COHOMOLOGICAL DIMENSION, CONNECTEDNESS PROPERTIES AND INITIAL IDEALS

COHOMOLOGICAL DIMENSION, CONNECTEDNESS PROPERTIES AND INITIAL IDEALS

Matteo Varbaro

Dipartimento di Matematica

Universit´ a di Genova

COHOMOLOGICAL DIMENSION, CONNECTEDNESS PROPERTIES AND INITIAL IDEALS

Matteo Varbaro

Dipartimento di Matematica

Universit´ a di Genova

COHOMOLOGICAL DIMENSION, CONNECTEDNESS PROPERTIES AND INITIAL IDEALS

Matteo Varbaro

Dipartimento di Matematica

Universit´ a di Genova

We start with a simple question

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

(a) a term order ≺ on P

(b) I ⊆ P homogeneus such that P/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 P := k[x , y , z, u, v , w , a] is it possible to find:

(a) a term order ≺ on P

(b) I ⊆ P homogeneus such that P/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 P := k[x , y , z, u, v , w , a] is it possible to find:

(a) a term order ≺ on P

(b) I ⊆ P homogeneus such that P/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 P := k[x , y , z, u, v , w , a] is it possible to find:

(a) a term order ≺ on P

(b) I ⊆ P homogeneus such that P/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 P := k[x , y , z, u, v , w , a] is it possible to find:

(a) a term order ≺ on P

(b) I ⊆ P homogeneus such that P/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 P := k[x , y , z, u, v , w , a] is it possible to find:

(a) a term order ≺ on P

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

pLT

(I ) = J ?

The answer to the question is NO

We will see this fact as a consequence of the main result.

However notice that:

the Hilbert series of P/J doesn’t obstruct the existence of I

In fact the h-vector of P/J is admissible, since

HS

(z) = h(z)

(1 − z)

= 1 + 4z + 2z

(1 − z)

The answer to the question is NO

We will see this fact as a consequence of the main result.

However notice that:

the Hilbert series of P/J doesn’t obstruct the existence of I

In fact the h-vector of P/J is admissible, since

HS

(z) = h(z)

(1 − z)

= 1 + 4z + 2z

(1 − z)

The answer to the question is NO

We will see this fact as a consequence of the main result.

However notice that:

the Hilbert series of P/J doesn’t obstruct the existence of I

In fact the h-vector of P/J is admissible, since

HS

(z) = h(z)

(1 − z)

= 1 + 4z + 2z

(1 − z)

The answer to the question is NO

We will see this fact as a consequence of the main result.

However notice that:

the Hilbert series of P/J doesn’t obstruct the existence of I

In fact the h-vector of P/J is admissible, since

HS

(z) = h(z)

(1 − z)

= 1 + 4z + 2z

(1 − z)

The answer to the question is NO

We will see this fact as a consequence of the main result.

However notice that:

the Hilbert series of P/J doesn’t obstruct the existence of I

In fact the h-vector of P/J is admissible, since

HS

(z) = h(z)

(1 − z)

= 1 + 4z + 2z

(1 − z)

The answer to the question is NO

We will see this fact as a consequence of the main result.

However notice that:

the Hilbert series of P/J doesn’t obstruct the existence of I

In fact the h-vector of P/J is admissible, since

HS

(z) = h(z)

(1 − z)

= 1 + 4z + 2z

(1 − z)

MAIN GOAL

We will generalize a theorem due to Kalkbrener and Sturmfels: [KaSt, 1995] k = ¯ k , I ⊆ P := k[x

, . . . , x

] and ω ∈ (Z

)

. If I is a prime ideal, then P/ in

(I ) is connected in codimension 1. So, in particular, they proved that for every term order ≺ on P

I prime ⇒ P/LT

(I ) connected in codimension 1

MAIN GOAL

We will generalize a theorem due to Kalkbrener and Sturmfels:

[KaSt, 1995] k = ¯ k , I ⊆ P := k[x

, . . . , x

] and ω ∈ (Z

)

. If I is a prime ideal, then P/ in

(I ) is connected in codimension 1. So, in particular, they proved that for every term order ≺ on P

I prime ⇒ P/LT

(I ) connected in codimension 1

MAIN GOAL

We will generalize a theorem due to Kalkbrener and Sturmfels:

[KaSt, 1995] k = ¯ k , I ⊆ P := k[x

, . . . , x

] and ω ∈ (Z

)

. If I is a prime ideal, then P/ in

(I ) is connected in codimension 1.

