Graded analogues of theorems in commutative algebra












15












$begingroup$


Many theorems in commutative algebra hold true in a ($mathbb{Z}$-)graded context. More precisely, we can take any theorem in commutative algebra and replace every occurrence of the word




  • commutative ring by commutative graded ring (without the sign for commutativity)

  • module by graded module

  • element by homogeneous element

  • ideal by homogeneous ideal (i.e. ideal generated by homogeneous elements)


This results in further substitutions, e.g. a $ast$local ring is a graded ring with a unique maximal homogeneous ideal, we get a notion of graded depth etc. After all these substitutions we can ask whether the theorem is still true.



One book that does some steps in this direction is Cohen-Macaulay rings by Bruns and Herzog, especially Section 1.5. For example, they have as Exercise 1.5.24 the following graded analogue of the Nakayama lemma:




Let $(R,mathfrak{m})$ be a $ast$local ring, $M$ be a finitely generated graded $R$-module and $N$ a graded submodule. Assume $M = N + mathfrak{m}M$. Then $M = N$.




Moreover, a student of mine recently showed that the graded analogue of Lazard's theorem (a module is flat if and only if it is a filtered colimit of free modules) is also true.



Usually one proves this kind of theorems essentially by a combination of two techniques:




  1. Copy the ungraded proof and just substitute ungraded for graded concepts in the manner sketched above.

  2. If one is annoyed by the length of the resulting argument, use some shortcuts by some translations between ungraded and graded. (E.g. a noetherian graded ring that is graded Cohen-Macaulay is also ungraded Cohen-Macaulay.)


Sometimes one can also be lucky and the statement is suitably algebro-geometry that one can argue geometrically with the stack $[Spec R/mathbb{G}_m]$ for a graded ring $R$, using that a $mathbb{Z}$-grading corresponds to a $mathbb{G}_m$-action.



In any case, my question is the following:




Is there any class of statements, where one knows automatically that the graded analogue is true if the original statement in ungraded commutative algebra is true, without going through the whole proof?




I am not sure whether one can hope here for a model-theoretic approach as I know almost nothing about model theory, but any such statement could save a lot of work in proving graded analogues of known theorems.










share|cite|improve this question









$endgroup$












  • $begingroup$
    Do you have a reference for how to argue geometrically using stacks as you allude to?
    $endgroup$
    – Avi Steiner
    2 days ago






  • 1




    $begingroup$
    Call a graded module over a graded ring $R$ graded flat if its tensor product with every exact sequence of graded modules is graded again. I claim that a module is graded flat if and only if its flat. Indeed, the category of graded modules is monoidally equivalent to quasi-coherent sheaves on $[Spec R/mathbb{G}_m]$. As $Spec R to [Spec R/mathbb{G}_m]$ is fpqc, exactness of a sequence can be tested on $Spec R$, which implies the result. Thus every theorem about flat modules also applies to graded flat modules.
    $endgroup$
    – Lennart Meier
    yesterday
















15












$begingroup$


Many theorems in commutative algebra hold true in a ($mathbb{Z}$-)graded context. More precisely, we can take any theorem in commutative algebra and replace every occurrence of the word




  • commutative ring by commutative graded ring (without the sign for commutativity)

  • module by graded module

  • element by homogeneous element

  • ideal by homogeneous ideal (i.e. ideal generated by homogeneous elements)


This results in further substitutions, e.g. a $ast$local ring is a graded ring with a unique maximal homogeneous ideal, we get a notion of graded depth etc. After all these substitutions we can ask whether the theorem is still true.



One book that does some steps in this direction is Cohen-Macaulay rings by Bruns and Herzog, especially Section 1.5. For example, they have as Exercise 1.5.24 the following graded analogue of the Nakayama lemma:




Let $(R,mathfrak{m})$ be a $ast$local ring, $M$ be a finitely generated graded $R$-module and $N$ a graded submodule. Assume $M = N + mathfrak{m}M$. Then $M = N$.




Moreover, a student of mine recently showed that the graded analogue of Lazard's theorem (a module is flat if and only if it is a filtered colimit of free modules) is also true.



Usually one proves this kind of theorems essentially by a combination of two techniques:




  1. Copy the ungraded proof and just substitute ungraded for graded concepts in the manner sketched above.

  2. If one is annoyed by the length of the resulting argument, use some shortcuts by some translations between ungraded and graded. (E.g. a noetherian graded ring that is graded Cohen-Macaulay is also ungraded Cohen-Macaulay.)


Sometimes one can also be lucky and the statement is suitably algebro-geometry that one can argue geometrically with the stack $[Spec R/mathbb{G}_m]$ for a graded ring $R$, using that a $mathbb{Z}$-grading corresponds to a $mathbb{G}_m$-action.



In any case, my question is the following:




Is there any class of statements, where one knows automatically that the graded analogue is true if the original statement in ungraded commutative algebra is true, without going through the whole proof?




