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Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
2019. Volume 53
MSC2010: 13D45, 14B15 © L. Lu
STRUCTURAL THEOREM FOR GR-INJECTIVE MODULES OVER GR-NOETHERIAN G-GRADED COMMUTATIVE RINGS AND LOCAL COHOMOLOGY FUNCTORS
It is well known that the decomposition of injective modules over noetherian rings is one of the most aesthetic and important results in commutative algebra. Our aim is to prove similar results for graded noetherian rings. In this paper, we will study the structure theorem for gr-injective modules over gr-noetherian G-graded commutative rings, give a definition of the gr-Bass numbers, and study their properties. We will show that every gr-injective module has an indecomposable decomposition. Let R be a gr-noetherian graded ring and M be a gr-finitely generated R-module, we will give a formula for expressing the Bass numbers using the functor Ext. We will define the section functor rV with support in a specialization-closed subset V of Specgr (R) and the abstract local cohomology functor. Finally, we will show that a left exact radical functor F is of the form rV for a specialization-closed subset V.
Keywords: graded commutative rings, gr-Bass numbers, local cohomology functors, derived categories, radical functors.
DOI: 10.20537/2226-3594-2019-53-11 Introduction
Recently, considerable interest has been noted in rings and other algebraic structures equipped with grading. This is explained by the fact that many important classes of rings, for example, polynomial rings, matrix rings, group rings, admit a natural grading.
A natural and important question in the theory of graded rings is to find graded analogues of some classical theorems. For example, C. Park in his work [17] proved Krull's principal ideal theorem, the Krull-Akizuki theorem and the Mori-Nagata theorem in the graded case. S. Paul Smith in his work [21] proved some category equivalences involving the quotient category QGr(kQ) := Gr(kQ)/Fdim(kQ) of graded kQ-modules modulo those that are the sum of their finite-dimensional submodules. J. Bell and J. Zhang in their work [2] proved, for two connected graded algebras A and B that are finitely generated in degree one, that if A is isomorphic to B as ungraded algebras, then they are also isomorphic to each other as graded algebras. J. Chen and Y. Kim in their work [3] show that if a graded submodule of a noetherian module cannot be written as a proper intersection of graded submodules, then it cannot be written as a proper intersection of submodules at all. In [4] I. DellAmbrogio and G. Stevenson establish an inclusion-preserving bijection between, on the one hand, the twist-closed localizing subcategories of the derived category, and, on the other hand, subsets of the homogeneous spectrum of prime ideals of the ring.
It is well known that the decomposition of injective modules over noetherian rings is one of the most aesthetic and important results in commutative algebra. Our aim is to prove similar results for graded noetherian rings. This is important for us to understand the structure of the noetherian modules on the graded rings.
We now give a brief outline of the paper.
In Section 1, we set notation and review some basics of graded commutative rings, torsion theories and t-structures.
In Section 2, we give a definition of gr-Bass numbers and obtained the following important theorem:
Theorem 2.1 Let R be a gr-noetherian G-graded ring and M € gr(R). Then
№ (P,g,M) = dimK(p) ExfR(p) (K(P) ,M(P) (g)).
In Section 3, we show that a left exact radical functor F is of the form rV for a specialization-closed subset V. This is a generalization of Yoshino and Yoshizawa's theorem in the case of graded rings (see also [23, Theorem 2.6]).
Theorem 3.1 The following conditions are equivalent for a left exact preradical functor F on Gr(R).
(1) F is a radical functor.
(2) F preserves injectivity.
(3) F is a section functor with support in a specialization-closed subset of Spec9T (R).
(4) RF is an abstract local cohomology functor.
Symbols
G totally ordered abelian group
R G-graded ring
Mod(R) category of R-modules
Gr(R) category of graded R-modules
gr(R) category of gr-noetherian graded R-modules
A abelian category
C(A) category of A-complexes
K(A) homotopy category of chain complexes in A
D (A) derived category of A
For more information on graded rings, see [8,15-17,19,20].