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

I prime ⇒ P/LT

(I ) connected in codimension 1

MAIN GOAL

We will generalize a theorem due to Kalkbrener and Sturmfels:

[KaSt, 1995] k = ¯ k , I ⊆ P := k[x

, . . . , x

] and ω ∈ (Z

)

. If I is a prime ideal, then P/ in

(I ) is connected in codimension 1.

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

I prime ⇒ P/LT

(I ) connected in codimension 1

MAIN GOAL

We will generalize a theorem due to Kalkbrener and Sturmfels:

[KaSt, 1995] k = ¯ k , I ⊆ P := k[x

, . . . , x

] and ω ∈ (Z

)

. If I is a prime ideal, then P/ in

(I ) is connected in codimension 1.

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

I prime ⇒ P/LT

(I ) connected in codimension 1

References

I

[HuTa, 2004] C. Huneke, A. Taylor, Lectures on Local Cohomology, notes for students of the University of Chicago, 2004.

I

[KaSt, 1995] M. Kalkbrener, B. Sturmfels, Initial Complexes of Prime Ideals, Advanced in Mathematics, V. 116, 1995.

I

[V, 2008] M. Varbaro, Cohomological dimension,

connectedness properties and initial ideals, arXiv:0802.1800,

2008.

References

I

[HuTa, 2004] C. Huneke, A. Taylor, Lectures on Local Cohomology, notes for students of the University of Chicago, 2004.

I

[KaSt, 1995] M. Kalkbrener, B. Sturmfels, Initial Complexes of Prime Ideals, Advanced in Mathematics, V. 116, 1995.

I

[V, 2008] M. Varbaro, Cohomological dimension,

connectedness properties and initial ideals, arXiv:0802.1800,

2008.

References

I

[HuTa, 2004] C. Huneke, A. Taylor, Lectures on Local Cohomology, notes for students of the University of Chicago, 2004.

I

[KaSt, 1995] M. Kalkbrener, B. Sturmfels, Initial Complexes of Prime Ideals, Advanced in Mathematics, V. 116, 1995.

I

[V, 2008] M. Varbaro, Cohomological dimension,

connectedness properties and initial ideals, arXiv:0802.1800,

2008.

References

I

[HuTa, 2004] C. Huneke, A. Taylor, Lectures on Local Cohomology, notes for students of the University of Chicago, 2004.

I

[KaSt, 1995] M. Kalkbrener, B. Sturmfels, Initial Complexes of Prime Ideals, Advanced in Mathematics, V. 116, 1995.

I

[V, 2008] M. Varbaro, Cohomological dimension,

connectedness properties and initial ideals, arXiv:0802.1800,

2008.

Structure of the seminary

Two principal chapters:

- Local cohomology and connectedness The main result joins:

cohomological dimension of an ideal in a local complete ring with connectedness properties of the spectrum of the quotient ring. Translate this result from complete to graded case.

- Applications to initial ideals

We will use the first part to prove the main result.

Structure of the seminary

Two principal chapters:

- Local cohomology and connectedness The main result joins:

cohomological dimension of an ideal in a local complete ring with connectedness properties of the spectrum of the quotient ring. Translate this result from complete to graded case.

- Applications to initial ideals

We will use the first part to prove the main result.

Structure of the seminary

Two principal chapters:

- Local cohomology and connectedness

The main result joins:

cohomological dimension of an ideal in a local complete ring with connectedness properties of the spectrum of the quotient ring. Translate this result from complete to graded case.

- Applications to initial ideals

We will use the first part to prove the main result.

Structure of the seminary

Two principal chapters:

- Local cohomology and connectedness The main result joins:

cohomological dimension of an ideal in a local complete ring with connectedness properties of the spectrum of the quotient ring.

Translate this result from complete to graded case. - Applications to initial ideals

We will use the first part to prove the main result.

Structure of the seminary

Two principal chapters:

- Local cohomology and connectedness The main result joins:

cohomological dimension of an ideal in a local complete ring with connectedness properties of the spectrum of the quotient ring.

Translate this result from complete to graded case.

- Applications to initial ideals

We will use the first part to prove the main result.