I am not sure whether one can hope here for a model-theoretic approach as I know almost nothing about model theory, but any such statement could save a lot of work in proving graded analogues of known theorems.










share|cite|improve this question









$endgroup$












  • $begingroup$
    Do you have a reference for how to argue geometrically using stacks as you allude to?
    $endgroup$
    – Avi Steiner
    2 days ago






  • 1




    $begingroup$
    Call a graded module over a graded ring $R$ graded flat if its tensor product with every exact sequence of graded modules is graded again. I claim that a module is graded flat if and only if its flat. Indeed, the category of graded modules is monoidally equivalent to quasi-coherent sheaves on $[Spec R/mathbb{G}_m]$. As $Spec R to [Spec R/mathbb{G}_m]$ is fpqc, exactness of a sequence can be tested on $Spec R$, which implies the result. Thus every theorem about flat modules also applies to graded flat modules.
    $endgroup$
    – Lennart Meier
    yesterday














15












15








15


8



$begingroup$


Many theorems in commutative algebra hold true in a ($mathbb{Z}$-)graded context. More precisely, we can take any theorem in commutative algebra and replace every occurrence of the word




  • commutative ring by commutative graded ring (without the sign for commutativity)

  • module by graded module

  • element by homogeneous element

  • ideal by homogeneous ideal (i.e. ideal generated by homogeneous elements)


This results in further substitutions, e.g. a $ast$local ring is a graded ring with a unique maximal homogeneous ideal, we get a notion of graded depth etc. After all these substitutions we can ask whether the theorem is still true.



One book that does some steps in this direction is Cohen-Macaulay rings by Bruns and Herzog, especially Section 1.5. For example, they have as Exercise 1.5.24 the following graded analogue of the Nakayama lemma:




Let $(R,mathfrak{m})$ be a $ast$local ring, $M$ be a finitely generated graded $R$-module and $N$ a graded submodule. Assume $M = N + mathfrak{m}M$. Then $M = N$.




Moreover, a student of mine recently showed that the graded analogue of Lazard's theorem (a module is flat if and only if it is a filtered colimit of free modules) is also true.



Usually one proves this kind of theorems essentially by a combination of two techniques:




  1. Copy the ungraded proof and just substitute ungraded for graded concepts in the manner sketched above.

  2. If one is annoyed by the length of the resulting argument, use some shortcuts by some translations between ungraded and graded. (E.g. a noetherian graded ring that is graded Cohen-Macaulay is also ungraded Cohen-Macaulay.)


Sometimes one can also be lucky and the statement is suitably algebro-geometry that one can argue geometrically with the stack $[Spec R/mathbb{G}_m]$ for a graded ring $R$, using that a $mathbb{Z}$-grading corresponds to a $mathbb{G}_m$-action.



In any case, my question is the following:




Is there any class of statements, where one knows automatically that the graded analogue is true if the original statement in ungraded commutative algebra is true, without going through the whole proof?




I am not sure whether one can hope here for a model-theoretic approach as I know almost nothing about model theory, but any such statement could save a lot of work in proving graded analogues of known theorems.










share|cite|improve this question









$endgroup$




Many theorems in commutative algebra hold true in a ($mathbb{Z}$-)graded context. More precisely, we can take any theorem in commutative algebra and replace every occurrence of the word




  • commutative ring by commutative graded ring (without the sign for commutativity)

  • module by graded module

  • element by homogeneous element

  • ideal by homogeneous ideal (i.e. ideal generated by homogeneous elements)


This results in further substitutions, e.g. a $ast$local ring is a graded ring with a unique maximal homogeneous ideal, we get a notion of graded depth etc. After all these substitutions we can ask whether the theorem is still true.



One book that does some steps in this direction is Cohen-Macaulay rings by Bruns and Herzog, especially Section 1.5. For example, they have as Exercise 1.5.24 the following graded analogue of the Nakayama lemma:




Let $(R,mathfrak{m})$ be a $ast$local ring, $M$ be a finitely generated graded $R$-module and $N$ a graded submodule. Assume $M = N + mathfrak{m}M$. Then $M = N$.




Moreover, a student of mine recently showed that the graded analogue of Lazard's theorem (a module is flat if and only if it is a filtered colimit of free modules) is also true.



Usually one proves this kind of theorems essentially by a combination of two techniques:




  1. Copy the ungraded proof and just substitute ungraded for graded concepts in the manner sketched above.

  2. If one is annoyed by the length of the resulting argument, use some shortcuts by some translations between ungraded and graded. (E.g. a noetherian graded ring that is graded Cohen-Macaulay is also ungraded Cohen-Macaulay.)


Sometimes one can also be lucky and the statement is suitably algebro-geometry that one can argue geometrically with the stack $[Spec R/mathbb{G}_m]$ for a graded ring $R$, using that a $mathbb{Z}$-grading corresponds to a $mathbb{G}_m$-action.



In any case, my question is the following:




Is there any class of statements, where one knows automatically that the graded analogue is true if the original statement in ungraded commutative algebra is true, without going through the whole proof?




