Graded Cohen-Macaulayness for commutative Brian Johnson rings - PowerPoint PPT Presentation

Graded Cohen- Macaulayness Graded Cohen-Macaulayness for commutative Brian Johnson rings graded by arbitrary abelian groups Introduction Primary De- composition Height & Brian Johnson Dimension Grade and Depth University Of Nebraska – Lincoln Main Theorem 14 October 2011 s-bjohns67@math.unl.edu 1 / 13

Introduction Outline Graded Cohen- Macaulayness Brian Johnson Notation Introduction Primary De- composition Height & Dimension Grade and Depth Main Theorem 2 / 13

Introduction Outline Graded Cohen- Macaulayness Brian Johnson Notation Introduction Primary De- composition Properties Height & Dimension Grade and Depth Main Theorem 2 / 13

Introduction Outline Graded Cohen- Macaulayness Brian Johnson Notation Introduction Primary De- composition Properties Height & Dimension Primary Decomposition Grade and Depth Main Theorem 2 / 13

Introduction Outline Graded Cohen- Macaulayness Brian Johnson Notation Introduction Primary De- composition Properties Height & Dimension Primary Decomposition Grade and Depth Height & Dimension Main Theorem 2 / 13

Introduction Outline Graded Cohen- Macaulayness Brian Johnson Notation Introduction Primary De- composition Properties Height & Dimension Primary Decomposition Grade and Depth Height & Dimension Main Theorem Grade & Depth 2 / 13

Introduction Notation Graded Cohen- Let G be an abelian group. A (commutative) ring R is Macaulayness G -graded if there is a family of subgroups of R , { R g } g ∈ G , such Brian Johnson that R = � g ∈ G R g , and R g R h ⊆ R g + h for all g, h ∈ G . Introduction Primary De- composition Height & Dimension Grade and Depth Main Theorem 3 / 13

Introduction Notation Graded Cohen- Let G be an abelian group. A (commutative) ring R is Macaulayness G -graded if there is a family of subgroups of R , { R g } g ∈ G , such Brian Johnson that R = � g ∈ G R g , and R g R h ⊆ R g + h for all g, h ∈ G . Introduction Primary De- For a subgroup H ≤ G , we set R H = � composition h ∈ H R h , which is a G - Height & and H -graded subring of R . More generally, Dimension � Grade and Depth R g + H := R g + h Main Theorem h ∈ H is a G -graded R H -submodule of R . 3 / 13

Introduction Notation Graded Cohen- Let G be an abelian group. A (commutative) ring R is Macaulayness G -graded if there is a family of subgroups of R , { R g } g ∈ G , such Brian Johnson that R = � g ∈ G R g , and R g R h ⊆ R g + h for all g, h ∈ G . Introduction Primary De- For a subgroup H ≤ G , we set R H = � composition h ∈ H R h , which is a G - Height & and H -graded subring of R . More generally, Dimension � Grade and Depth R g + H := R g + h Main Theorem h ∈ H is a G -graded R H -submodule of R . Note: the previous definition defines a G/H -grading on the ring R , using the family { R x } x ∈ G/H . 3 / 13

Introduction Notation/Properties Graded Cohen- Macaulayness One can define analogues of many usual properties. For Brian Johnson example, a G -field is a ring in which every homogeneous element is a unit, and a G -maximal ideal is a homogeneous Introduction ideal I such that R/I is a G -field (but we omit the G Primary De- composition whenever possible). Height & Dimension Grade and Depth Main Theorem 4 / 13

Introduction Notation/Properties Graded Cohen- Macaulayness One can define analogues of many usual properties. For Brian Johnson example, a G -field is a ring in which every homogeneous element is a unit, and a G -maximal ideal is a homogeneous Introduction ideal I such that R/I is a G -field (but we omit the G Primary De- composition whenever possible). Height & Dimension Grade and Depth Proposition Main Theorem Let R be a G -graded ring and H a torsion-free subgroup of G . Then 1 R is a domain if and only if R is a G/H -domain. 2 R is reduced if and only if R is G/H -reduced. 4 / 13

Introduction A crucial property Graded Cohen- Macaulayness Theorem Brian Johnson Suppose R is a G -graded ring. If H ≤ G is a finitely generated Introduction subgroup, the following are equivalent: Primary De- composition 1 R is Noetherian. Height & 2 R is G/H -Noetherian. Dimension Grade and Depth Main Theorem 5 / 13

Introduction A crucial property Graded Cohen- Macaulayness Theorem Brian Johnson Suppose R is a G -graded ring. If H ≤ G is a finitely generated Introduction subgroup, the following are equivalent: Primary De- composition 1 R is Noetherian. Height & 2 R is G/H -Noetherian. Dimension Grade and Depth One more basic piece of notation is the following: If R is Main Theorem G -graded, M is a G -graded R -module, and N is a 0 -graded R -submodule (i.e, not necessarily G -homogeneous) of M , we let N ∗ G denote the R -submodule of M generated by all the G -homogeneous elements contained in N . 5 / 13