Structure of the seminary

Two principal chapters:

- Local cohomology and connectedness The main result joins:

cohomological dimension of an ideal in a local complete ring with connectedness properties of the spectrum of the quotient ring.

Translate this result from complete to graded case.

- Applications to initial ideals

We will use the first part to prove the main result.

Structure of the seminary

Two principal chapters:

- Local cohomology and connectedness The main result joins:

cohomological dimension of an ideal in a local complete ring with connectedness properties of the spectrum of the quotient ring.

Translate this result from complete to graded case.

- Applications to initial ideals

We will use the first part to prove the main result.

PART 1

LOCAL COHOMOLOGY AND CONNECTEDNESS

Notations, definitions and ”basic” results

Notations, definitions and ”basic” results

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

Notations, definitions and ”basic” results

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

- cd(R; a) is the infimum d ∈ N s.t. H

(M) = 0 for every R-module

M and i ≥ d .

Notations, definitions and ”basic” results

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

- cd(R; a) is the infimum d ∈ N s.t. H

(R) = 0 for every i ≥ d .

Notations, definitions and ”basic” results

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

- cd(R; a) is the infimum d ∈ N s.t. H

(R) = 0 for every i ≥ d . - ara

(a) is the infimum r ∈ N such that there exist f

, . . . , f

∈ R such that √

a = p(f

, . . . , f

).

Notations, definitions and ”basic” results

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

- cd(R; a) is the infimum d ∈ N s.t. H

(R) = 0 for every i ≥ d . - ara(a) is the infimum r ∈ N such that there exist f

, . . . , f

∈ R such that √

a = p(f

, . . . , f

).

Notations, definitions and ”basic” results

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

- cd(R; a) is the infimum d ∈ N s.t. H

(R) = 0 for every i ≥ d . - ara(a) is the infimum r ∈ N such that there exist f

, . . . , f

∈ R such that √

a = p(f

, . . . , f

).

- ht(a) ≤ cd(R; a) ≤ ara(a).

Notations, definitions and ”basic” results

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

- cd(R; a) is the infimum d ∈ N s.t. H

(R) = 0 for every i ≥ d . - ara(a) is the infimum r ∈ N such that there exist f

, . . . , f

∈ R such that √

a = p(f

, . . . , f

).

- ht(a) ≤ cd(R; a) ≤ ara(a).

- ht(a) ≤ cd(R; a) ≤ dim R.

Notations, definitions and ”basic” results

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

- cd(R; a) is the infimum d ∈ N s.t. H

(R) = 0 for every i ≥ d . - ara(a) is the infimum r ∈ N such that there exist f

, . . . , f

∈ R such that √

a = p(f

, . . . , f

).

- ht(a) ≤ cd(R; a) ≤ ara(a).

- ht(a) ≤ cd(R; a) ≤ dim R.

- ht(a) ≤ ara(a) ≤ dim R + 1.

Notations, definitions and ”basic” results

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

- cd(R; a) is the infimum d ∈ N s.t. H

(R) = 0 for every i ≥ d . - ara(a) is the infimum r ∈ N such that there exist f

, . . . , f

∈ R such that √

a = p(f

, . . . , f

).

- If R is local or a polinomial ring , then

ht(a) ≤ cd(R; a) ≤ ara(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

. 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.:

c(R) ≥ d ;

(2)

∀℘

, ℘

∈ Min R, are d-connected, i.e ∃℘

= ℘

, ℘

,

. . . , ℘

= ℘

, ℘

∈ Min R and dim R/(℘

+ ℘

) ≥ 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)

∀℘

, ℘

∈ Min R, are d-connected, i.e ∃℘

= ℘

, ℘

,

. . . , ℘

= ℘

, ℘

∈ Min R and dim R/(℘

+ ℘

) ≥ 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)

∀℘

, ℘

∈ Min R, are d-connected, i.e ∃℘

= ℘

, ℘

,

. . . , ℘

= ℘

, ℘

∈ Min R and dim R/(℘

+ ℘

) ≥ 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)

∀℘

, ℘

∈ Min R, are d-connected, i.e ∃℘

= ℘

, ℘

, . . . , ℘

= ℘

, ℘

∈ Min R and dim R/(℘

+ ℘

) ≥ 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)

