F-stable secondary representations and deformation of F-injectivity
aa r X i v : . [ m a t h . A C ] S e p F-STABLE SECONDARY REPRESENTATIONS AND DEFORMATION OFF-INJECTIVITY
ALESSANDRO DE STEFANI AND LINQUAN MA
Abstract.
We prove that deformation of F-injectivity holds for local rings ( R, m ) that ad-mit secondary representations of H i m ( R ) which are stable under the natural Frobenius action.As a consequence, F-injectivity deforms when ( R, m ) is sequentially Cohen–Macaulay (ormore generally when all the local cohomology modules H i m ( R ) have no embedded attachedprimes). We obtain some additional cases if R/ m is perfect or if R is N -graded. Introduction
Throughout this article, all rings are commutative, Noetherian, and with multiplicativeidentity. For rings containing a field of characteristic p > , the seminal work of Hochsterand Huneke on tight closure, and subsequent works of many others, has led to a systematicstudy of the so-called F-singularities. Roughly speaking, these are singularities that can bedefined using the Frobenius endomorphism F : R → R , which is the map that raises everyelement of R to its p -th power. One of the most studied F-singularities is F-injectivity, whichis defined in terms of injectivity of the natural Frobenius actions on the local cohomologymodules H i m ( R ) . It was first introduced and studied by Fedder in [Fed83].We say that a property P of local rings deforms if, whenever ( R, m ) is a local ring and x ∈ m is a nonzerodivisor such that R/ ( x ) satisfies P , then R satisfies P . While this deformationproblem for other classical F-singularities has been settled [Fed83, HH94, Sin99b, Sin99a],whether F-injectivity deforms or not in general is still an open question. Fedder proved thatF-injectivity deforms when R is Cohen–Macaulay [Fed83, Theorem 3.4], and Horiuchi, Miller,Shimomoto proved that F-injectivity deforms either if R/ ( x ) is F-split [HMS14, Theorem4.13], or if H i m ( R/ ( x )) has finite length for all i = dim( R ) and R/ m is perfect [HMS14,Theorem 4.7]. More recently, the second author and Pham [MQ18] extended some of theseresults by further relaxing the assumptions on R/ ( x ) .In this paper, we consider secondary representations of the local cohomology modules H i m ( R ) (see subsection 2.2 for definitions and basic properties of secondary representationsof Artinian modules). It seems natural to ask how the Frobenius action on H i m ( R ) interactswith a given secondary representation. Our first main result is that F-injectivity deformswhen each local cohomology module H i m ( R ) admits a secondary representation which isstable under the natural Frobenius action (see Definition 3.1 for details). Theorem A (Theorem 3.4) . Let ( R, m ) be a d -dimensional local ring of characteristic p > and let x ∈ m be a nonzerodivisor on R . Suppose for each i = d , H i m ( R ) has an F-stablesecondary representation. If R/ ( x ) is F-injective, then R is F-injective. The second author was supported by NSF Grant DMS e prove that secondary components that correspond to minimal attached primes of H i m ( R ) are always F-stable, see Lemma 3.2. As a consequence, F-injectivity deforms whenthe attached primes of H i m ( R ) are all minimal, see Corollary 3.5 for a slightly strongerstatement. In particular, we obtain the following: Corollary B (Corollary 3.6) . Let ( R, m ) be a d -dimensional sequentially Cohen–Macaulaylocal ring of characteristic p > and let x ∈ m be a nonzerodivisor on R . If R/ ( x ) isF-injective, then R is F-injective. We can further relax our assumptions if either the residue field of R is perfect, or if R is N -graded over a field, by only putting conditions on those secondary components of H i m ( R ) whose attached primes are not equal to m . We refer to Definition 3.7 for the precise meaningof F ◦ -stable secondary representations. Theorem C (Theorem 3.8 and Theorem 3.10) . Let ( R, m , k ) be a d -dimensional local ringof characteristic p > that is either local with perfect residue field or N -graded over a field k , and let x ∈ m be a nonzerodivisor on R (homogeneous in the graded case). Suppose foreach i = d , H i m ( R ) has an F ◦ -stable secondary representation. If R/ ( x ) is F-injective, then R is F-injective. Acknowledgments.