§ 1. Preliminaries
§ 1.1. Graded commutative rings and graded modules
In this article we always assume that G is a totally ordered abelian group. For the definition of G-graded commutative rings and modules see, for example, [15, 16]. Let R be a G-graded ring and M = ®gSG Mg be a graded R-module, we let h(M) := U g^cMg be the set of homogeneous elements. We denote by Mod(R) the category of R-modules, and by Gr(R) the category of graded R-modules whose objects are graded R-modules, and the morphisms are homomorphisms preserving grading:
HomGr(R) (M, N) = {f € Homn(M,N) : f (Mg) C Ng for all g € G}.
Obviously, Gr(R) is a Grothendieck category [16, Proposition 2.2].
Gr-noetherian module is a graded R-module M in which every sequence of graded submodules Ni c N2 C ■ ■ ■ C Nn c ... is stabilized. If R as a R-module is gr-noetherian, then we call R gr-noetherian ring. If R is a gr-noetherian ring, then the category of gr-noetherian R-modules gr(R) is an abelian category [16, Proposition 2.2].
A gr-multiplicative set is a subset S of a graded ring R such that the following two conditions hold:
(1) 0 € S, 1 € S, S C h(R).
(2) For all x and y in S, the product xy is in S.
Let M = (&geG Mg be a graded R-module. The grading of the module M induces a grading on the modulus of quotients S-1M over the gr-multiplicative system S:
(S~1M)g = {™e 5_1M I m <E Mh, seSnRk, g = M;"1} .
A graded ring R is a gr-local ring if R has a unique graded maximal ideal.
Let R be a gr-noetherian G-graded commutative ring, and M a graded module over R. We say that p € Specgr(R) is a gr-associated prime of M if p is the annihilator of some x € h(M). The set of gr-associated primes of M is denoted by AssaRi(M) or Assgr (M).
§ 1.2. Gr-injective modules over gr-noetherian rings
Definition 1.1. Let R be a G-graded ring.
(1) A graded R-module M is gr-injective if it is injective as an object of the category Gr(R).
(2) Let N be a graded submodule of M. N is said to be an gr-essential submodule of M if for every gr-submodule H of M the equality H n N = 0 implies that H = 0.
(3) The gr-injective hull Egr (M) of a graded module M is both the smallest gr-injective module containing it and the largest gr-essential extension of it.
(4) A graded module is gr-indecomposable if it is nonzero and cannot be written as a direct sum of two nonzero graded submodules.
Proposition 1.1. Ifp € Spec9T(R), then Ass9r(f) = {p} and Ass9r(E9r(f)) = {p}.
Proof. p is a gr-associated prime of R/p because there is a monomorphism from R/p to itself. If
Q € -1-s.s',r(y ). we must show that Q p. Suppose that Q is the kernel of an epimorphism I! —> Rx. Then s € Q iff sx € p iff s € p (because p is prime).
Let q e Ass9r (E9r (f)), then q = AnnR(x) and h(E9r(f)). By the definition of the gr-
injective hull, j is a gr-essential submodule of E9T(j). There is an element r € h(R), and 0 / rx € j. obviously, r € q. We have Ann(rx) = p D Ann(x) = q. For any p € p, we have prx = 0, so pr € q, but r € q, hence p € q, that is, p = q. □
Theorem 1.1. Let R be a gr-noetherian G-graded commutative ring.
(1) For every family { ja}asA of gr-injective modules, the direct sum 0 AeA J\ is also gr-injective.
(2) Every gr-injective module has an indecomposable decomposition.
(3) IfO / E is a gr-indecomposable gr-injective module, then E = E9T(j(g)), where p € Specgr(R) and g € G.
(4) let p, q € Spec9r(R), g, h € G and let p / q, then E9r(f (g)) / E9r(j(h)). Proof.
(1) See [10, Theorem 5.9].
(2) See [10, Theorem 5.9].
(3) Let p € Ass9'r(E), there exist x € h(E) and g € G such that Rx ^ f(g). Hence E = E9r(Rx).
f<
(4) This is Proposition 1.1. □
§ 1.3. Torsion theories
Our main references for the torsion theories are [5,12,18].
