a),

℘ 6 = (y , u, v , a), ℘ 7 = (z, u, v , a).

Note that dim S /(℘ ₁ + ℘ _i ) = 1 whereas dim S /J = 3.

So S /J is not connected in codimension 1, therefore

cannot exist I ⊆ S Cohen-Macaulay and ≺ s. t. pLT ≺ (I ) = J!

Examples

S := C[x 1 , . . . , x 7 ]

X :=





x ₁ + x ₂ x ₅ x ₄ x 6 − x ₅ x 3 + x 7 x 5

x ₄ + x ₇ x ₁ − x ₃ x ₅ + x ₇





I := I 2 (X ) is a prime ideal of codimension 4, so S /I is a Gorenstein domain of dimension 3.

Moreover Proj(S /I ) is a smooth surface. If ≺ is the lexicographic order, then

LT ≺ (I ) 6= pLT ≺ (I ) = (x ₁ , x ₃ x ₅ , x ₂ x ₅ , x ₃ x ₄ , x ₄ x ₅ , x ₂ x ₃ x ₆ , x ₂ x ₄ x ₇ , x ₂ x ₆ x ₇ )

S /pLT _≺ (I ) is not Cohen-Macaulay !

Examples

S := C[x 1 , . . . , x 7 ]

X :=





x ₁ + x ₂ x ₅ x ₄ x 6 − x ₅ x 3 + x 7 x 5

x ₄ + x ₇ x ₁ − x ₃ x ₅ + x ₇





I := I 2 (X ) is a prime ideal of codimension 4, so S /I is a Gorenstein domain of dimension 3.

Moreover Proj(S /I ) is a smooth surface. If ≺ is the lexicographic order, then

LT ≺ (I ) 6= pLT ≺ (I ) = (x ₁ , x ₃ x ₅ , x ₂ x ₅ , x ₃ x ₄ , x ₄ x ₅ , x ₂ x ₃ x ₆ , x ₂ x ₄ x ₇ , x ₂ x ₆ x ₇ )

S /pLT _≺ (I ) is not Cohen-Macaulay !

Examples

S := C[x 1 , . . . , x 7 ]

X :=





x ₁ + x ₂ x ₅ x ₄ x 6 − x ₅ x 3 + x 7 x 5

x ₄ + x ₇ x ₁ − x ₃ x ₅ + x ₇





I := I 2 (X ) is a prime ideal of codimension 4, so S /I is a Gorenstein domain of dimension 3.

Moreover Proj(S /I ) is a smooth surface. If ≺ is the lexicographic order, then

LT ≺ (I ) 6= pLT ≺ (I ) = (x ₁ , x ₃ x ₅ , x ₂ x ₅ , x ₃ x ₄ , x ₄ x ₅ , x ₂ x ₃ x ₆ , x ₂ x ₄ x ₇ , x ₂ x ₆ x ₇ )

S /pLT _≺ (I ) is not Cohen-Macaulay !

Examples

S := C[x 1 , . . . , x 7 ]

X :=





x ₁ + x ₂ x ₅ x ₄ x 6 − x ₅ x 3 + x 7 x 5

x ₄ + x ₇ x ₁ − x ₃ x ₅ + x ₇





I := I 2 (X ) is a prime ideal of codimension 4,

so S /I is a Gorenstein domain of dimension 3. Moreover Proj(S /I ) is a smooth surface. If ≺ is the lexicographic order, then

LT ≺ (I ) 6= pLT ≺ (I ) = (x ₁ , x ₃ x ₅ , x ₂ x ₅ , x ₃ x ₄ , x ₄ x ₅ , x ₂ x ₃ x ₆ , x ₂ x ₄ x ₇ , x ₂ x ₆ x ₇ )

S /pLT _≺ (I ) is not Cohen-Macaulay !

Examples

S := C[x 1 , . . . , x 7 ]

X :=





x ₁ + x ₂ x ₅ x ₄ x 6 − x ₅ x 3 + x 7 x 5

x ₄ + x ₇ x ₁ − x ₃ x ₅ + x ₇





I := I 2 (X ) is a prime ideal of codimension 4, so S /I is a Gorenstein domain of dimension 3.