Primary Decomposition Graded Cohen- Let R be a G -graded ring and N ⊆ M graded R -modules. Say Macaulayness N is G -irreducible if whenever N = N 1 ∩ N 2 ( N 1 , N 2 graded) Brian Johnson then N 1 = N or N 2 = N . Introduction Primary De- composition Height & Dimension Grade and Depth Main Theorem 6 / 13

Primary Decomposition Graded Cohen- Let R be a G -graded ring and N ⊆ M graded R -modules. Say Macaulayness N is G -irreducible if whenever N = N 1 ∩ N 2 ( N 1 , N 2 graded) Brian Johnson then N 1 = N or N 2 = N . Introduction Call N G -primary if for all homogeneous r ∈ R the map Primary De- composition r M/N → M/N induced by multiplication by r is either Height & injective or nilpotent. Dimension Grade and Depth Main Theorem 6 / 13

Primary Decomposition Graded Cohen- Let R be a G -graded ring and N ⊆ M graded R -modules. Say Macaulayness N is G -irreducible if whenever N = N 1 ∩ N 2 ( N 1 , N 2 graded) Brian Johnson then N 1 = N or N 2 = N . Introduction Call N G -primary if for all homogeneous r ∈ R the map Primary De- composition r M/N → M/N induced by multiplication by r is either Height & injective or nilpotent. Dimension If N = � N i is a primary decomposition, then the prime ideals Grade and Depth Main Theorem P i that occur as radicals of the Ann( M/N i ) depend only on M and N . 6 / 13

Primary Decomposition Graded Cohen- Let R be a G -graded ring and N ⊆ M graded R -modules. Say Macaulayness N is G -irreducible if whenever N = N 1 ∩ N 2 ( N 1 , N 2 graded) Brian Johnson then N 1 = N or N 2 = N . Introduction Call N G -primary if for all homogeneous r ∈ R the map Primary De- composition r M/N → M/N induced by multiplication by r is either Height & injective or nilpotent. Dimension If N = � N i is a primary decomposition, then the prime ideals Grade and Depth Main Theorem P i that occur as radicals of the Ann( M/N i ) depend only on M and N . If R is Noetherian, P ∈ Ass R if and only if P = (0 : f ) for some homogeneous element f ∈ R . Also, the union of the associated primes of R is, in general, strictly contained in the collection of zerodivisors of R . 6 / 13

Height & Dimension Graded Cohen- Macaulayness Brian Johnson Dimension of a G -graded ring and height of a Introduction ( G -homogeneous) ideal are defined in an expected way: Primary De- composition Height & Dimension Results on Height Grade and Depth Main Theorem 7 / 13

Height & Dimension Graded Cohen- Macaulayness Brian Johnson Dimension of a G -graded ring and height of a Introduction ( G -homogeneous) ideal are defined in an expected way: Primary De- composition Height & Dimension dim G R ( R ) := sup { n | P 0 � P 1 � · · · � P n Results on Height is a chain of prime ideals of R } Grade and Depth Main Theorem 7 / 13

Height & Dimension Graded Cohen- Macaulayness Brian Johnson Dimension of a G -graded ring and height of a Introduction ( G -homogeneous) ideal are defined in an expected way: Primary De- composition Height & Dimension dim G R ( R ) := sup { n | P 0 � P 1 � · · · � P n Results on Height is a chain of prime ideals of R } Grade and Depth Main Theorem ht G R ( I ) := min { dim( R ( P ) ) | P ⊇ I and P is prime } 7 / 13

Results on Height Graded Cohen- The following fact is a generalization of a result of Macaulayness Matijevic-Roberts (1973). Brian Johnson Let R be a G -graded ring, and suppose H ≤ G is a Introduction torsion-free subgroup. If P ∈ Spec G/H ( R ) and P ∗ = P ∗ G , Primary De- composition then ht G/H ( P/P ∗ ) ≤ rank H . Height & Dimension Results on Height Grade and Depth Main Theorem 8 / 13

Results on Height Graded Cohen- The following fact is a generalization of a result of Macaulayness Matijevic-Roberts (1973). Brian Johnson Let R be a G -graded ring, and suppose H ≤ G is a Introduction torsion-free subgroup. If P ∈ Spec G/H ( R ) and P ∗ = P ∗ G , Primary De- composition then ht G/H ( P/P ∗ ) ≤ rank H . Height & Dimension Results on This was extended to Z d -graded rings and sharpened by Height Uliczka (2009). A further generalization is: Grade and Depth Main Theorem Theorem Let R be a G -graded ring and H ≤ G a torsion-free subgroup of finite rank, and set P ∗ = P ∗ G . If P ∈ Spec G/H ( R ) , then ht G/H ( P ) = ht G/H ( P ∗ ) + ht G/H ( P/P ∗ ) . 8 / 13