A torsion theory in a Grothendieck category A is a couple (T, F) of strictly full additive subcategories called the torsion class T and the torsion free class F such that the following conditions hold:
(1) Hom(T, F) = 0.
(2) For all M € Obj(A), there exists N c M, N e Obj(T) and M/N € Obj(F).
For every object M there exist the largest subobject t(M) c M which is in T and it is called the torsion part of M.
t : M ^ t(M)
is an additive functor. A torsion theory is hereditary if T is closed under subobjects, or equivalently, t is left exact functor.
A radical functor, or more generally, a preradical functor, has its own long history in the theory of categories and functors. See [7] or [13] for the case of module category. Let F,G : A ^ A be functors. Recall that F is said to be a subfunctor of G, denoted by F c G, if F (M ) is a subobject of G(M ) for all M e A and if F (f ) is a restriction of G(f ) to F (M ) for all f € HomA(M, N ).
Definition 1.2. A functor F : A ^ A is called a preradical functor if F is a subfunctor of 1.
Lemma 1.1 (see [23, Lemma 1.1]). Let F : A ^ A be a preradical functor and assume that F is a left exact functor on A. If N is a subobject of M, then the equality F (N ) = N n F (M ) holds.
Definition 1.3. A preradical functor F is called a radical functor if F (M/F (M )) = 0 for all M e Obj(A).
If F : A ^ A is a left exact radical functor, then there is a hereditary torsion theory (Tf , Ff ) by setting
Tf = {M eA| F (M) = M},
Ff = {M e A | F (M ) =0}. ( . )
§1.4. ¿-structures
The notion of a t-structure arose in the work [1] of Beilinson, Bernstein, Deligne, and Gabber on perverse sheaves.
Let D be a triangulated category. t-structure in D is a pair t = (U, W) of full subcategories, closed under taking direct summands in D, which satisfy the following properties:
(t-S.1) Homv (U, W [-1]) = 0, for all U e U and W e W ;
(t-S.2) U[1] C U, W[-1] C W;
(t-S.3) for each Y e D, there is a triangle A ^ Y ^ B ^ A[1] in D, where A eU and B e W [-1].
A t-structure t = (U, W) in D is called a stable t-structure on D if U and W are triangulated subcategories.
Theorem 1.2 (see [14, Proposition 2.6]). Let D be a triangulated category and U a triangulated subcategory of D. Then the following conditions are equivalent for U.
(1) There is a triangulated subcategory W of D such that (U, W) is a stable t-structure on D.
(2) The natural embedding functor i : U ^ D has a right adjoint p : D ^ U.
If this is the case, setting 5 = i o p : D ^ D,we have the equalities U = Im(5) and W = Ker(S). There is an natural morphism $ : 5 ^ 1, where 1 is the identity functor on D. Every C e D can be embedded in a triangle of the form
5(C) C ^ D ^ 5(C)[1].
§ 2. Gr-Bass numbers
Let R be a gr-noetherian G-graded ring. For each p e Specgr (R), S = h(R) — p is a gr-multiplicative
system and R(p) := S-1R is a gr-local ring. pR(p) is the unique gr-maximal ideal of R(p). We define
_ Rw HV) - pR(p)-
Proposition 2.1. Suppose that f : M ^ N is a monomorphism in Gr(R). Then N is an essential
extension of M if and only if for every p e Specgr (R) the induced morphism
f(p) : HomR(p) (K(p) ,M(p)) ^ HomR(p) (K(p) ,N(p))
is an isomorphism.
Proof. Since localization is exact and HomR(p) (K(p), —) is left exact, f(p) is always a monomorphism. Let S = h(R)— p. Since y is finitely presented, we have S~1Hom.R(j,M) = Hom.Rw(K^, M^) and S~lHornR(f,N) ^ HomRw(n{p),N{p)).
First, let N be an essential extension of M and let 0 / | € S~lHornR{^, N). We have 0 / /(1) e N, so there is an element r € h(R) such that 0 = rf (1) = f (r) € M. It is obvious that r € p, that is, r £ S, so ^ € S~1HomR(j, M) and its image in S~1HomR(j, N) is so /(P) is an isomorphism.