∀℘

, ℘

∈ Min R, are d-connected, i.e ∃℘

= ℘

, ℘

,

. . . , ℘

= ℘

, ℘

∈ Min R and dim R/(℘

+ ℘

) ≥ 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)

∀℘

, ℘

∈ Min R, are d-connected, i.e ∃℘

= ℘

, ℘

,

. . . , ℘

= ℘

, ℘

∈ Min R and dim R/(℘

+ ℘

) ≥ 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)

∀℘

, ℘

∈ Min R, are d-connected, i.e ∃℘

= ℘

, ℘

, . . . , ℘

= ℘

, ℘

∈ Min R and dim R/(℘

+ ℘

) ≥ 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)

∀℘

, ℘

∈ Min R, are d-connected, i.e ∃℘

= ℘

, ℘

, . . . , ℘

= ℘

, ℘

∈ Min R and dim R/(℘

+ ℘

) ≥ 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 ) = ℘

∩ ℘

∩ ℘

A characterization of connectivity dimension

T.F.A.E.:

(1)

c(R) ≥ d ;

(2)

∀℘

, ℘

∈ Min R, are d-connected, i.e ∃℘

= ℘

, ℘

, . . . , ℘

= ℘

, ℘

∈ Min R and dim R/(℘

+ ℘

) ≥ 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 ) = ℘

∩ ℘

∩ ℘

dim R = 2, dim S /(℘

+ ℘

) = 1,

dim S /(℘

+ ℘

) = 0, dim S /(℘

+ ℘

) = 0.

A characterization of connectivity dimension

T.F.A.E.:

(1)

c(R) ≥ d ;

(2)

∀℘

, ℘

∈ Min R, are d-connected, i.e ∃℘

= ℘

, ℘

, . . . , ℘

= ℘

, ℘

∈ Min R and dim R/(℘

+ ℘

) ≥ 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)

∀℘

, ℘

∈ Min R, are d-connected, i.e ∃℘

= ℘

, ℘

, . . . , ℘

= ℘

, ℘

∈ Min R and dim R/(℘

+ ℘

) ≥ 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 ) = ℘

∩ ℘

∩ ℘

∩ ℘

A characterization of connectivity dimension

T.F.A.E.:

(1)

c(R) ≥ d ;

(2)

∀℘

, ℘

∈ Min R, are d-connected, i.e ∃℘

= ℘

, ℘

, . . . , ℘

= ℘

, ℘

∈ Min R and dim R/(℘

+ ℘

) ≥ 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 ) = ℘

∩ ℘

∩ ℘

∩ ℘

dim R = 2, dim S /(℘

+ ℘

) = 1,

dim S /(℘

+ ℘

) = 1, dim S /(℘

+ ℘

) = 1.

The main result of the first part

The main result of the first part

Let (R, m) be local and complete.

The main result of the first part

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

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

The main result of the first part

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

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

Corollary

The main result of the first part

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 main result of the first part

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 main result of the first part

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 main result of the first part

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

(R) indecomposable.

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

The main result of the first part

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

(R) indecomposable.

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

Proof of Corollary

The main result of the first part

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

(R) indecomposable.

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

Proof of Corollary

H

(R) indecomposable iff R connected in codimension 1.

The main result of the first part

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

(R) indecomposable.

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

Proof of Corollary

H

(R) indecomposable iff R connected in codimension 1.

c(Spec R/a \ m) = c(R/a) − 1.

Sketch of the proof

Sketch of the proof

Proof Grothendieck’s theor. in the book of Brodmann and Sharp.

Sketch of the proof

Proof Grothendieck’s theor. in the book of Brodmann and Sharp.

There are some easy changes...

Sketch of the proof

Proof Grothendieck’s theor. in the book of Brodmann and Sharp.

There are some easy changes...

- ara(a) ≥ ht(a)

Sketch of the proof

Proof Grothendieck’s theor. in the book of Brodmann and Sharp.

There are some easy changes...

- cd(R; a) ≥ ht a

Sketch of the proof

Proof Grothendieck’s theor. in the book of Brodmann and Sharp.

There are some easy changes...

- cd(R; a) ≥ ht a

- If R −→ S , then ara

(a) ≥ ara

(aS )

Sketch of the proof

Proof Grothendieck’s theor. in the book of Brodmann and Sharp.

There are some easy changes...

