Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Graded rings

Brian Johnson

University of Nebraska-Lincoln

22 April 2012

s-bjohns67@math.unl.edu

1 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

IntroductionOutline

Polynomial rings

Notation/background

Graded properties

Chain conditions

Free and projective modules

Prime avoidance

Grade

2 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

IntroductionOutline

Polynomial rings

Notation/background

Graded properties

Chain conditions

Free and projective modules

Prime avoidance

Grade

2 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

IntroductionOutline

Polynomial rings

Notation/background

Graded properties

Chain conditions

Free and projective modules

Prime avoidance

Grade

2 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

IntroductionOutline

Polynomial rings

Notation/background

Graded properties

Chain conditions

Free and projective modules

Prime avoidance

Grade

2 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

IntroductionOutline

Polynomial rings

Notation/background

Graded properties

Chain conditions

Free and projective modules

Prime avoidance

Grade

2 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

IntroductionOutline

Polynomial rings

Notation/background

Graded properties

Chain conditions

Free and projective modules

Prime avoidance

Grade

2 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

IntroductionOutline

Polynomial rings

Notation/background

Graded properties

Chain conditions

Free and projective modules

Prime avoidance

Grade

2 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

IntroductionPolynomial rings

A Z-grading for R[x] can be given by:

R[x] = Rdeg 0

⊕ Rxdeg 1

⊕ Rx2

deg 2⊕ Rx3

deg 3⊕ · · ·

A Z-grading for R[x, y] can be given by:

R[x, y] = Rdeg 0

⊕ (Rx+ Ry)︸ ︷︷ ︸deg 1

⊕ (Rx2 + Rxy + Ry2)︸ ︷︷ ︸deg 2

⊕ · · ·

But why not decompose R[x, y] as follows:

R[x, y] = R(0,0)⊕ Rx

(1,0)⊕ Ry

(0,1)⊕Rx2

(2,0)⊕Rxy

(1,1)⊕Ry2

(0,2)⊕Rx3

(3,0)⊕Rxy2

(1,2)⊕· · ·

3 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

IntroductionPolynomial rings

A Z-grading for R[x] can be given by:

R[x] = Rdeg 0

⊕ Rxdeg 1

⊕ Rx2

deg 2⊕ Rx3

deg 3⊕ · · ·

A Z-grading for R[x, y] can be given by:

R[x, y] = Rdeg 0

⊕ (Rx+ Ry)︸ ︷︷ ︸deg 1

⊕ (Rx2 + Rxy + Ry2)︸ ︷︷ ︸deg 2

⊕ · · ·

But why not decompose R[x, y] as follows:

R[x, y] = R(0,0)⊕ Rx

(1,0)⊕ Ry

(0,1)⊕Rx2

(2,0)⊕Rxy

(1,1)⊕Ry2

(0,2)⊕Rx3

(3,0)⊕Rxy2

(1,2)⊕· · ·

3 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

IntroductionPolynomial rings

A Z-grading for R[x] can be given by:

R[x] = Rdeg 0

⊕ Rxdeg 1

⊕ Rx2

deg 2⊕ Rx3

deg 3⊕ · · ·

A Z-grading for R[x, y] can be given by:

R[x, y] = Rdeg 0

⊕ (Rx+ Ry)︸ ︷︷ ︸deg 1

⊕ (Rx2 + Rxy + Ry2)︸ ︷︷ ︸deg 2

⊕ · · ·

But why not decompose R[x, y] as follows:

R[x, y] = R(0,0)⊕ Rx

(1,0)⊕ Ry

(0,1)⊕Rx2

(2,0)⊕Rxy

(1,1)⊕Ry2

(0,2)⊕Rx3

(3,0)⊕Rxy2

(1,2)⊕· · ·

3 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

IntroductionPolynomial rings

One can also decompose R[x] over the group Z/2Z as

R + Rx2 + Rx4 + · · ·︸ ︷︷ ︸deg 0

⊕Rx+ Rx3 + Rx5 + · · ·︸ ︷︷ ︸deg 1

Two common themes:

Adding two elements of the same degree gives anotherelement of the same degree.