I am not sure whether one can hope here for a model-theoretic approach as I know almost nothing about model theory, but any such statement could save a lot of work in proving graded analogues of known theorems.







ac.commutative-algebra graded-rings-modules






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share|cite|improve this question











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share|cite|improve this question










asked Jan 7 at 8:31









Lennart MeierLennart Meier

6,28424572




6,28424572












  • $begingroup$
    Do you have a reference for how to argue geometrically using stacks as you allude to?
    $endgroup$
    – Avi Steiner
    2 days ago






  • 1




    $begingroup$
    Call a graded module over a graded ring $R$ graded flat if its tensor product with every exact sequence of graded modules is graded again. I claim that a module is graded flat if and only if its flat. Indeed, the category of graded modules is monoidally equivalent to quasi-coherent sheaves on $[Spec R/mathbb{G}_m]$. As $Spec R to [Spec R/mathbb{G}_m]$ is fpqc, exactness of a sequence can be tested on $Spec R$, which implies the result. Thus every theorem about flat modules also applies to graded flat modules.
    $endgroup$
    – Lennart Meier
    yesterday


















  • $begingroup$
    Do you have a reference for how to argue geometrically using stacks as you allude to?
    $endgroup$
    – Avi Steiner
    2 days ago






  • 1




    $begingroup$
    Call a graded module over a graded ring $R$ graded flat if its tensor product with every exact sequence of graded modules is graded again. I claim that a module is graded flat if and only if its flat. Indeed, the category of graded modules is monoidally equivalent to quasi-coherent sheaves on $[Spec R/mathbb{G}_m]$. As $Spec R to [Spec R/mathbb{G}_m]$ is fpqc, exactness of a sequence can be tested on $Spec R$, which implies the result. Thus every theorem about flat modules also applies to graded flat modules.
    $endgroup$
    – Lennart Meier
    yesterday
















$begingroup$
Do you have a reference for how to argue geometrically using stacks as you allude to?
$endgroup$
– Avi Steiner
2 days ago




$begingroup$
Do you have a reference for how to argue geometrically using stacks as you allude to?
$endgroup$
– Avi Steiner
2 days ago




1




1




$begingroup$
Call a graded module over a graded ring $R$ graded flat if its tensor product with every exact sequence of graded modules is graded again. I claim that a module is graded flat if and only if its flat. Indeed, the category of graded modules is monoidally equivalent to quasi-coherent sheaves on $[Spec R/mathbb{G}_m]$. As $Spec R to [Spec R/mathbb{G}_m]$ is fpqc, exactness of a sequence can be tested on $Spec R$, which implies the result. Thus every theorem about flat modules also applies to graded flat modules.
$endgroup$
– Lennart Meier
yesterday




$begingroup$
Call a graded module over a graded ring $R$ graded flat if its tensor product with every exact sequence of graded modules is graded again. I claim that a module is graded flat if and only if its flat. Indeed, the category of graded modules is monoidally equivalent to quasi-coherent sheaves on $[Spec R/mathbb{G}_m]$. As $Spec R to [Spec R/mathbb{G}_m]$ is fpqc, exactness of a sequence can be tested on $Spec R$, which implies the result. Thus every theorem about flat modules also applies to graded flat modules.
$endgroup$
– Lennart Meier
yesterday










3 Answers
3






active

oldest

votes


















3












$begingroup$

Let $mathbb P(A,M)$ be a property of a finitely generated module $M$ over a
Noetherian local ring $A$.
Let $R$ be a $mathbb Z^n$-graded ring, $N$ a finitely generated graded $R$-module, and
$P$ a prime ideal of $R$.
Let $P^*$ be the largest homogeneous ideal contained in $P$ (it is a prime ideal).
For many important $mathbb P$, we have that $mathbb P(R_P,N_P)$ is equivalent to
$mathbb P(R_{P^*},N_{P^*})$.



For example, the property $M=0$ satisfies the equivalence, and the graded Nakayama
follows from this.



Indeed, if $mathcal C$ and $mathcal D$ are classes of Noetherian local rings and
$mathbb P(A,M)$ is a property of a finitely generated module over a Noetherian
local ring, and assume that



(1) If $Ainmathcal C$, $M$ a finite $A$-module, $mathbb P(A,M)$, and $Arightarrow B$ is a regular local
homomorphism essentially of finite type, then $Binmathcal D$ and $mathbb P(B,B
otimes_AM)$
holds.



(2) If $A$ is a Noetherian local ring and $M$ a finite $A$-module, $Arightarrow B$ is a regular local homomorphism essentially of finite type, $Binmathcal D$ and $mathbb P(B,Botimes_AM)$ holds, then $Ainmathcal D$ and $mathbb P(A,M)$ holds.



Then for any $mathbb Z^n$-graded Noetherian ring $R$ and a finitely generated graded $R$-module
$N$, if $R_{P^*}inmathcal C$ and $mathbb P(R_{P^*},N_{P^*})$ holds, then $R_Pinmathcal D$ and $mathbb P(R_P,N_P)$.



For example, letting $mathcal C=mathcal D=text{Cohen-Macaulay}$ and $mathbb P$ be
anything, we have that $R_{P^*}$ CM implies $R_P$ CM.
So if $(R,mathfrak m)$ is a ${}^*$local ring and $R$ is graded CM, then
$R_{mathfrak m}$ is CM, and so $R_Q$ is CM for any graded prime ideal $Q$.
So $R_P$ is CM for any prime, since $R_{P^*}$ is CM.
We have proved that $R$ is CM.