Now let f(p) be an isomorphism for any p € Specgr(R). Let x € h(N). There is a graded prime ideal p € Assgr(Rx) and an element r € h(R) such that y = rx and AnnR(y) = p. We define a homomorphism / € HornR(j,N) such that /(1) = y. There is a homomorphism h € lloninij. M) and an element
s € S such that ^(y) = y(y) = j. There is an element u € S such that h( 1) = usy = usrx € M, hence N is an essential extension of M. □
Let M be a graded R-module. A gr-injective resolution is an exact sequence
E* = 0 ^ M ^ E0 A E1 A E2 A ...,
where E% is a gr-injective module. If Ker(d%) ^ E% is an essential extension, then we say that E* is a
gr-minimal injective resolution.
Proposition 2.2. Let R be a gr-noetherian ring and M be an R-module with a gr-injective resolution E*. Then E* is a gr-minimal injective resolution if and only if for all p € Specgr (R) and all i, the induced morphisms HomR(p) (K(p) , E^p ) ^ HomR(p) (K(p), E^1) are zero.
Proof. Let Z% = ker(d%) = Im(d%-1). E* is gr-minimal if and only if E% is an essential extension of Z%, if and only if the induced morphism HomR(p) (K(p), Z^) ) ^ HomR{p) (K(p), E(p)) is an isomorphism by Proposition 2.1. Since there are exact sequences 0 ^ Z% ^ E% ^ Ei+1, and HomR{p) (K(p), —) is left exact, the latter condition holds if and only if the induced morphisms HomR{p)(K(p),E(p)) ^ HomR{p)(K(p),E(+)1) areO. □
Proposition 2.3. Let R be a gr-noetherian G-graded ring and S be a gr-multiplicative system. If M is a gr-injective R-module, then S-1M is a gr-injective S-1 R-module.
Proof. In the nongraded case this is a classical result (see [9, Proposition 2.8]). In the graded case, it is possible to repeat this proof. □
Proposition 2.4. Let R be a gr-noetherian G-graded ring and M be a graded R-module with a gr-minimal injective resolution E*. Then for any p € Specgr (R), E*p) are gr-minimal injective resolutions
of M(p).
Proof. We know that E(p) are gr-injective R(p)-module by Proposition 2.3. By Proposition 2.2 it is only necessary to prove that HomR(q) (k^, (E(p))(q)) ^ HomR(q) (k^, (E(p+1)(q)) are 0 for all q c p. We know that (E(p))(q) = E(q), hence E*p) are gr-minimal injective resolutions of M(p) by Proposition 2.2. □
Definition 2.1. (Gr-Bass numbers) Let R be a gr-noetherian G-graded ring and let M € gr(R) have a gr-minimal injective resolution E*. By Theorem 1.1, we have
Ei= © ©Wfto)]'*"^
peSpec3r(r) g&G p
^gr(p,g, M) are called gr-Bass numbers.
Proposition 2.5. Let R be a gr-noetherian G-graded ring and M € gr(R). Then, for any i,
^gr (p,g, m ) = ^gr (p(p) ,g, M(p)).
Proof. Take a gr-minimal injective resolution E*. By Proposition 2.4, E*p) is the gr-minimal
injective resolution of M(p). Since Ei = 0p eSpecgr{R) ®geG[E9^ (f)(g)Ylir {v'gM\ we make localization and get the result. □
Theorem 2.1. Let R be a gr-noetherian G-graded ring and M € gr(R). Then
№(P,g,M) = dimK(p)Ext\p) (K(P),M(P)(g)).
Proof. By Proposition 2.5, it suffices to consider the case when (R, m, k) is a gr-noetherian gr-local ring and p = m. Let E* be a minimal gr-injective resolution of M. Then ExtGr(R) (k, M) is the ith cohomology of the complex HomGr(R) (k,E*). By the minimality of the resolution all maps in this complex are zero, so that ExttGr(R)(K, M) = HomGr(R) (k,E1). To finish the proof it suffices to show that
HomGr{R)(n, EgR (-)) =
p [k (if p = m).