- cd(R; a) ≥ ht a

- If R −→ S , then cd(R, a) ≥ cd(S , aS )

Sketch of the proof

Proof Grothendieck’s theor. in the book of Brodmann and Sharp.

There are some easy changes...

- cd(R; a) ≥ ht a

- If R −→ S , then cd(R, a) ≥ cd(S , aS )

And one change more difficult:

Sketch of the proof

Proof Grothendieck’s theor. in the book of Brodmann and Sharp.

There are some easy changes...

- cd(R; a) ≥ ht a

- If R −→ S , then cd(R, a) ≥ cd(S , aS ) And one change more difficult:

- x ∈ R, then ara(a + (x )) ≤ ara(a) + 1

Sketch of the proof

Proof Grothendieck’s theor. in the book of Brodmann and Sharp.

There are some easy changes...

- cd(R; a) ≥ ht a

- If R −→ S , then cd(R, a) ≥ cd(S , aS ) And one change more difficult:

- x ∈ R, then cd(R, a + (x )) ≤ cd(R, a) + 1 ?

Sketch of the proof

Proof Grothendieck’s theor. in the book of Brodmann and Sharp.

There are some easy changes...

- cd(R; a) ≥ ht a

- If R −→ S , then cd(R, a) ≥ cd(S , aS ) And one change more difficult:

- x ∈ R, then cd(R, a + (x )) ≤ cd(R, a) + 1 ? Yes!

Sketch of the proof

Proof Grothendieck’s theor. in the book of Brodmann and Sharp.

There are some easy changes...

- cd(R; a) ≥ ht a

- If R −→ S , then cd(R, a) ≥ cd(S , aS ) And one change more difficult:

- x ∈ R, then cd(R, a + (x )) ≤ cd(R, a) + 1 ? Yes!

More generally, one can prove that,

given another ideal b, there is a spectral sequence:

Sketch of the proof

Proof Grothendieck’s theor. in the book of Brodmann and Sharp.

There are some easy changes...

- cd(R; a) ≥ ht a

- If R −→ S , then cd(R, a) ≥ cd(S , aS ) And one change more difficult:

- x ∈ R, then cd(R, a + (x )) ≤ cd(R, a) + 1 ? Yes!

More generally, one can prove that,

given another ideal b, there is a spectral sequence:

H

(H

(R)) ⇒ H

(R)

Sketch of the proof

Proof Grothendieck’s theor. in the book of Brodmann and Sharp.

There are some easy changes...

- cd(R; a) ≥ ht a

- If R −→ S , then cd(R, a) ≥ cd(S , aS ) And one change more difficult:

- x ∈ R, then cd(R, a + (x )) ≤ cd(R, a) + 1 ? Yes!

More generally, one can prove that,

given another ideal b, there is a spectral sequence:

H

(H

(R)) ⇒ H

(R)

from which cd(R, a + b) ≤ cd(R, a) + cd(R, b)!

The graded case

R k-algebra finitely generated positively graded, a homogeneus. Then c(R/a) ≥ min{c(R), dim R − 1} − cd(R, a).

Idea of the proof Let be m = ⊕

R

.

- R −→ c R

faithfully flat ⇒

local cohomology in R ! local cohomology in c R

- R positively graded ⇒

”connectedness in R” ! ”connectedness in c R

”

The graded case

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

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

Let be m = ⊕

R

.

- R −→ c R

faithfully flat ⇒

local cohomology in R ! local cohomology in c R

- R positively graded ⇒

”connectedness in R” ! ”connectedness in c R

”

The graded case

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

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

Idea of the proof Let be m = ⊕

R

.

- R −→ c R

faithfully flat ⇒

local cohomology in R ! local cohomology in c R

- R positively graded ⇒

”connectedness in R” ! ”connectedness in c R

”

The graded case

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

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

Idea of the proof

Let be m = ⊕

R

.

- R −→ c R

faithfully flat ⇒

local cohomology in R ! local cohomology in c R

- R positively graded ⇒

”connectedness in R” ! ”connectedness in c R

”

The graded case

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

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

Idea of the proof Let be m = ⊕

R

.

- R −→ c R

faithfully flat ⇒

local cohomology in R ! local cohomology in c R

- R positively graded ⇒

”connectedness in R” ! ”connectedness in c R

”

The graded case

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

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

Idea of the proof Let be m = ⊕

R

.