Similar argument is applicable to Gorenstein, local complete intersection, and regular.
It is also applicable to some $F$-singularities.



On the other hand, letting $mathcal C=mathcal D=text{Noetherian local}$, and
letting $mathbb P(A,M)$ to be finite projective dimension,'finite injective
dimension,' torsionless,'reflexive' and so on, the condition is satisfied.



Maybe this does not produce any graded "analogue," but I hope this is useful.
Please see section 7 of M. Hashimoto and M. Miyazaki, Comm. Algebra 2013 for details.






share|cite|improve this answer









$endgroup$





















    2












    $begingroup$

    Maybe not really what you are after, but anyway: This holds for all those statements that depend only on the structure of abelian category on the category of graded modules. Namely, if $G$ is any commutative group, then the category of $G$-graded $R$-modules, where $R$ is a $G$-graded ring, is abelian and fulfills AB5 and AB4, hence has a projective generator and an injective cogenerator.



    However, one has to be careful with this approach, since there are still many differences between these categories for different groups $G$. For example, suppose that $R$ is a noetherian $G$-graded ring. If $G$ is finite, then the category of $G$-graded $R$-modules has a noetherian generator, but if $G$ is infinite then it does not.






    share|cite|improve this answer









    $endgroup$





















      1












      $begingroup$

      Since the OP is asking for some "class of statements" which can be readily transferred from the ungraded to the graded case, let me outline a couple of thoughts, which are not really tied to neither the commutative case, nor the $mathbb{Z}$-gradation itself:




      • On the one hand, if $R$ is a $k$-algebra and $G$ is a group, then a $G$-gradation on $R$ can be equivalently viewed either as a $(kG)^*$-action or as a $kG$-coaction (where $kG$ is the group hopf algebra and $(kG)^*$ its dual hopf algebra). So, the graded algebra $R$ is alternatively viewed as an algebra in the category of $(kG)^*$-modules or an algebra in the category of $kG$-comodules. In this sense, the statements and the properties which are "naturally" transferred between the graded and the ungraded case are those which are in some sense "compatible" or "invariant" with respect to the $kG$-coaction (or the $(kG)^*$-action). I do not know if it sounds like an abuse of terminology to speak about


        properties in the category of $(kG)^*$-modules or properties in the category of $kG$-comodules





      • on the other hand, the Category $R_{gr}$-$mod$ of graded modules over the $G$-graded ring $R$, is equivalent (actually i think it is isomorphic in most cases) to a closed subcategory of modules (in the ungraded sense) over the smash product algebra $Rsharp kG$. (Here the smash product is understood in the sense of the one associated to a Hopf module algebra).

        In this sense, we could say that the statements and properties which one might expect to "naturally transfer" over between graded and ungraded, are




        the ones which are preserved under the smash product between $R$ and the group hopf algebra $kG$.




        (In some sense, this implies that it might be meaningful to consider properties which "transfer" between the $R$-modules and the $Rsharp kG$-modules).




      One might reasonably expect that




      all those statements that depend only on the structure of abelian category on the category of graded modules




      mentioned in Fred Rohrer's answer fall into the classes mentioned above, however i can't tell for sure on the exact overlap between these classes.






      share|cite|improve this answer











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        3 Answers
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        3 Answers
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        $begingroup$

        Let $mathbb P(A,M)$ be a property of a finitely generated module $M$ over a
        Noetherian local ring $A$.
        Let $R$ be a $mathbb Z^n$-graded ring, $N$ a finitely generated graded $R$-module, and
        $P$ a prime ideal of $R$.
        Let $P^*$ be the largest homogeneous ideal contained in $P$ (it is a prime ideal).
        For many important $mathbb P$, we have that $mathbb P(R_P,N_P)$ is equivalent to
        $mathbb P(R_{P^*},N_{P^*})$.



        For example, the property $M=0$ satisfies the equivalence, and the graded Nakayama
        follows from this.



        Indeed, if $mathcal C$ and $mathcal D$ are classes of Noetherian local rings and
        $mathbb P(A,M)$ is a property of a finitely generated module over a Noetherian
        local ring, and assume that



        (1) If $Ainmathcal C$, $M$ a finite $A$-module, $mathbb P(A,M)$, and $Arightarrow B$ is a regular local
        homomorphism essentially of finite type, then $Binmathcal D$ and $mathbb P(B,B
        otimes_AM)$
        holds.



        (2) If $A$ is a Noetherian local ring and $M$ a finite $A$-module, $Arightarrow B$ is a regular local homomorphism essentially of finite type, $Binmathcal D$ and $mathbb P(B,Botimes_AM)$ holds, then $Ainmathcal D$ and $mathbb P(A,M)$ holds.



        Then for any $mathbb Z^n$-graded Noetherian ring $R$ and a finitely generated graded $R$-module
        $N$, if $R_{P^*}inmathcal C$ and $mathbb P(R_{P^*},N_{P^*})$ holds, then $R_Pinmathcal D$ and $mathbb P(R_P,N_P)$.