Moreover Proj(S /I ) is a smooth surface. If ≺ is the lexicographic order, then

LT ≺ (I ) 6= pLT ≺ (I ) = (x ₁ , x ₃ x ₅ , x ₂ x ₅ , x ₃ x ₄ , x ₄ x ₅ , x ₂ x ₃ x ₆ , x ₂ x ₄ x ₇ , x ₂ x ₆ x ₇ )

S /pLT _≺ (I ) is not Cohen-Macaulay !

Examples

S := C[x 1 , . . . , x 7 ]

X :=





x ₁ + x ₂ x ₅ x ₄ x 6 − x ₅ x 3 + x 7 x 5

x ₄ + x ₇ x ₁ − x ₃ x ₅ + x ₇





I := I 2 (X ) is a prime ideal of codimension 4, so S /I is a Gorenstein domain of dimension 3.

Moreover Proj(S /I ) is a smooth surface.

If ≺ is the lexicographic order, then

LT ≺ (I ) 6= pLT ≺ (I ) = (x ₁ , x ₃ x ₅ , x ₂ x ₅ , x ₃ x ₄ , x ₄ x ₅ , x ₂ x ₃ x ₆ , x ₂ x ₄ x ₇ , x ₂ x ₆ x ₇ )

S /pLT _≺ (I ) is not Cohen-Macaulay !

Examples

S := C[x 1 , . . . , x 7 ]

X :=





x ₁ + x ₂ x ₅ x ₄ x 6 − x ₅ x 3 + x 7 x 5

x ₄ + x ₇ x ₁ − x ₃ x ₅ + x ₇





I := I 2 (X ) is a prime ideal of codimension 4, so S /I is a Gorenstein domain of dimension 3.

Moreover Proj(S /I ) is a smooth surface.

If ≺ is the lexicographic order, then

LT ≺ (I ) 6= pLT ≺ (I ) = (x ₁ , x ₃ x ₅ , x ₂ x ₅ , x ₃ x ₄ , x ₄ x ₅ , x ₂ x ₃ x ₆ , x ₂ x ₄ x ₇ , x ₂ x ₆ x ₇ )

S /pLT _≺ (I ) is not Cohen-Macaulay !

Examples

S := C[x 1 , . . . , x 7 ]

X :=





x ₁ + x ₂ x ₅ x ₄ x 6 − x ₅ x 3 + x 7 x 5

x ₄ + x ₇ x ₁ − x ₃ x ₅ + x ₇





I := I 2 (X ) is a prime ideal of codimension 4, so S /I is a Gorenstein domain of dimension 3.

Moreover Proj(S /I ) is a smooth surface.

If ≺ is the lexicographic order, then

LT ≺ (I ) 6= pLT ≺ (I ) = (x ₁ , x ₃ x ₅ , x ₂ x ₅ , x ₃ x ₄ , x ₄ x ₅ , x ₂ x ₃ x ₆ , x ₂ x ₄ x ₇ , x ₂ x ₆ x ₇ )

S /pLT _≺ (I ) is not Cohen-Macaulay !

Examples

S := C[x 1 , . . . , x 6 ]

I := (x ₁ x ₅ + x ₂ x ₆ + x ₄ ² , x ₁ x ₄ + x ₃ ² − x ₄ x ₅ , x ₁ ² + x ₁ x ₂ + x ₂ x ₅ ) I is a prime complete intersection of codimension 3, Proj(S /I ) is a normal surface with 3 singular points.

If ≺ is the lexicographic order, then

LT ≺ (I ) 6= pLT ≺ (I ) = (x ₁ , x ₃ x ₅ , x ₂ x ₃ x ₄ , x ₂ x ₄ x ₆ , x ₂ x ₅ x ₆ ) S /pLT _≺ (I ) is not Cohen-Macaulay !

In this case we have also that cd(S , I ) < cd(S , LT ≺ (I ))!

Examples

S := C[x 1 , . . . , x ₆ ]

I := (x ₁ x ₅ + x ₂ x ₆ + x ₄ ² , x ₁ x ₄ + x ₃ ² − x ₄ x ₅ , x ₁ ² + x ₁ x ₂ + x ₂ x ₅ ) I is a prime complete intersection of codimension 3, Proj(S /I ) is a normal surface with 3 singular points.

If ≺ is the lexicographic order, then

LT ≺ (I ) 6= pLT ≺ (I ) = (x ₁ , x ₃ x ₅ , x ₂ x ₃ x ₄ , x ₂ x ₄ x ₆ , x ₂ x ₅ x ₆ ) S /pLT _≺ (I ) is not Cohen-Macaulay !