- R −→ c R

faithfully flat ⇒

local cohomology in R ! local cohomology in c R

- R positively graded ⇒

”connectedness in R” ! ”connectedness in c R

”

The graded case

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

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

Idea of the proof Let be m = ⊕

R

.

- R −→ c R

faithfully flat ⇒

local cohomology in R ! local cohomology in c R

- R positively graded ⇒

”connectedness in R” ! ”connectedness in c R

”

The graded case

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

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

Idea of the proof Let be m = ⊕

R

.

- R −→ c R

faithfully flat ⇒

local cohomology in R ! local cohomology in c R

- R positively graded ⇒

”connectedness in R” ! ”connectedness in c R

”

The graded case

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

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

Idea of the proof Let be m = ⊕

R

.

- R −→ c R

faithfully flat ⇒

local cohomology in R ! local cohomology in c R

- R positively graded ⇒

”connectedness in R” ! ”connectedness in c R

”

An example of application

X = ProjR projective scheme over k connected in codimension 1, C = V

(a) ⊆ X curve of X .

If C set-theoretic complete intersection in X (ara

(a) = ht(a) = codim

C ), then C connected. In fact, (1) c(C ) = c(R/a) − 1, c(X ) = c(R) − 1 and (2) min{c(R), dim R − 1} = dim X , so

c(C ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − ara

(a) − 1= 0.

An example of application

X = ProjR projective scheme over k connected in codimension 1, C = V

(a) ⊆ X curve of X .

If C set-theoretic complete intersection in X (ara

(a) = ht(a) = codim

C ), then C connected. In fact, (1) c(C ) = c(R/a) − 1, c(X ) = c(R) − 1 and (2) min{c(R), dim R − 1} = dim X , so

c(C ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − ara

(a) − 1= 0.

An example of application

X = ProjR projective scheme over k connected in codimension 1, C = V

(a) ⊆ X curve of X .

If C set-theoretic complete intersection in X (ara

(a) = ht(a) = codim

C ),

then C connected. In fact, (1) c(C ) = c(R/a) − 1, c(X ) = c(R) − 1 and (2) min{c(R), dim R − 1} = dim X , so

c(C ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − ara

(a) − 1= 0.

An example of application

X = ProjR projective scheme over k connected in codimension 1, C = V

(a) ⊆ X curve of X .

If C set-theoretic complete intersection in X (ara

(a) = ht(a) = codim

C ), then C connected.

In fact, (1) c(C ) = c(R/a) − 1, c(X ) = c(R) − 1 and (2) min{c(R), dim R − 1} = dim X , so

c(C ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − ara

(a) − 1= 0.

An example of application

X = ProjR projective scheme over k connected in codimension 1, C = V

(a) ⊆ X curve of X .

If C set-theoretic complete intersection in X (ara

(a) = ht(a) = codim

C ), then C connected.

In fact, (1) c(C ) = c(R/a) − 1, c(X ) = c(R) − 1

and (2) min{c(R), dim R − 1} = dim X , so

c(C ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − ara

(a) − 1= 0.

An example of application

X = ProjR projective scheme over k connected in codimension 1, C = V

(a) ⊆ X curve of X .

If C set-theoretic complete intersection in X (ara

(a) = ht(a) = codim

C ), then C connected.

In fact, (1) c(C ) = c(R/a) − 1, c(X ) = c(R) − 1 and (2) min{c(R), dim R − 1} = dim X

, so c(C ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − ara

(a) − 1= 0.

An example of application

X = ProjR projective scheme over k connected in codimension 1, C = V

(a) ⊆ X curve of X .

If C set-theoretic complete intersection in X (ara

(a) = ht(a) = codim

C ), then C connected.

In fact, (1) c(C ) = c(R/a) − 1, c(X ) = c(R) − 1 and (2) min{c(R), dim R − 1} = dim X , so

c(C ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − ara

(a) − 1= 0.

An example of application

X = ProjR projective scheme over k connected in codimension 1, C = V

(a) ⊆ X curve of X .

If C set-theoretic complete intersection in X (ara

(a) = ht(a) = codim

C ), then C connected.