        For example, letting $mathcal C=mathcal D=text{Cohen-Macaulay}$ and $mathbb P$ be
        anything, we have that $R_{P^*}$ CM implies $R_P$ CM.
        So if $(R,mathfrak m)$ is a ${}^*$local ring and $R$ is graded CM, then
        $R_{mathfrak m}$ is CM, and so $R_Q$ is CM for any graded prime ideal $Q$.
        So $R_P$ is CM for any prime, since $R_{P^*}$ is CM.
        We have proved that $R$ is CM.



        Similar argument is applicable to Gorenstein, local complete intersection, and regular.
        It is also applicable to some $F$-singularities.



        On the other hand, letting $mathcal C=mathcal D=text{Noetherian local}$, and
        letting $mathbb P(A,M)$ to be finite projective dimension,'finite injective
        dimension,' torsionless,'reflexive' and so on, the condition is satisfied.



        Maybe this does not produce any graded "analogue," but I hope this is useful.
        Please see section 7 of M. Hashimoto and M. Miyazaki, Comm. Algebra 2013 for details.






        share|cite|improve this answer









        $endgroup$


















          3












          $begingroup$

          Let $mathbb P(A,M)$ be a property of a finitely generated module $M$ over a
          Noetherian local ring $A$.
          Let $R$ be a $mathbb Z^n$-graded ring, $N$ a finitely generated graded $R$-module, and
          $P$ a prime ideal of $R$.
          Let $P^*$ be the largest homogeneous ideal contained in $P$ (it is a prime ideal).
          For many important $mathbb P$, we have that $mathbb P(R_P,N_P)$ is equivalent to
          $mathbb P(R_{P^*},N_{P^*})$.



          For example, the property $M=0$ satisfies the equivalence, and the graded Nakayama
          follows from this.



          Indeed, if $mathcal C$ and $mathcal D$ are classes of Noetherian local rings and
          $mathbb P(A,M)$ is a property of a finitely generated module over a Noetherian
          local ring, and assume that



          (1) If $Ainmathcal C$, $M$ a finite $A$-module, $mathbb P(A,M)$, and $Arightarrow B$ is a regular local
          homomorphism essentially of finite type, then $Binmathcal D$ and $mathbb P(B,B
          otimes_AM)$
          holds.



          (2) If $A$ is a Noetherian local ring and $M$ a finite $A$-module, $Arightarrow B$ is a regular local homomorphism essentially of finite type, $Binmathcal D$ and $mathbb P(B,Botimes_AM)$ holds, then $Ainmathcal D$ and $mathbb P(A,M)$ holds.



          Then for any $mathbb Z^n$-graded Noetherian ring $R$ and a finitely generated graded $R$-module
          $N$, if $R_{P^*}inmathcal C$ and $mathbb P(R_{P^*},N_{P^*})$ holds, then $R_Pinmathcal D$ and $mathbb P(R_P,N_P)$.



          For example, letting $mathcal C=mathcal D=text{Cohen-Macaulay}$ and $mathbb P$ be
          anything, we have that $R_{P^*}$ CM implies $R_P$ CM.
          So if $(R,mathfrak m)$ is a ${}^*$local ring and $R$ is graded CM, then
          $R_{mathfrak m}$ is CM, and so $R_Q$ is CM for any graded prime ideal $Q$.
          So $R_P$ is CM for any prime, since $R_{P^*}$ is CM.
          We have proved that $R$ is CM.



          Similar argument is applicable to Gorenstein, local complete intersection, and regular.
          It is also applicable to some $F$-singularities.



          On the other hand, letting $mathcal C=mathcal D=text{Noetherian local}$, and
          letting $mathbb P(A,M)$ to be finite projective dimension,'finite injective
          dimension,' torsionless,'reflexive' and so on, the condition is satisfied.



          Maybe this does not produce any graded "analogue," but I hope this is useful.
          Please see section 7 of M. Hashimoto and M. Miyazaki, Comm. Algebra 2013 for details.






          share|cite|improve this answer









          $endgroup$
















            3












            3








            3





            $begingroup$

            Let $mathbb P(A,M)$ be a property of a finitely generated module $M$ over a
            Noetherian local ring $A$.
            Let $R$ be a $mathbb Z^n$-graded ring, $N$ a finitely generated graded $R$-module, and
            $P$ a prime ideal of $R$.
            Let $P^*$ be the largest homogeneous ideal contained in $P$ (it is a prime ideal).
            For many important $mathbb P$, we have that $mathbb P(R_P,N_P)$ is equivalent to
            $mathbb P(R_{P^*},N_{P^*})$.



            For example, the property $M=0$ satisfies the equivalence, and the graded Nakayama
            follows from this.



            Indeed, if $mathcal C$ and $mathcal D$ are classes of Noetherian local rings and
            $mathbb P(A,M)$ is a property of a finitely generated module over a Noetherian
            local ring, and assume that



            (1) If $Ainmathcal C$, $M$ a finite $A$-module, $mathbb P(A,M)$, and $Arightarrow B$ is a regular local
            homomorphism essentially of finite type, then $Binmathcal D$ and $mathbb P(B,B
            otimes_AM)$
            holds.



            (2) If $A$ is a Noetherian local ring and $M$ a finite $A$-module, $Arightarrow B$ is a regular local homomorphism essentially of finite type, $Binmathcal D$ and $mathbb P(B,Botimes_AM)$ holds, then $Ainmathcal D$ and $mathbb P(A,M)$ holds.