When multiplying (homogeneous) elements together,degree is additive.

4 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

IntroductionPolynomial rings

One can also decompose R[x] over the group Z/2Z as

R + Rx2 + Rx4 + · · ·︸ ︷︷ ︸deg 0

⊕Rx+ Rx3 + Rx5 + · · ·︸ ︷︷ ︸deg 1

Two common themes:

Adding two elements of the same degree gives anotherelement of the same degree.

When multiplying (homogeneous) elements together,degree is additive.

4 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

IntroductionPolynomial rings

One can also decompose R[x] over the group Z/2Z as

R + Rx2 + Rx4 + · · ·︸ ︷︷ ︸deg 0

⊕Rx+ Rx3 + Rx5 + · · ·︸ ︷︷ ︸deg 1

Two common themes:

Adding two elements of the same degree gives anotherelement of the same degree.

When multiplying (homogeneous) elements together,degree is additive.

4 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Notation/backgroundGraded rings

Let G be an abelian group. A (commutative) ring R isG-graded if there is a family of subgroups of R, {Rg}g∈G, suchthat R =

⊕g∈GRg, and RgRh ⊆ Rg+h for all g, h ∈ G.

Then for r ∈ R, we can write a unique expression

r =∑g∈G

rg

where rg ∈ Rg and almost all rg = 0. The element rg is calledthe gth homogeneous component, and if r = rg, we say r ishomogeneous of degree g.

5 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Notation/backgroundGraded rings

Let G be an abelian group. A (commutative) ring R isG-graded if there is a family of subgroups of R, {Rg}g∈G, suchthat R =

⊕g∈GRg, and RgRh ⊆ Rg+h for all g, h ∈ G.

Then for r ∈ R, we can write a unique expression

r =∑g∈G

rg

where rg ∈ Rg and almost all rg = 0. The element rg is calledthe gth homogeneous component, and if r = rg, we say r ishomogeneous of degree g.

5 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Notation/backgroundExamples

With the Z-grading for R[x] above, 1 + x2 is nothomogeneous. On the other hand, with the Z/2Z-gradinggiven, 1 + x2 is homogeneous, since both terms have evendegree.

With the Z-grading given for R[x, y] above, 2x2 − y2 ishomogeneous (this is the usual definition of ahomogeneous polynomial).

Under the Z⊕Z-grading for R[x, y] the only homogeneouselements now are monomials.

6 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Notation/backgroundExamples

With the Z-grading for R[x] above, 1 + x2 is nothomogeneous. On the other hand, with the Z/2Z-gradinggiven, 1 + x2 is homogeneous, since both terms have evendegree.

With the Z-grading given for R[x, y] above, 2x2 − y2 ishomogeneous (this is the usual definition of ahomogeneous polynomial).

Under the Z⊕Z-grading for R[x, y] the only homogeneouselements now are monomials.

6 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Notation/backgroundExamples

With the Z-grading for R[x] above, 1 + x2 is nothomogeneous. On the other hand, with the Z/2Z-gradinggiven, 1 + x2 is homogeneous, since both terms have evendegree.

With the Z-grading given for R[x, y] above, 2x2 − y2 ishomogeneous (this is the usual definition of ahomogeneous polynomial).

Under the Z⊕Z-grading for R[x, y] the only homogeneouselements now are monomials.

6 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Notation/backgroundHomogeneous modules and ideals

A G-graded module M satisfies M =⊕

g∈GMg andRgMh ⊆Mg+h for all g, h ∈ G. Equivalently, M is G-graded ifit has a G-homogeneous generating set.

A G-homogeneous ideal I of R is then a G-graded submoduleof R. Again, we can use the equivalent characterization that Iis homogeneous if I has a homogeneous generating set.