In fact, (1) c(C ) = c(R/a) − 1, c(X ) = c(R) − 1 and (2) min{c(R), dim R − 1} = dim X , so

c(C ) = c(R/a) − 1

≥ min{c(R), dim R − 1} − cd(R, a) − 1

=

= dim X − cd(R, a) − 1≥

≥ dim X − ara

(a) − 1= 0.

An example of application

X = ProjR projective scheme over k connected in codimension 1, C = V

(a) ⊆ X curve of X .

If C set-theoretic complete intersection in X (ara

(a) = ht(a) = codim

C ), then C connected.

In fact, (1) c(C ) = c(R/a) − 1, c(X ) = c(R) − 1 and (2) min{c(R), dim R − 1} = dim X , so

c(C ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1

≥

≥ dim X − ara

(a) − 1= 0.

An example of application

X = ProjR projective scheme over k connected in codimension 1, C = V

(a) ⊆ X curve of X .

If C set-theoretic complete intersection in X (ara

(a) = ht(a) = codim

C ), then C connected.

In fact, (1) c(C ) = c(R/a) − 1, c(X ) = c(R) − 1 and (2) min{c(R), dim R − 1} = dim X , so

c(C ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − ara

(a) − 1

= 0.

An example of application

X = ProjR projective scheme over k connected in codimension 1, C = V

(a) ⊆ X curve of X .

If C set-theoretic complete intersection in X (ara

(a) = ht(a) = codim

C ), then C connected.

In fact, (1) c(C ) = c(R/a) − 1, c(X ) = c(R) − 1 and (2) min{c(R), dim R − 1} = dim X , so

c(C ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − ara

(a) − 1= 0.

Another example of application

X = ProjR projective scheme over k connected in codimension 1, D effective divisor on X .

D ample ⇒ Z := Supp D connected in codimension 1. i = i

: X −→ P

closed immersion for n >> 0, so

X \ Z = X \ Supp nD affine.

Affineness Serre’s criterion ⇒ cd(R, a) ≤ 1, where Z = V

(a) , so c(Z ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − 2 = dim Z − 1.

Another example of application

X = ProjR projective scheme over k connected in codimension 1, D effective divisor on X .

D ample ⇒ Z := Supp D connected in codimension 1. i = i

: X −→ P

closed immersion for n >> 0, so

X \ Z = X \ Supp nD affine.

Affineness Serre’s criterion ⇒ cd(R, a) ≤ 1, where Z = V

(a) , so c(Z ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − 2 = dim Z − 1.

Another example of application

X = ProjR projective scheme over k connected in codimension 1, D effective divisor on X .

D ample ⇒ Z := Supp D connected in codimension 1.

i = i

: X −→ P

closed immersion for n >> 0, so X \ Z = X \ Supp nD affine.

Affineness Serre’s criterion ⇒ cd(R, a) ≤ 1, where Z = V

(a) , so c(Z ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − 2 = dim Z − 1.

Another example of application

X = ProjR projective scheme over k connected in codimension 1, D effective divisor on X .

D ample ⇒ Z := Supp D connected in codimension 1.

i = i

: X −→ P

closed immersion for n >> 0

, so X \ Z = X \ Supp nD affine.

Affineness Serre’s criterion ⇒ cd(R, a) ≤ 1, where Z = V

(a) , so c(Z ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − 2 = dim Z − 1.

Another example of application

X = ProjR projective scheme over k connected in codimension 1, D effective divisor on X .

D ample ⇒ Z := Supp D connected in codimension 1.

i = i

: X −→ P

closed immersion for n >> 0, so X \ Z = X \ Supp nD affine.

Affineness Serre’s criterion ⇒ cd(R, a) ≤ 1, where Z = V

(a) , so c(Z ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − 2 = dim Z − 1.

Another example of application

X = ProjR projective scheme over k connected in codimension 1, D effective divisor on X .

D ample ⇒ Z := Supp D connected in codimension 1.

i = i

: X −→ P

closed immersion for n >> 0, so X \ Z = X \ Supp nD affine.

Affineness Serre’s criterion ⇒ cd(R, a) ≤ 1, where Z = V

(a)

, so c(Z ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − 2 = dim Z − 1.

Another example of application

X = ProjR projective scheme over k connected in codimension 1, D effective divisor on X .

D ample ⇒ Z := Supp D connected in codimension 1.

i = i

: X −→ P

closed immersion for n >> 0, so X \ Z = X \ Supp nD affine.