            Then for any $mathbb Z^n$-graded Noetherian ring $R$ and a finitely generated graded $R$-module
            $N$, if $R_{P^*}inmathcal C$ and $mathbb P(R_{P^*},N_{P^*})$ holds, then $R_Pinmathcal D$ and $mathbb P(R_P,N_P)$.



            For example, letting $mathcal C=mathcal D=text{Cohen-Macaulay}$ and $mathbb P$ be
            anything, we have that $R_{P^*}$ CM implies $R_P$ CM.
            So if $(R,mathfrak m)$ is a ${}^*$local ring and $R$ is graded CM, then
            $R_{mathfrak m}$ is CM, and so $R_Q$ is CM for any graded prime ideal $Q$.
            So $R_P$ is CM for any prime, since $R_{P^*}$ is CM.
            We have proved that $R$ is CM.



            Similar argument is applicable to Gorenstein, local complete intersection, and regular.
            It is also applicable to some $F$-singularities.



            On the other hand, letting $mathcal C=mathcal D=text{Noetherian local}$, and
            letting $mathbb P(A,M)$ to be finite projective dimension,'finite injective
            dimension,' torsionless,'reflexive' and so on, the condition is satisfied.



            Maybe this does not produce any graded "analogue," but I hope this is useful.
            Please see section 7 of M. Hashimoto and M. Miyazaki, Comm. Algebra 2013 for details.






            share|cite|improve this answer









            $endgroup$



            Let $mathbb P(A,M)$ be a property of a finitely generated module $M$ over a
            Noetherian local ring $A$.
            Let $R$ be a $mathbb Z^n$-graded ring, $N$ a finitely generated graded $R$-module, and
            $P$ a prime ideal of $R$.
            Let $P^*$ be the largest homogeneous ideal contained in $P$ (it is a prime ideal).
            For many important $mathbb P$, we have that $mathbb P(R_P,N_P)$ is equivalent to
            $mathbb P(R_{P^*},N_{P^*})$.



            For example, the property $M=0$ satisfies the equivalence, and the graded Nakayama
            follows from this.



            Indeed, if $mathcal C$ and $mathcal D$ are classes of Noetherian local rings and
            $mathbb P(A,M)$ is a property of a finitely generated module over a Noetherian
            local ring, and assume that



            (1) If $Ainmathcal C$, $M$ a finite $A$-module, $mathbb P(A,M)$, and $Arightarrow B$ is a regular local
            homomorphism essentially of finite type, then $Binmathcal D$ and $mathbb P(B,B
            otimes_AM)$
            holds.



            (2) If $A$ is a Noetherian local ring and $M$ a finite $A$-module, $Arightarrow B$ is a regular local homomorphism essentially of finite type, $Binmathcal D$ and $mathbb P(B,Botimes_AM)$ holds, then $Ainmathcal D$ and $mathbb P(A,M)$ holds.



            Then for any $mathbb Z^n$-graded Noetherian ring $R$ and a finitely generated graded $R$-module
            $N$, if $R_{P^*}inmathcal C$ and $mathbb P(R_{P^*},N_{P^*})$ holds, then $R_Pinmathcal D$ and $mathbb P(R_P,N_P)$.



            For example, letting $mathcal C=mathcal D=text{Cohen-Macaulay}$ and $mathbb P$ be
            anything, we have that $R_{P^*}$ CM implies $R_P$ CM.
            So if $(R,mathfrak m)$ is a ${}^*$local ring and $R$ is graded CM, then
            $R_{mathfrak m}$ is CM, and so $R_Q$ is CM for any graded prime ideal $Q$.
            So $R_P$ is CM for any prime, since $R_{P^*}$ is CM.
            We have proved that $R$ is CM.



            Similar argument is applicable to Gorenstein, local complete intersection, and regular.
            It is also applicable to some $F$-singularities.



            On the other hand, letting $mathcal C=mathcal D=text{Noetherian local}$, and
            letting $mathbb P(A,M)$ to be finite projective dimension,'finite injective
            dimension,' torsionless,'reflexive' and so on, the condition is satisfied.



            Maybe this does not produce any graded "analogue," but I hope this is useful.
            Please see section 7 of M. Hashimoto and M. Miyazaki, Comm. Algebra 2013 for details.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Jan 7 at 14:37









            P. GrapeP. Grape

            814




            814























                2












                $begingroup$

                Maybe not really what you are after, but anyway: This holds for all those statements that depend only on the structure of abelian category on the category of graded modules. Namely, if $G$ is any commutative group, then the category of $G$-graded $R$-modules, where $R$ is a $G$-graded ring, is abelian and fulfills AB5 and AB4, hence has a projective generator and an injective cogenerator.