7 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Notation/backgroundHomogeneous modules and ideals

A G-graded module M satisfies M =⊕

g∈GMg andRgMh ⊆Mg+h for all g, h ∈ G. Equivalently, M is G-graded ifit has a G-homogeneous generating set.

A G-homogeneous ideal I of R is then a G-graded submoduleof R. Again, we can use the equivalent characterization that Iis homogeneous if I has a homogeneous generating set.

7 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Notation/backgroundTwo constructions

1 For a subgroup H ≤ G, we set RH =⊕

h∈H Rh, which isa G- and H-graded subring of R.

2 More generally,

Rg+H :=⊕h∈H

Rg+h

is a G-graded RH -submodule of R. This allows us todefine an induced G/H-grading on the ring R, using thefamily {Rx}x∈G/H .

8 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Notation/backgroundTwo constructions

1 For a subgroup H ≤ G, we set RH =⊕

h∈H Rh, which isa G- and H-graded subring of R.

2 More generally,

Rg+H :=⊕h∈H

Rg+h

is a G-graded RH -submodule of R. This allows us todefine an induced G/H-grading on the ring R, using thefamily {Rx}x∈G/H .

8 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Notation/backgroundExamples

The Z/2Z-grading on R[x] above is induced from theZ-grading in this way by taking G = Z and H = 2Z.

In a similar way, the Z-grading on R[x, y] above can beinduced from the Z2-grading: let G = Z2 and defineH = ker(φ), where φ : Z2 → Z is given by φ(a, b) = a+ b.

9 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Notation/backgroundExamples

The Z/2Z-grading on R[x] above is induced from theZ-grading in this way by taking G = Z and H = 2Z.

In a similar way, the Z-grading on R[x, y] above can beinduced from the Z2-grading: let G = Z2 and defineH = ker(φ), where φ : Z2 → Z is given by φ(a, b) = a+ b.

9 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded properties

Goal: Define analogues of usual ring-theoretic properties interms of only homogeneous or graded objects and see if (orunder what assumptions) these properties descend from theG-grading to the G/H-grading (or ascend from theG/H-grading to the G-grading).

One can define analogues of many usual properties. Forexample, a G-field (resp., G-domain) is a ring in which everyG-homogeneous element is invertible (resp., not a zerodivisor).A G-maximal (resp., G-prime) ideal is a G-homogeneous idealI such that R/I is a G-field (resp., G-domain).

10 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded properties

Goal: Define analogues of usual ring-theoretic properties interms of only homogeneous or graded objects and see if (orunder what assumptions) these properties descend from theG-grading to the G/H-grading (or ascend from theG/H-grading to the G-grading).

One can define analogues of many usual properties. Forexample, a G-field (resp., G-domain) is a ring in which everyG-homogeneous element is invertible (resp., not a zerodivisor).A G-maximal (resp., G-prime) ideal is a G-homogeneous idealI such that R/I is a G-field (resp., G-domain).

10 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesExamples

With deg(x±1) = ±1, R[x, x−1] is a Z-field, but theunderlying ring is not a field.

The ring R[x]/(xn − 1) can be given a Z/nZ-grading bysetting deg(x) = 1. This is then an example of aZ/nZ-field, yet the underlying ring is not even a domain.

Consider the ring R[x, y] with a Z-grading given bydeg(x) = deg(y) = 1.

The ideals (x), (y), and (x+ y) are all prime and Z-prime.

The ideal (x, y) is the unique Z-maximal ideal.

The ideal (x, y + 1) is maximal, but not Z-maximal (sinceit’s not homogeneous).

11 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesExamples

With deg(x±1) = ±1, R[x, x−1] is a Z-field, but theunderlying ring is not a field.

The ring R[x]/(xn − 1) can be given a Z/nZ-grading bysetting deg(x) = 1. This is then an example of aZ/nZ-field, yet the underlying ring is not even a domain.

Consider the ring R[x, y] with a Z-grading given bydeg(x) = deg(y) = 1.