We thank Pham Hung Quy and Ilya Smirnov for several useful dis-cussions on the topics of this article.2.
Preliminaries
Frobenius actions on local cohomology and F-injectivity.
Let R be a ring ofcharacteristic p > . A Frobenius action on an R -module W is an additive map F : W → W such that F ( rη ) = r p F ( η ) for all r ∈ R and η ∈ W .Let I = ( f , . . . , f n ) be an ideal of R , then we have the Čech complex: C • ( f , . . . , f n ; R ) := 0 → R → ⊕ i R f i → · · · → R f f ··· f n → . Since the Frobenius endomorphism on R induces the Frobenius endomorphism on all local-izations of R , it induces a natural Frobenius action on C • ( f , . . . , f n ; R ) , and hence it inducesa natural Frobenius action on each H iI ( R ) . In particular, there is a natural Frobenius action F : H i m ( R ) → H i m ( R ) on each local cohomology module of R supported at a maximal ideal m . A local ring ( R, m ) is called F-injective if F : H i m ( R ) → H i m ( R ) is injective for all i .2.2. Secondary representations.
We recall some well-known facts on secondary represen-tations that we will use throughout this article. For unexplained facts, or further details, werefer the reader to [BS13, Section 7.2].
Definition 2.1.
Let R be a ring. An R -module W is called secondary if W = 0 and foreach x ∈ R the multiplication by x map on W is either surjective or nilpotent.One can easily check that, if W is a secondary R -module, then p = p ann R ( W ) is a primeideal, and ann R ( W ) is p -primary. Definition 2.2.
Let R be a ring and W be an R -module. A secondary representation of W is an expression of W as a sum of secondary submodules, W = P ti =1 W i , where each W i iscalled a secondary component of this representation. secondary representation of W is called irredundant if the prime ideals p i = p ann R ( W i ) are all distinct and none of the summands W i can be removed from the sum. The set { p , . . . , p t } is independent of the irredundant secondary representation and is called the setof attached primes of W , denoted by Att R ( W ) .Clearly a secondary module has a unique attached prime. Moreover, over a local ring ( R, m ) , if a nonzero module W has finite length, then W is secondary with Att R ( W ) = { m } .A key fact is that every Artinian R -module admits an irredundant secondary representa-tion. In particular, all local cohomology modules H i m ( R ) have an irredundant secondaryrepresentation. Remark . When ( R, m ) is a complete local ring, Matlis duality induces a correspondencebetween (irredundant) secondary representations of Artinian modules and (irredundant) pri-mary decompositions of Noetherian modules. In particular, if ( R, m ) is complete, and S is an n -dimensional regular local ring mapping onto R , then Att R ( H i m ( R )) = Ass R (Ext n − iS ( R, S )) ,as the Matlis dual of H i m ( R ) is isomorphic to Ext n − iS ( R, S ) .We conclude this section by recalling the definition of surjective element and strictly filterregular element. Definition 2.4.