In this case we have also that cd(S , I ) < cd(S , LT ≺ (I ))!

Examples

S := C[x 1 , . . . , x ₆ ]

I := (x ₁ x ₅ + x ₂ x ₆ + x ₄ ² , x ₁ x ₄ + x ₃ ² − x ₄ x ₅ , x ₁ ² + x ₁ x ₂ + x ₂ x ₅ )

I is a prime complete intersection of codimension 3, Proj(S /I ) is a normal surface with 3 singular points.

If ≺ is the lexicographic order, then

LT ≺ (I ) 6= pLT ≺ (I ) = (x ₁ , x ₃ x ₅ , x ₂ x ₃ x ₄ , x ₂ x ₄ x ₆ , x ₂ x ₅ x ₆ ) S /pLT _≺ (I ) is not Cohen-Macaulay !

In this case we have also that cd(S , I ) < cd(S , LT ≺ (I ))!

Examples

S := C[x 1 , . . . , x ₆ ]

I := (x ₁ x ₅ + x ₂ x ₆ + x ₄ ² , x ₁ x ₄ + x ₃ ² − x ₄ x ₅ , x ₁ ² + x ₁ x ₂ + x ₂ x ₅ ) I is a prime complete intersection of codimension 3,

Proj(S /I ) is a normal surface with 3 singular points.

If ≺ is the lexicographic order, then

LT ≺ (I ) 6= pLT ≺ (I ) = (x ₁ , x ₃ x ₅ , x ₂ x ₃ x ₄ , x ₂ x ₄ x ₆ , x ₂ x ₅ x ₆ ) S /pLT _≺ (I ) is not Cohen-Macaulay !

In this case we have also that cd(S , I ) < cd(S , LT ≺ (I ))!

Examples

S := C[x 1 , . . . , x ₆ ]

I := (x ₁ x ₅ + x ₂ x ₆ + x ₄ ² , x ₁ x ₄ + x ₃ ² − x ₄ x ₅ , x ₁ ² + x ₁ x ₂ + x ₂ x ₅ ) I is a prime complete intersection of codimension 3, Proj(S /I ) is a normal surface with 3 singular points.

If ≺ is the lexicographic order, then

LT ≺ (I ) 6= pLT ≺ (I ) = (x ₁ , x ₃ x ₅ , x ₂ x ₃ x ₄ , x ₂ x ₄ x ₆ , x ₂ x ₅ x ₆ )

S /pLT _≺ (I ) is not Cohen-Macaulay !

In this case we have also that cd(S , I ) < cd(S , LT ≺ (I ))!

Examples

S := C[x 1 , . . . , x ₆ ]

I := (x ₁ x ₅ + x ₂ x ₆ + x ₄ ² , x ₁ x ₄ + x ₃ ² − x ₄ x ₅ , x ₁ ² + x ₁ x ₂ + x ₂ x ₅ ) I is a prime complete intersection of codimension 3, Proj(S /I ) is a normal surface with 3 singular points.

If ≺ is the lexicographic order, then

LT ≺ (I ) 6= pLT ≺ (I ) = (x ₁ , x ₃ x ₅ , x ₂ x ₃ x ₄ , x ₂ x ₄ x ₆ , x ₂ x ₅ x ₆ ) S /pLT _≺ (I ) is not Cohen-Macaulay !

In this case we have also that cd(S , I ) < cd(S , LT ≺ (I ))!

Examples

S := C[x 1 , . . . , x ₆ ]

I := (x ₁ x ₅ + x ₂ x ₆ + x ₄ ² , x ₁ x ₄ + x ₃ ² − x ₄ x ₅ , x ₁ ² + x ₁ x ₂ + x ₂ x ₅ ) I is a prime complete intersection of codimension 3, Proj(S /I ) is a normal surface with 3 singular points.

If ≺ is the lexicographic order, then

LT ≺ (I ) 6= pLT ≺ (I ) = (x ₁ , x ₃ x ₅ , x ₂ x ₃ x ₄ , x ₂ x ₄ x ₆ , x ₂ x ₅ x ₆ ) S /pLT _≺ (I ) is not Cohen-Macaulay !

In this case we have also that cd(S , I ) < cd(S , LT ≺ (I ))!