The ideals (x), (y), and (x+ y) are all prime and Z-prime.

The ideal (x, y) is the unique Z-maximal ideal.

The ideal (x, y + 1) is maximal, but not Z-maximal (sinceit’s not homogeneous).

11 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesExamples

With deg(x±1) = ±1, R[x, x−1] is a Z-field, but theunderlying ring is not a field.

The ring R[x]/(xn − 1) can be given a Z/nZ-grading bysetting deg(x) = 1. This is then an example of aZ/nZ-field, yet the underlying ring is not even a domain.

Consider the ring R[x, y] with a Z-grading given bydeg(x) = deg(y) = 1.

The ideals (x), (y), and (x+ y) are all prime and Z-prime.

The ideal (x, y) is the unique Z-maximal ideal.

The ideal (x, y + 1) is maximal, but not Z-maximal (sinceit’s not homogeneous).

11 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesExamples

With deg(x±1) = ±1, R[x, x−1] is a Z-field, but theunderlying ring is not a field.

The ring R[x]/(xn − 1) can be given a Z/nZ-grading bysetting deg(x) = 1. This is then an example of aZ/nZ-field, yet the underlying ring is not even a domain.

Consider the ring R[x, y] with a Z-grading given bydeg(x) = deg(y) = 1.

The ideals (x), (y), and (x+ y) are all prime and Z-prime.

The ideal (x, y) is the unique Z-maximal ideal.

The ideal (x, y + 1) is maximal, but not Z-maximal (sinceit’s not homogeneous).

11 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesExamples

With deg(x±1) = ±1, R[x, x−1] is a Z-field, but theunderlying ring is not a field.

The ring R[x]/(xn − 1) can be given a Z/nZ-grading bysetting deg(x) = 1. This is then an example of aZ/nZ-field, yet the underlying ring is not even a domain.

Consider the ring R[x, y] with a Z-grading given bydeg(x) = deg(y) = 1.

The ideals (x), (y), and (x+ y) are all prime and Z-prime.

The ideal (x, y) is the unique Z-maximal ideal.

The ideal (x, y + 1) is maximal, but not Z-maximal (sinceit’s not homogeneous).

11 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesExamples

With deg(x±1) = ±1, R[x, x−1] is a Z-field, but theunderlying ring is not a field.

The ring R[x]/(xn − 1) can be given a Z/nZ-grading bysetting deg(x) = 1. This is then an example of aZ/nZ-field, yet the underlying ring is not even a domain.

Consider the ring R[x, y] with a Z-grading given bydeg(x) = deg(y) = 1.

The ideals (x), (y), and (x+ y) are all prime and Z-prime.

The ideal (x, y) is the unique Z-maximal ideal.

The ideal (x, y + 1) is maximal, but not Z-maximal (sinceit’s not homogeneous).

11 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesA crucial property

We call a graded ring G-Noetherian (G-Artinian) if it satisfiesthe ascending (descending) chain condition on G-homogeneousideals. That is, any chain

I0 ⊆ I1 ⊆ I2 ⊆ · · ·(I0 ⊇ I1 ⊇ I2 ⊇ · · · )

eventually stabilizes.

Theorem (-)

Suppose R is a G-graded ring. If H ≤ G is a finitely generated(finite) subgroup, the following are equivalent:

1 R is G-Noetherian (G-Artinian).

2 R is G/H-Noetherian (G/H-Artinian).

12 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesA crucial property

We call a graded ring G-Noetherian (G-Artinian) if it satisfiesthe ascending (descending) chain condition on G-homogeneousideals. That is, any chain

I0 ⊆ I1 ⊆ I2 ⊆ · · ·(I0 ⊇ I1 ⊇ I2 ⊇ · · · )

eventually stabilizes.