If p = m, then, by Proposition 2.1, HomGr(R)(K, ERR(k)) = HomGr(R)(K, k) = k.
If p / m. then there is a homogeneous element x of m not in p. Let / € HornGr^r)(k,j), then 0 = f(x) = xf( 1), but x in y is not a zero divisor, so /(1) = 0, that is, Hom.Qr(R)(K, y) = 0. By Proposition 2.1, HomGr{R)(n, E9R{ f)) = HomGr{R)(n, f) = 0. □
§ 3. Local cohomology
D e f i n i t i o n 3.1. Let R be a gr-noetherian G-graded ring. Let M € Gr(R).
(1) The small gr-support (following [6]) of M is
suppgr (M) = {p € Specgr (R) | TorR(p) (M(p), K(p)) = 0}.
(2) The (usual) gr-support of M is
Suppgr (M) = {p € Specgr (R) | M(p) = 0}.
Remark 3.1. Note that suppgr(M) c Suppgr(M) and the equality holds if M € gr(R) (see, e.g., [6, Lemma 2.6]).
Definition 3.2. Let R be a gr-noetherian G-graded ring. Let M € Gr(R).
(1) For any subset V c Specgr (R) we say that V is a specialization-closed subset if for any p € V and any q € Specgr(R) we have q € V whenever p c q.
(2) Let V be a specialization-closed subset of Specgr(R). We can define the section functor rV with support in V as
rv(M) = |J{N c M | Suppgr(N) c V} = |J{N c M | suppgr(N) c V} for all M € Gr(R).
Definition 3.3. We denote C = D+(Gr(R)) in this definition. Let 5 : C ^ C be a triangle functor. We say that 5 is an abstract local cohomology functor if the following conditions are satisfied.
(1) The natural embedding functor i : Im(5) ^ C has a right adjoint p : C ^ Im(5) and 5 = i o p.
(2) The t-structure (Im(5), Ker(5)) divides indecomposable injective objects, by which we mean that each indecomposable injective object belongs to either Im(5) or Ker(5).
Theorem 3.1. The following conditions are equivalent for a left exact preradical functor F on Gr(R).
(1) F is a radical functor.
(2) F preserves injectivity.
(3) F is a section functor with support in a specialization-closed subset of Spec9r (R).
(4) RF is an abstract local cohomology functor.
The proof of Theorem 3.1 consists of a succession of relatively short lemmas.
L e m m a 3.1. Let R be a gr-noetherian G-graded ring, and V be a specialization-closed subset of Spec9r (R).
(1) If N is a graded submodule of M, then the equality ry (N ) = N n ry (M ) holds.
(2) ry(M/ry(M)) = 0 for every M € Gr(R).
(3) ry is a left exact radical functor. Proof. (1) If H € Gr(R), then
H C ry (N) ^ H C N and Supp9r (H) C V ^ H C ry (M) n N.
(2) If H € Gr(R), then
H C ry (M/ry (M)) ^ H C M/ry (M) and Supp9r (H) C V ^ H = 0.
(3) Let 0 A K A M A N be an exact sequence in Gr(R). By (1) we have ry(K) = K n ry(M), hence 0 A ry(K) a ry(M) is an exact sequence. Let H € Gr(R), we have
H C Kerry (g) ^ H C Kerg n ry (M ) = Imf n ry (M ) ^ H C Imry (f ),
hence 0 A ry (K) a ry (M) a ry (N) is an exact sequence. □
Lemma 3.2. Let R be a gr-noetherian G-graded ring. Let F : Gr(R) A Gr(R) be a left exact radical functor.
(1) Let p € Specgr(R), then F{EaR{f)) is identical to either EaR{f) or 0.
(2) F preserves injectivity.