                However, one has to be careful with this approach, since there are still many differences between these categories for different groups $G$. For example, suppose that $R$ is a noetherian $G$-graded ring. If $G$ is finite, then the category of $G$-graded $R$-modules has a noetherian generator, but if $G$ is infinite then it does not.






                share|cite|improve this answer









                $endgroup$


















                  2












                  $begingroup$

                  Maybe not really what you are after, but anyway: This holds for all those statements that depend only on the structure of abelian category on the category of graded modules. Namely, if $G$ is any commutative group, then the category of $G$-graded $R$-modules, where $R$ is a $G$-graded ring, is abelian and fulfills AB5 and AB4, hence has a projective generator and an injective cogenerator.



                  However, one has to be careful with this approach, since there are still many differences between these categories for different groups $G$. For example, suppose that $R$ is a noetherian $G$-graded ring. If $G$ is finite, then the category of $G$-graded $R$-modules has a noetherian generator, but if $G$ is infinite then it does not.






                  share|cite|improve this answer









                  $endgroup$
















                    2












                    2








                    2





                    $begingroup$

                    Maybe not really what you are after, but anyway: This holds for all those statements that depend only on the structure of abelian category on the category of graded modules. Namely, if $G$ is any commutative group, then the category of $G$-graded $R$-modules, where $R$ is a $G$-graded ring, is abelian and fulfills AB5 and AB4, hence has a projective generator and an injective cogenerator.



                    However, one has to be careful with this approach, since there are still many differences between these categories for different groups $G$. For example, suppose that $R$ is a noetherian $G$-graded ring. If $G$ is finite, then the category of $G$-graded $R$-modules has a noetherian generator, but if $G$ is infinite then it does not.






                    share|cite|improve this answer









                    $endgroup$



                    Maybe not really what you are after, but anyway: This holds for all those statements that depend only on the structure of abelian category on the category of graded modules. Namely, if $G$ is any commutative group, then the category of $G$-graded $R$-modules, where $R$ is a $G$-graded ring, is abelian and fulfills AB5 and AB4, hence has a projective generator and an injective cogenerator.



                    However, one has to be careful with this approach, since there are still many differences between these categories for different groups $G$. For example, suppose that $R$ is a noetherian $G$-graded ring. If $G$ is finite, then the category of $G$-graded $R$-modules has a noetherian generator, but if $G$ is infinite then it does not.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Jan 7 at 12:24









                    Fred RohrerFred Rohrer

                    4,39111734




                    4,39111734























                        1












                        $begingroup$

                        Since the OP is asking for some "class of statements" which can be readily transferred from the ungraded to the graded case, let me outline a couple of thoughts, which are not really tied to neither the commutative case, nor the $mathbb{Z}$-gradation itself:




                        • On the one hand, if $R$ is a $k$-algebra and $G$ is a group, then a $G$-gradation on $R$ can be equivalently viewed either as a $(kG)^*$-action or as a $kG$-coaction (where $kG$ is the group hopf algebra and $(kG)^*$ its dual hopf algebra). So, the graded algebra $R$ is alternatively viewed as an algebra in the category of $(kG)^*$-modules or an algebra in the category of $kG$-comodules. In this sense, the statements and the properties which are "naturally" transferred between the graded and the ungraded case are those which are in some sense "compatible" or "invariant" with respect to the $kG$-coaction (or the $(kG)^*$-action). I do not know if it sounds like an abuse of terminology to speak about


                          properties in the category of $(kG)^*$-modules or properties in the category of $kG$-comodules





                        • on the other hand, the Category $R_{gr}$-$mod$ of graded modules over the $G$-graded ring $R$, is equivalent (actually i think it is isomorphic in most cases) to a closed subcategory of modules (in the ungraded sense) over the smash product algebra $Rsharp kG$. (Here the smash product is understood in the sense of the one associated to a Hopf module algebra).

                          In this sense, we could say that the statements and properties which one might expect to "naturally transfer" over between graded and ungraded, are




                          the ones which are preserved under the smash product between $R$ and the group hopf algebra $kG$.




                          (In some sense, this implies that it might be meaningful to consider properties which "transfer" between the $R$-modules and the $Rsharp kG$-modules).




                        One might reasonably expect that




                        all those statements that depend only on the structure of abelian category on the category of graded modules




                        mentioned in Fred Rohrer's answer fall into the classes mentioned above, however i can't tell for sure on the exact overlap between these classes.






                        share|cite|improve this answer











                        $endgroup$


















                          1












                          $begingroup$

                          Since the OP is asking for some "class of statements" which can be readily transferred from the ungraded to the graded case, let me outline a couple of thoughts, which are not really tied to neither the commutative case, nor the $mathbb{Z}$-gradation itself:




                          • On the one hand, if $R$ is a $k$-algebra and $G$ is a group, then a $G$-gradation on $R$ can be equivalently viewed either as a $(kG)^*$-action or as a $kG$-coaction (where $kG$ is the group hopf algebra and $(kG)^*$ its dual hopf algebra). So, the graded algebra $R$ is alternatively viewed as an algebra in the category of $(kG)^*$-modules or an algebra in the category of $kG$-comodules. In this sense, the statements and the properties which are "naturally" transferred between the graded and the ungraded case are those which are in some sense "compatible" or "invariant" with respect to the $kG$-coaction (or the $(kG)^*$-action). I do not know if it sounds like an abuse of terminology to speak about


                            properties in the category of $(kG)^*$-modules or properties in the category of $kG$-comodules





                          • on the other hand, the Category $R_{gr}$-$mod$ of graded modules over the $G$-graded ring $R$, is equivalent (actually i think it is isomorphic in most cases) to a closed subcategory of modules (in the ungraded sense) over the smash product algebra $Rsharp kG$. (Here the smash product is understood in the sense of the one associated to a Hopf module algebra).