Let ( R, m ) be a local ring of dimension d . An element x ∈ m is called asurjective element if x / ∈ p for all p ∈ S di =0 Att R ( H i m ( R )) , and x is called a strictly filterregular element if x / ∈ p for all p ∈ (cid:16)S di =0 Att R ( H i m ( R )) (cid:17) r { m } . Remark . The definition of surjective element we give is not the original one introducedin [HMS14]. However, note that
Ass R ( R ) ⊆ ∪ dim( R ) i =0 Att R ( H i m ( R )) by [BS13, 11.3.9] and thussurjective elements are always nonzerodivisors. Moreover, it follows from the definition that x is a surjective element if and only if H i m ( R ) · x −→ H i m ( R ) is surjective for each i . Therefore ourdefinition is equivalent to the original definition of surjective element by [MQ18, Proposition3.3]. Similarly, it is easy to see that x is a strictly filter regular element if and only if coker( H i m ( R ) x −→ H i m ( R )) has finite length for each i .Surjective elements are important in the study of the deformation problem for F-injectivity.For instance, it was first proved in [HMS14, Theorem 3.7] that if R/ ( x ) is F-injective and x is a surjective element, then R is F-injective (see also [MQ18, Corollary 3.8] or the proofof Theorem 3.4 in the next section). In fact, we do not know any example that R/ ( x ) isF-injective but x is not a surjective element, see Question 4.3.3. F-stable secondary representation
We introduce the key concept of this article.
Definition 3.1.
Let R be a ring of characteristic p > , and let W be an R -module witha Frobenius action F . We say that W admits an F-stable secondary representation if thereexists a secondary representation W = P ti =1 W i such that each W i is F-stable, i.e., F ( W i ) ⊆ W i for all i .Observe that, even though we are not explicitly asking that the F-stable secondary repre-sentation is irredundant, this can always be achieved, whenever such a representation exists.It seems natural to ask when a secondary component of an Artianina module is F-stable, we how this is always the case for secondary components whose attached primes are minimalin the set of all attached primes. Lemma 3.2.
Let R be a ring of characteristic p > , and let W be an Artinian R -modulewith a Frobenius action F . Let W = P ti =1 W i be an irredundant secondary representation,with p i = p ann R ( W i ) . If p i ∈ MinAtt R ( W ) , then W i is F-stable.Proof. Since p i ∈ Min Att( W ) , we can pick y ∈ ∩ j = i p j but y / ∈ p i . Then yW i = W i and y N W j = 0 for all j = i and N ≫ . Therefore we have y N W = W i for all N ≫ , and thus F ( W i ) = F ( y N W i ) ⊆ F ( y N W ) = y pN F ( W ) ⊆ y pN W = W i . (cid:3) For secondary components whose attached primes are not necessarily minimal, the corre-sponding secondary components may not be F-stable. However, we do not know whetherthis can happen when W is a local cohomology module with its natural Frobenius action,see Question 4.1. Example 3.3.
Let R = F p J x, y K and let W = F p ⊕ H m ( R ) . Consider the Frobenius action F on W that sends (1 , to (1 , x − p y − ) and is the natural one on H m ( R ) . Then F is injective on W , but we claim that H m ( R ) is the only proper nontrivial F-stable submodule of W . Indeed,let = W ′ be an F-stable submodule of W , it is enough to show that ⊕ H m ( R ) ⊆ W ′ .Choose a = ( b, c ) = 0 inside W ′ . If c = 0 , then b = 0 . By replacing a with F ( a ) , we canassume that c = 0 . Note that yF ( a ) = yF ( b,
0) + (0 , yF ( c )) = (0 , yF ( c )) = 0 since theaction yF : H m ( R ) → H m ( R ) is injective. Moreover H m ( R ) is simple as an R -module with aFrobenius action, so ⊕ H m ( R ) ⊆ W ′ . Since W is not secondary, this implies that there isno secondary representation of W which is stable with respect to the given Frobenius action(any secondary component with attached prime m is not F-stable).We let V ( x ) denote the set of primes of R which contain x . Our first main result is thefollowing. Theorem 3.4.