Theorem (-)

Suppose R is a G-graded ring. If H ≤ G is a finitely generated(finite) subgroup, the following are equivalent:

1 R is G-Noetherian (G-Artinian).

2 R is G/H-Noetherian (G/H-Artinian).

12 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesExamples

For any group G, a G-graded ring where the underlyingring is Noetherian is automatically G-Noetherian.

By the theorem, for a finitely generated group G, a ringbeing G-Noetherian is equivalent to being Noetherian (inthe usual sense).

Define R := R[x±11 , x±1

2 , . . . ] and G :=⊕∞

i=1 Z. Underthe natural G-grading, R is a G-field (and soG-Noetherian). However, R is not Noetherian.

13 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesExamples

For any group G, a G-graded ring where the underlyingring is Noetherian is automatically G-Noetherian.

By the theorem, for a finitely generated group G, a ringbeing G-Noetherian is equivalent to being Noetherian (inthe usual sense).

Define R := R[x±11 , x±1

2 , . . . ] and G :=⊕∞

i=1 Z. Underthe natural G-grading, R is a G-field (and soG-Noetherian). However, R is not Noetherian.

13 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesExamples

For any group G, a G-graded ring where the underlyingring is Noetherian is automatically G-Noetherian.

By the theorem, for a finitely generated group G, a ringbeing G-Noetherian is equivalent to being Noetherian (inthe usual sense).

Define R := R[x±11 , x±1

2 , . . . ] and G :=⊕∞

i=1 Z. Underthe natural G-grading, R is a G-field (and soG-Noetherian). However, R is not Noetherian.

13 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesFree and projective modules

A G-graded module F is G-free if F possesses aG-homogeneous basis. We say a graded module P isG-projective if HomG

R(P,−) is exact.

Suppose M is a G-graded R-module. It is easy to see that Mis G-free =⇒ M is G/H-free for any subgroup H ≤ G. Onecan also show that M is G-projective =⇒ M isG/H-projective for any subgroup H ≤ G.

What about the reverse implications?

14 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesFree and projective modules

A G-graded module F is G-free if F possesses aG-homogeneous basis. We say a graded module P isG-projective if HomG

R(P,−) is exact.

Suppose M is a G-graded R-module. It is easy to see that Mis G-free =⇒ M is G/H-free for any subgroup H ≤ G. Onecan also show that M is G-projective =⇒ M isG/H-projective for any subgroup H ≤ G.

What about the reverse implications?

14 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesFree and projective modules

A G-graded module F is G-free if F possesses aG-homogeneous basis. We say a graded module P isG-projective if HomG

R(P,−) is exact.

Suppose M is a G-graded R-module. It is easy to see that Mis G-free =⇒ M is G/H-free for any subgroup H ≤ G. Onecan also show that M is G-projective =⇒ M isG/H-projective for any subgroup H ≤ G.

What about the reverse implications?

14 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesFree and projective modules

Let R = k × k for some field k. Consider R as a Z-graded ringwhere deg(R) = 0 and deg(x) = 1. Set I =

((0, 1)x, (1, 0)x2

)

Under the trivial grading, one can showI =

((0, 1)x+ (1, 0)x2

), and {(0, 1)x+ (1, 0)x2} is a basis for

I. Thus I is a free R-module. However, there is noZ-homogeneous basis for I, and so I cannot be a Z-freemodule.

On the other hand, under certain hypotheses (finitepresentation), it is true that for a G-graded module P , P beingG/H-projective implies that P is G-projective!

15 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesFree and projective modules

Let R = k × k for some field k. Consider R as a Z-graded ringwhere deg(R) = 0 and deg(x) = 1. Set I =

((0, 1)x, (1, 0)x2

)Under the trivial grading, one can showI =

((0, 1)x+ (1, 0)x2

), and {(0, 1)x+ (1, 0)x2} is a basis for

I. Thus I is a free R-module. However, there is noZ-homogeneous basis for I, and so I cannot be a Z-freemodule.

On the other hand, under certain hypotheses (finitepresentation), it is true that for a G-graded module P , P beingG/H-projective implies that P is G-projective!