                            In this sense, we could say that the statements and properties which one might expect to "naturally transfer" over between graded and ungraded, are




                            the ones which are preserved under the smash product between $R$ and the group hopf algebra $kG$.




                            (In some sense, this implies that it might be meaningful to consider properties which "transfer" between the $R$-modules and the $Rsharp kG$-modules).




                          One might reasonably expect that




                          all those statements that depend only on the structure of abelian category on the category of graded modules




                          mentioned in Fred Rohrer's answer fall into the classes mentioned above, however i can't tell for sure on the exact overlap between these classes.






                          share|cite|improve this answer











                          $endgroup$
















                            1












                            1








                            1





                            $begingroup$

                            Since the OP is asking for some "class of statements" which can be readily transferred from the ungraded to the graded case, let me outline a couple of thoughts, which are not really tied to neither the commutative case, nor the $mathbb{Z}$-gradation itself:




                            • On the one hand, if $R$ is a $k$-algebra and $G$ is a group, then a $G$-gradation on $R$ can be equivalently viewed either as a $(kG)^*$-action or as a $kG$-coaction (where $kG$ is the group hopf algebra and $(kG)^*$ its dual hopf algebra). So, the graded algebra $R$ is alternatively viewed as an algebra in the category of $(kG)^*$-modules or an algebra in the category of $kG$-comodules. In this sense, the statements and the properties which are "naturally" transferred between the graded and the ungraded case are those which are in some sense "compatible" or "invariant" with respect to the $kG$-coaction (or the $(kG)^*$-action). I do not know if it sounds like an abuse of terminology to speak about


                              properties in the category of $(kG)^*$-modules or properties in the category of $kG$-comodules





                            • on the other hand, the Category $R_{gr}$-$mod$ of graded modules over the $G$-graded ring $R$, is equivalent (actually i think it is isomorphic in most cases) to a closed subcategory of modules (in the ungraded sense) over the smash product algebra $Rsharp kG$. (Here the smash product is understood in the sense of the one associated to a Hopf module algebra).

                              In this sense, we could say that the statements and properties which one might expect to "naturally transfer" over between graded and ungraded, are




                              the ones which are preserved under the smash product between $R$ and the group hopf algebra $kG$.




                              (In some sense, this implies that it might be meaningful to consider properties which "transfer" between the $R$-modules and the $Rsharp kG$-modules).




                            One might reasonably expect that




                            all those statements that depend only on the structure of abelian category on the category of graded modules




                            mentioned in Fred Rohrer's answer fall into the classes mentioned above, however i can't tell for sure on the exact overlap between these classes.






                            share|cite|improve this answer











                            $endgroup$



                            Since the OP is asking for some "class of statements" which can be readily transferred from the ungraded to the graded case, let me outline a couple of thoughts, which are not really tied to neither the commutative case, nor the $mathbb{Z}$-gradation itself:




                            • On the one hand, if $R$ is a $k$-algebra and $G$ is a group, then a $G$-gradation on $R$ can be equivalently viewed either as a $(kG)^*$-action or as a $kG$-coaction (where $kG$ is the group hopf algebra and $(kG)^*$ its dual hopf algebra). So, the graded algebra $R$ is alternatively viewed as an algebra in the category of $(kG)^*$-modules or an algebra in the category of $kG$-comodules. In this sense, the statements and the properties which are "naturally" transferred between the graded and the ungraded case are those which are in some sense "compatible" or "invariant" with respect to the $kG$-coaction (or the $(kG)^*$-action). I do not know if it sounds like an abuse of terminology to speak about


                              properties in the category of $(kG)^*$-modules or properties in the category of $kG$-comodules





                            • on the other hand, the Category $R_{gr}$-$mod$ of graded modules over the $G$-graded ring $R$, is equivalent (actually i think it is isomorphic in most cases) to a closed subcategory of modules (in the ungraded sense) over the smash product algebra $Rsharp kG$. (Here the smash product is understood in the sense of the one associated to a Hopf module algebra).

                              In this sense, we could say that the statements and properties which one might expect to "naturally transfer" over between graded and ungraded, are




                              the ones which are preserved under the smash product between $R$ and the group hopf algebra $kG$.




                              (In some sense, this implies that it might be meaningful to consider properties which "transfer" between the $R$-modules and the $Rsharp kG$-modules).




                            One might reasonably expect that




                            all those statements that depend only on the structure of abelian category on the category of graded modules




                            mentioned in Fred Rohrer's answer fall into the classes mentioned above, however i can't tell for sure on the exact overlap between these classes.







                            share|cite|improve this answer














                            share|cite|improve this answer



                            share|cite|improve this answer








                            edited Jan 8 at 21:47

























                            answered Jan 8 at 2:44









                            Konstantinos KanakoglouKonstantinos Kanakoglou

                            3,18521233




                            3,18521233






























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