Let ( R, m ) be a d -dimensional local ring of characteristic p > and let x ∈ m be a nonzerodivisor on R . Suppose for each i = d , H i m ( R ) admits a secondaryrepresentation in which the secondary components whose attached primes belong to V ( x ) areF-stable (e.g., H i m ( R ) has an F-stable secondary representation). If R/ ( x ) is F-injective,then x is a surjective element and R is F-injective.Proof. We prove by induction on i > − that multiplication by x is surjective on H i m ( R ) andthat x p e − F e is injective on H i m ( R ) for all e > . This will conclude the proof, since the firstassertion implies x is a surjective element and the second assertion implies F is injective on H i m ( R ) for all i . The base case i = − is trivial. Suppose both assertions hold for i − ; weshow them for i . Consider the following commutative diagram: / / H i − m ( R/ ( x )) F e (cid:15) (cid:15) / / H i m ( R ) · x / / x pe − F e (cid:15) (cid:15) H i m ( R ) / / F e (cid:15) (cid:15) H i m ( R/ ( x )) F e (cid:15) (cid:15) / / . . . / / H i − m ( R/ ( x )) / / H i m ( R ) · x / / H i m ( R ) / / H i m ( R/ ( x )) / / . . . where injectivity on the left of the rows follows from our inductive hypotheses. Let u ∈ soc( H i m ( R )) ∩ ker( x p e − F e ) . Then xu = 0 , and thus u is the image of an element v ∈ H i − m ( R/ ( x )) . Chasing the diagram shows that F e ( v ) = 0 . But since R/ ( x ) is F-injective, F e s injective on H i − m ( R/ ( x )) for all e > , so v = 0 and thus u = 0 . This shows that x p e − F e is injective on H i m ( R ) for all e > .It remains to show that multiplication by x is surjective on H i m ( R ) . We distinguish twocases. If i = d then this is clear because H d m ( R/ ( x )) = 0 . Therefore we assume i < d .Let H i m ( R ) = P W j be the secondary representation that satisfies the conditions of thetheorem. If there exists W j = 0 whose attached prime p j ∈ V ( x ) , then it follows from theassumptions that W j is F-stable. Thus x p e − F e ( W j ) ⊆ x p e − W j = 0 for all e ≫ (since x ∈ p j = p ann R ( W j ) ). However, we have proved that x p e − F e is injective on H i m ( R ) forall e > , this implies W j = 0 and we arrive at a contradiction. Therefore x / ∈ p for all p ∈ Att R ( H i m ( R )) , i.e., multiplication by x is surjective on H i m ( R ) . (cid:3) Corollary 3.5.
Let ( R, m ) be a d -dimensional local ring of characteristic p > and let x ∈ m be a nonzerodivisor on R . Suppose that Att R ( H i m ( R )) ∩ V ( x ) ⊆ MinAtt R ( H i m ( R )) forall i = d (e.g., when each H i m ( R ) has no embedded attached primes). If R/ ( x ) is F-injective,then x is a surjective element and R is F-injective.Proof. By Lemma 3.2, every irredundant secondary representation of H i m ( R ) satisfies theassumptions of Theorem 3.4 so the conclusion follows. (cid:3) We next exhibit an explicit new class of rings for which deformation of F-injectivity holds.Recall that a finitely generated R -module M is called sequentially Cohen–Macaulay if thereexists a finite filtration M ⊆ M ⊆ M ⊆ · · · ⊆ M n = M such that each M i +1 /M i is Cohen–Macaulay and dim( M i /M i − ) < dim( M i +1 /M i ) . A local ring ( R, m ) is calledsequentially Cohen–Macaulay if R is sequentially Cohen–Macaulay as an R -module. Corollary 3.6.
Let ( R, m ) be a d -dimensional sequentially Cohen–Macaulay local ring ofcharacteristic p > and let x ∈ m be a nonzerodivisor on R . If R/ ( x ) is F-injective, then x is a surjective element and R is F-injective.Proof. First we observe that R is sequentially Cohen–Macaulay implies b R is sequentiallyCohen–Macaulay and whether R is F-injective (and whether x is a surjective element) isunaffected by passing to the completion. Therefore we may assume R is complete and thus R is a homomorphic image of a regular local ring S . By [HS02, Theorem 1.4], R is sequentiallyCohen–Macaulay is equivalent to saying that, for each i d , Ext dim( S ) − iS ( R, S ) is eitherzero or Cohen–Macaulay of dimension i . In particular, Ext dim( S ) − iS ( R, S ) has no embeddedassociated primes and hence by Remark 2.3, H i m ( R ) has no embedded attached primes foreach i d , that is, Att R ( H i m ( R )) = Min Att R ( H i m ( R )) . The conclusion now followsfrom Corollary 3.5. (cid:3) Results on local rings with perfect residue field.