15 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesFree and projective modules

Let R = k × k for some field k. Consider R as a Z-graded ringwhere deg(R) = 0 and deg(x) = 1. Set I =

((0, 1)x, (1, 0)x2

)Under the trivial grading, one can showI =

((0, 1)x+ (1, 0)x2

), and {(0, 1)x+ (1, 0)x2} is a basis for

I. Thus I is a free R-module. However, there is noZ-homogeneous basis for I, and so I cannot be a Z-freemodule.

On the other hand, under certain hypotheses (finitepresentation), it is true that for a G-graded module P , P beingG/H-projective implies that P is G-projective!

15 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesPrime avoidance

Lemma (Prime avoidance)

Suppose R is a commutative ring and P1, P2, . . . , Pn are primeideals of R. If I is an ideal of R such that I * Pi for each i,

then I *n⋃

i=1

Pi. That is, there exists x ∈ I \⋃Pi.

16 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesPrime avoidance

What is true in the G-graded case?

Proposition

Suppose R is a G-graded ring and P1, P2, . . . , Pn are G-primeideals. If I is a G-homogeneous ideal such that I * Pi for each

i, then I *n⋃

i=1

Pi.

17 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesPrime avoidance

What is true in the G-graded case?

Proposition

Suppose R is a G-graded ring and P1, P2, . . . , Pn are G-primeideals. If I is a G-homogeneous ideal such that I * Pi for each

i, then I *n⋃

i=1

Pi.

17 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesPrime avoidance

The following result is “among the most useful in the theory ofcommutative rings” (Kaplansky).

Proposition

Let R be Noetherian and M a finitely generated R-module. Ifan ideal I of R consists of zerodivisors of M , then I ⊆ P forsome P ∈ Ass(M).

Question: Suppose R is G-Noetherian, and M is a finitelygenerated graded R-module. If I is a G-homogeneous idealconsisting of zerodivisors, is I ⊆ P for some P ∈ AssG

R(M)?

18 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesPrime avoidance

The following result is “among the most useful in the theory ofcommutative rings” (Kaplansky).

Proposition

Let R be Noetherian and M a finitely generated R-module. Ifan ideal I of R consists of zerodivisors of M , then I ⊆ P forsome P ∈ Ass(M).

Question: Suppose R is G-Noetherian, and M is a finitelygenerated graded R-module. If I is a G-homogeneous idealconsisting of zerodivisors, is I ⊆ P for some P ∈ AssG

R(M)?

18 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesPrime avoidance

The following result is “among the most useful in the theory ofcommutative rings” (Kaplansky).

Proposition

Let R be Noetherian and M a finitely generated R-module. Ifan ideal I of R consists of zerodivisors of M , then I ⊆ P forsome P ∈ Ass(M).

Question: Suppose R is G-Noetherian, and M is a finitelygenerated graded R-module. If I is a G-homogeneous idealconsisting of zerodivisors, is I ⊆ P for some P ∈ AssG

R(M)?

18 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesGrade & depth

Classically, one way of defining the grade of an ideal I on R isin terms of regular sequences. An element x ∈ R is regular (onR) if xR 6= R and x is a non-zerodivisor on R. A sequencex1, . . . , xn is regular if x1 is regular on R and for i = 2, . . . , n,xi is regular on R/(x1, . . . , xi−1)R.

One then sets

gradeI(R) = max{n | x1, . . . , xn is a regular

sequence contained in I}.

There are problems (and solutions!) with this definition in thegraded case.

19 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesGrade & depth

Classically, one way of defining the grade of an ideal I on R isin terms of regular sequences. An element x ∈ R is regular (onR) if xR 6= R and x is a non-zerodivisor on R. A sequencex1, . . . , xn is regular if x1 is regular on R and for i = 2, . . . , n,xi is regular on R/(x1, . . . , xi−1)R.

One then sets

gradeI(R) = max{n | x1, . . . , xn is a regular

sequence contained in I}.