Affineness Serre’s criterion ⇒ cd(R, a) ≤ 1, where Z = V

(a) , so c(Z ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − 2 = dim Z − 1.

Another example of application

X = ProjR projective scheme over k connected in codimension 1, D effective divisor on X .

D ample ⇒ Z := Supp D connected in codimension 1.

i = i

: X −→ P

closed immersion for n >> 0, so X \ Z = X \ Supp nD affine.

Affineness Serre’s criterion ⇒ cd(R, a) ≤ 1, where Z = V

(a) , so c(Z ) = c(R/a) − 1

≥ min{c(R), dim R − 1} − cd(R, a) − 1

=

= dim X − cd(R, a) − 1≥

≥ dim X − 2 = dim Z − 1.

Another example of application

X = ProjR projective scheme over k connected in codimension 1, D effective divisor on X .

D ample ⇒ Z := Supp D connected in codimension 1.

i = i

: X −→ P

closed immersion for n >> 0, so X \ Z = X \ Supp nD affine.

Affineness Serre’s criterion ⇒ cd(R, a) ≤ 1, where Z = V

(a) , so c(Z ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1

≥

≥ dim X − 2 = dim Z − 1.

Another example of application

X = ProjR projective scheme over k connected in codimension 1, D effective divisor on X .

D ample ⇒ Z := Supp D connected in codimension 1.

i = i

: X −→ P

closed immersion for n >> 0, so X \ Z = X \ Supp nD affine.

Affineness Serre’s criterion ⇒ cd(R, a) ≤ 1, where Z = V

(a) , so c(Z ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − 2

= dim Z − 1.

Another example of application

X = ProjR projective scheme over k connected in codimension 1, D effective divisor on X .

D ample ⇒ Z := Supp D connected in codimension 1.

i = i

: X −→ P

closed immersion for n >> 0, so X \ Z = X \ Supp nD affine.

Affineness Serre’s criterion ⇒ cd(R, a) ≤ 1, where Z = V

(a) , so c(Z ) = c(R/a) − 1

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

= dim X − cd(R, a) − 1≥

≥ dim X − 2 = dim Z − 1.

PART 2

APPLICATIONS TO INITIAL IDEALS

Notations

I

P := k[x

, . . . , x

];

I

I ⊆ P ideal;

I

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

I

≺ term order on P;

I

LT

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

I

ω = (ω

, . . . , ω

) ∈ (Z

)

weight vector;

Notations

I

P := k[x

, . . . , x

];

I

I ⊆ P ideal;

I

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

I

≺ term order on P;

I

LT

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

I

ω = (ω

, . . . , ω

) ∈ (Z

)

weight vector;

Notations

I

P := k[x

, . . . , x

];

I

I ⊆ P ideal;

I

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

I

≺ term order on P;

I

LT

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

I

ω = (ω

, . . . , ω

) ∈ (Z

)

weight vector;

Notations

I

P := k[x

, . . . , x

];

I

I ⊆ P ideal;

I

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

I

≺ term order on P;

I

LT

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

I

ω = (ω

, . . . , ω

) ∈ (Z

)

weight vector;

Notations

I

P := k[x

, . . . , x

];

I

I ⊆ P ideal;

I

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

I

≺ term order on P;

I

LT

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

I

ω = (ω

, . . . , ω

) ∈ (Z

)

weight vector;

Notations

I

P := k[x

, . . . , x

];

I

I ⊆ P ideal;

I

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

I

≺ term order on P;

I

LT

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

I

ω = (ω

, . . . , ω

) ∈ (Z

)

weight vector;

Notations

I

P := k[x

, . . . , x

];

I

I ⊆ P ideal;

I

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

I

≺ term order on P;

I

LT

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

I

ω = (ω

, . . . , ω

) ∈ (Z

)

weight vector;

Initial ideals with respect to weight vectors

Initial ideals with respect to weight vectors

- f ∈ P, in

(f ) ∈ P is the leading coefficient of

f (x

t

, . . . , x

t

) ∈ P[t]

Initial ideals with respect to weight vectors

- f ∈ P, in

(f ) ∈ P is the leading coefficient of f (x

t

, . . . , x

t

) ∈ P[t]

E. g., f = 2x

x

+ x

x

+ 3x

x

∈ k[x

, x

, x

] and ω = (3, 2, 1);