If we assume the residue field of ( R, m ) is perfect, then we can prove some slight stronger results. The arguments are basedon appropriate modifications of the proof of Theorem 3.4, together with some ideas employedin [MQ18, Section 5]. First, we make a modification of the definition of F-stable secondaryrepresentation. Definition 3.7.
Let R be a ring of characteristic p > and m be a maximal ideal of R .Let W be an R -module with a Frobenius action F . We say that W admits an F ◦ -stablesecondary representation if there exists a secondary representation W = P ti =1 W i such that W i is F-stable for all i such that Att R ( W i ) = { m } . heorem 3.8. Let ( R, m ) be a d -dimensional local ring of characteristic p > with perfectresidue field, and let x ∈ m be a nonzerodivisor on R . Suppose for each i = d , H i m ( R ) = 0 admits a secondary representation in which the secondary components whose attached primesbelong to V ( x ) r { m } are F-stable (e.g., H i m ( R ) has an F ◦ -stable secondary representation).If R/ ( x ) is F-injective, then x is a strictly filter regular element and R is F-injective.Proof. For every i , we let L i = coker( H i m ( R ) x −→ H i m ( R )) . We prove by induction on i > − that L i has finite length and that the Frobenius action x p e − F e on H i m ( R ) is injective forall e > . This will conclude the proof, since the first assertion implies x is a strictly filterregular element and the second assertion implies F is injective on H i m ( R ) for all i . The initialcase i = − is trivial. Suppose both assertions hold for i − ; we show them for i . Considerthe following commutative diagram: / / H i − m ( R/ ( x )) /L i − F e (cid:15) (cid:15) / / H i m ( R ) · x / / x pe − F e (cid:15) (cid:15) H i m ( R ) / / F e (cid:15) (cid:15) H i m ( R/ ( x )) F e (cid:15) (cid:15) / / . . . / / H i − m ( R/ ( x )) /L i − / / H i m ( R ) · x / / H i m ( R ) / / H i m ( R/ ( x )) / / . . . Since L i − has finite length, F e is injective on H i − m ( R/ ( x )) by assumption, and R/ m isperfect, we have that F e induces a bijection on L i − ⊆ H i − m ( R/ ( x )) . Thus, F e induces aninjection on H i − m ( R/ ( x )) /L i − for all e > . Therefore, chasing the diagram above as in theproof of Theorem 3.4 we know that x p e − F e is injective on H i m ( R ) for all e > .It remains to show that L i has finite length. If i = d then this is clear because L d ⊆ H d m ( R/ ( x )) = 0 . Therefore we assume i < d . Let H i m ( R ) = P W j be the secondaryrepresentation that satisfies the conditions of the theorem. If there exists W j = 0 whoseattached prime p j ∈ V ( x ) r { m } , then it follows from the assumptions that W is F-stable.Thus x p e − F e ( W j ) ⊆ x p e − W j = 0 for all e ≫ (since x ∈ p j = p ann R ( W j ) ). However,we have proved that x p e − F e is injective on H i m ( R ) for all e > , this implies W j = 0 and we arrive at a contradiction. Therefore x / ∈ p for all p ∈ Att R ( H i m ( R )) r { m } , i.e., L i = coker( H i m ( R ) x −→ H i m ( R )) has finite length. (cid:3) Corollary 3.9.