There are problems (and solutions!) with this definition in thegraded case.

19 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesGrade & depth

Classically, one way of defining the grade of an ideal I on R isin terms of regular sequences. An element x ∈ R is regular (onR) if xR 6= R and x is a non-zerodivisor on R. A sequencex1, . . . , xn is regular if x1 is regular on R and for i = 2, . . . , n,xi is regular on R/(x1, . . . , xi−1)R.

One then sets

gradeI(R) = max{n | x1, . . . , xn is a regular

sequence contained in I}.

There are problems (and solutions!) with this definition in thegraded case.

19 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesGrade & depth

A graded ring R with a unique G-maximal ideal is calledG-local, and we usually denote such a ring by (R,m), where mis the G-maximal ideal.

Then depth is defined in the usual way. If (R,m) is a G-localG-Noetherian graded ring, we set

depthG(R) := gradeGm(R).

20 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Chain conditions

Free andprojectivemodules

Prime avoidance

Grade & depth

Cohen-Macaulayness

Graded propertiesGrade & depth

A graded ring R with a unique G-maximal ideal is calledG-local, and we usually denote such a ring by (R,m), where mis the G-maximal ideal.

Then depth is defined in the usual way. If (R,m) is a G-localG-Noetherian graded ring, we set

depthG(R) := gradeGm(R).

20 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Cohen-Macaulayness

It has been said by Hochster and Huneke that “theCohen-Macaulay condition (possibly on the local rings of avariety) is just what is needed to make the theory work.”

If (R,m) is a G-local graded ring, we say R isG-Cohen-Macaulay if

depthG(R) = dimG(R).

Theorem (-)

Let R be a G-Noetherian graded ring, and suppose H ≤ G is afinitely generated torsion-free subgroup. TFAE:

1 R is G-Cohen-Macaulay.

2 R is G/H-Cohen-Macaulay.

21 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Cohen-Macaulayness

It has been said by Hochster and Huneke that “theCohen-Macaulay condition (possibly on the local rings of avariety) is just what is needed to make the theory work.”

If (R,m) is a G-local graded ring, we say R isG-Cohen-Macaulay if

depthG(R) = dimG(R).

Theorem (-)

Let R be a G-Noetherian graded ring, and suppose H ≤ G is afinitely generated torsion-free subgroup. TFAE:

1 R is G-Cohen-Macaulay.

2 R is G/H-Cohen-Macaulay.

21 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Cohen-Macaulayness

It has been said by Hochster and Huneke that “theCohen-Macaulay condition (possibly on the local rings of avariety) is just what is needed to make the theory work.”

If (R,m) is a G-local graded ring, we say R isG-Cohen-Macaulay if

depthG(R) = dimG(R).

Theorem (-)

Let R be a G-Noetherian graded ring, and suppose H ≤ G is afinitely generated torsion-free subgroup. TFAE:

1 R is G-Cohen-Macaulay.

2 R is G/H-Cohen-Macaulay.

21 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Future work

Regular rings

Gorenstein rings

Examples and counterexamples(e.g., if G = Z/nZ is there a G-graded ring withcharacteristic 0 such that R is G-reduced but notreduced?)

22 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Future work

Regular rings

Gorenstein rings

Examples and counterexamples(e.g., if G = Z/nZ is there a G-graded ring withcharacteristic 0 such that R is G-reduced but notreduced?)

22 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Future work

Regular rings

Gorenstein rings

Examples and counterexamples

(e.g., if G = Z/nZ is there a G-graded ring withcharacteristic 0 such that R is G-reduced but notreduced?)

22 / 22

Graded rings

Brian Johnson

Introduction

Notation/background

Gradedproperties

Cohen-Macaulayness

Future work

Regular rings

Gorenstein rings

Examples and counterexamples(e.g., if G = Z/nZ is there a G-graded ring withcharacteristic 0 such that R is G-reduced but notreduced?)

22 / 22