Let ( R, m ) be a d -dimensional local ring of characteristic p > with perfectresidue field, and let x ∈ m be a nonzerodivisor on R . Suppose that Att R ( H i m ( R )) ∩ V ( x ) ⊆ MinAtt R ( H i m ( R )) ∪ { m } for all i = d . If R/ ( x ) is F-injective, then x is a strictly filter regularelement and R is F-injective. In particular, F-injectivity deforms if dim( R/ ann R ( H i m ( R ))) for all i = d and R/ m is perfect.Proof. By Lemma 3.2, every irredundant secondary representation of H i m ( R ) satisfies theassumptions of Theorem 3.8 so the first conclusion follows. To see the second conclusion,it is enough to observe that when dim( R/ ann R ( H i m ( R ))) , we have Att R ( H i m ( R )) ⊆ MinAtt R ( H i m ( R )) ∪ { m } . (cid:3) Results on N -graded rings. For the rest of this section, we assume that ( R, m , k ) is an N -graded algebra over a field k of characteristic p > ( k is not necessarily perfect).Given a graded module W = L j W j and a ∈ Z , we denote by W ( a ) the shift of W by a , that is, the graded R -module such that W ( a ) j = W a + j . In this context, when talkingabout a Frobenius action F on a graded module W , we insist that deg( F ( η )) = p · deg( η ) or all homogeneous η ∈ W . This is the case for the natural Frobenius action F on the localcohomology modules H i m ( R ) .The goal of this subsection is to extend Theorem 3.8 in this N -graded setting, by removingthe assumption that the residue field k is perfect and by strengthening the conclusion to that x is actually a surjective element. Theorem 3.10.
Let ( R, m , k ) be a d -dimensional N -graded k -algebra of characteristic p > and let x ∈ m be a homogeneous nonzerodivisor on R . Suppose for each i = d , H i m ( R ) admitsa secondary representation in which the secondary components whose attached primes belongto V ( x ) r { m } are F-stable (e.g., H i m ( R ) has an F ◦ -stable secondary representation). If R/ ( x ) is F-injective, then x is a surjective element and R is F-injective.Proof. Let deg( x ) = t > . We have a graded long exact sequence of local cohomology,induced by the short exact sequence → R ( − t ) x −→ R → R/ ( x ) → . Moreover, this exactsequence fits in the commutative diagram: . . . / / H i − m ( R/ ( x )) F e (cid:15) (cid:15) / / H i m ( R )( − t ) · x / / x pe − F e (cid:15) (cid:15) H i m ( R ) / / F e (cid:15) (cid:15) H i m ( R/ ( x )) F e (cid:15) (cid:15) / / . . .. . . / / H i − m ( R/ ( x )) / / H i m ( R )( − t ) · x / / H i m ( R ) / / H i m ( R/ ( x )) / / . . . Observe that all the Frobenius actions are compatible with the grading. We show by induc-tion on i > − that the map H i m ( R )( − t ) x −→ H i m ( R ) is surjective and that x p e − F e is injectiveon H i m ( R )( − t ) . This will conclude the proof, since the first assertion implies x is a surjectiveelement and the second assertion implies F is injective on H i m ( R ) for all i . The base case i = − is trivial. Suppose both assertions hold for i − ; we show them for i . By the sameargument as in the proof of Theorem 3.4, we have that x p e − F e is injective on H i m ( R )( − t ) for all e > .It remains to show that multiplication by x map H i m ( R )( − t ) x −→ H i m ( R ) is surjective. If i = d then this is clear because H d m ( R/ ( x )) = 0 . Therefore we assume i < d . Now by the sameargument as in the proof of Theorem 3.8, we know that L i = coker( H i m ( R )( − t ) x −→ H i m ( R )) has finite length (note that we can ignore the graded structure here). Finally, consider thefollowing commutative diagram: / / L iF e (cid:15) (cid:15) / / H i m ( R/ ( x )) F e (cid:15) (cid:15) / / H i +1 m ( R )( − t ) x pe − F e (cid:15) (cid:15) x / / . . . / / L i / / H i m ( R/ ( x )) / / H i +1 m ( R )( − t ) x / / . . . Since F e is injective on H i m ( R/ ( x )) by assumption, it is also injective on L i . But since thefinite length module L i is graded and the Frobenius action is compatible with the grading(as the action is induced from H i m ( R/ ( x )) ), this forces L i to be concentrated in degree zero.If L i = 0 , then [ L i ] ∼ = [ H i m ( R ) /xH i m ( R )( − t )] = 0 , in particular [ H i m ( R )] = 0 . However,this implies the existence of a nonzero element u ∈ [ H i m ( R )( − t )] t . Since we have provedthat x p e − F e is injective on H i m ( R )( − t ) , this gives a nonzero element x p e − F e ( u ) in degree p e t > for all e > , which is a contradiction because [ H i m ( R )( − t )] ≫ = 0 (here we are usingthat the Frobenius action x p e − F e is compatible with the grading on H i m ( R )( − t ) , that is, eg( x p e − F ( η )) = p e deg( η ) for all η ∈ H i m ( R )( − t ) ). Therefore L i = 0 , i.e., the multiplicationby x map H i m ( R )( − t ) x −→ H i m ( R ) is surjective. (cid:3) Corollary 3.11.
Let ( R, m , k ) be a d -dimensional N -graded k -algebra of characteristic p > and let x ∈ m be a homogeneous nonzerodivisor on R . Suppose that Att R ( H i m ( R )) ∩ V ( x ) ⊆ MinAtt R ( H i m ( R )) ∪ { m } for all i = d . If R/ ( x ) is F-injective, then x is a surjective elementand R is F-injective.Proof. By Lemma 3.2, every irredundant secondary representation of H i m ( R ) satisfies theassumptions of Theorem 3.10 so the conclusion follows. (cid:3) Ending questions and Remarks
We end by collecting some questions that arise from the results in this article. Motivatedby Definition 3.1 and Theorem 3.4, it is natural to ask the following.
Question 4.1.
Let ( R, m ) be a local ring of characteristic p > . If H i m ( R ) = 0 , does itadmit an F-stable secondary representation?By Theorem 3.4, a positive answer to Question 4.1 implies that F-injectivity deforms. Question 4.2.
Let ( R, m ) be a local ring of characteristic p > . If H i m ( R ) = 0 , does itadmit a secondary representation such that the secondary component with attached prime m , if not zero, is F-stable?This is weaker than Question 4.1, but an affirmative answer also implies that F-injectivitydeforms. Suppose R/ ( x ) is F-injective, we will show x is a surjective element and thus R isF-injective by [HMS14, Theorem 3.7] (or use the same argument as in Theorem 3.4). In fact,if x ∈ p for some p ∈ Att R ( H i m ( R )) , then x ∈ p R p ∈ Att R p ( H j p R p ( R p )) for some j and R p /xR p is still F-injective. Now an affirmative answer to Question 4.2 applied to ( R p , p R p ) impliesthat there exists a nonzero secondary component of H j p R p ( R p ) with attached prime p R p thatis F-stable, and we can argue as in the proof of Theorem 3.4 to arrive at a contradiction. Question 4.3.
Let ( R, m ) be a local ring of characteristic p > , and let x ∈ m be anonzerodivisor on R . If R/ ( x ) is F-injective, is it true that m / ∈ Att( H i m ( R )) for all i ?Similar to the discussion above, we point out that an affirmative answer to Question 4.3also implies that x is a surjective element (and hence implies that F-injectivity deforms):if not, then x ∈ p for some p ∈ Att R ( H i m ( R )) , but then R p /xR p is still F-injective and p R p ∈ Att R p ( H j p R p ( R p )) for some j , which contradicts Question 4.3 for ( R p , p R p ) . References [BS13] M. P. Brodmann and R. Y. Sharp.
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Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova,Italy
E-mail address : [email protected] Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
E-mail address : [email protected]@purdue.edu