Proof. (1) Since F is a left exact radical functor, there is a hereditary torsion theory (Tf, Ff),
which is defined in (1.1). Hence there is an exact sequence
R
o N a a H a 0
with N € TF and H € 7>. If N = 0, then E9R(f) ^ H € Ff, therefore F(E9Rr(f)) = 0. If N / 0, since Assgr{EaR{f)) = {p}, we have Assgr{N) = {p}, hence R/p C N. Since Tf is a
localizing subcategory and N € Tf, we have /i/p e Tf- By Proposition 1.1, ER(^) e Tf- Therefore,
(2) For an injective J € Gr(R), by Theorem 1.1 it has an indecomposable decomposition J =
We set Ji = ©teXi and J2 = ©teX2 where
pt pt
R
Pt
By (1) we have J = J\ ® J2. Since Tf is closed under taking direct sums and Ff is closed under taking direct products and subsheaves, we have J\ € Tf, and J2 € Ff . Therefore, we have an equality F (J ) = F ( Ji ) ® F ( J2 ) = Ji, which is a gr-injective module. □
Proposition 3.1. Let R be a gr-noetherian G-graded ring. Let F : Gr(R) ^ Gr(R) be a left exact preradical functor which preserves injectivity. Then
(1) F{E9k {f)) is identical to either E9^{|) or 0.
(2) F(j) is identical to either y or 0.
>R\
Proof. (1) Since F(Er(j)) is a gr-injective submodule of a gr-indecomposable injective module (y), it is a direct summand of E9^(j). Thus the indecomposability of fyj'> (j) forces F(E^[(j)) is er E%(f) or 0.
It follows from Lemma 1.1 that F(|) = f n F(E9Rr(§)), therefore F(|) is either f or 0 by (1). □
p p R p p p
For a left exact preradical functor F which preserves injectivity, we define a subset VF of Specgr(R)
as follows:
VF = jp € Spec9r(R) : F(^) = ^ Note from the proof of Proposition 3.1 that VF is the same as the set
jp G Spe^(R) : F{E9^)) = ¿S£(|)} .
R p R * p
Proposition 3.2. Let F be a left exact preradical functor which preserves injectivity. Then VF is a specialization-closed subset.
Proof. Let p € VF and q D p. There is an exact sequence
0 —>■ ii —>■ — —>■ —.
pq
Since F is a left exact preradical functor, there is an exact sequence
RR 0 F(K) ^ F(-) ^ F(-).
We have F{§) = § and F(K) = KCiF{§) = K by Lemma 1.1, therefore F{§) = f, hence q € VF. □
p p p q q
L e m m a 3.3. Let F be a left exact preradical functor which preserves injectivity. Then the equality F = rVF holds as subfunctors of 1, where VF is a specialization-closed subset of Specgr (R) defined in Proposition 3.2.
Proof. First of all, we consider the case that M is a finite direct sum of indecomposabe injective
n rpg r t R i=l ^R
objects 0"=1 f:)(gi) in Gr(R). Then we have an equality
R
F(M) = 0 E%(-)(gi)=TVF(M)
Pi&Vf Pi
by Proposition 3.1.
Next, we consider the case that M € gr(R). Since the gr-injective hull Egr (M) of M is a finite direct sum of indecomposable gr-injective modules, we have already shown that F(Egr (M)) = Tvf (Egr (M)). Thus, using Lemma 1.1, we have
F(M) = M n F(Egr(M)) = M n rvf(Egr(M)) = Tvf(M).
Finally, we show the claimed equality for an object M in Gr(R) without any assumption. We should notice that a graded submodule N c M belongs to F(M) if and only if the equality F(N) = N holds. In fact, this equivalence is easily observed from the equality F(N) = N n F(M) by Lemma 1.1. This equivalence is true for the section functor Tvf as well. So N c M belongs to Tvf (M) if and only if rvf (N) = N. Therefore, we see that N c F(M) if and only if N c Tvf (M), and the proof is completed. □
Lemma 3.4. Let M* € D(Gr(R)) and let V be a specialization-closed subset of Specgr(R). Then
(1) M* belongs to Im(RrV) if and only if M* is quasi-isomorphic to a gr-injective complex whose components are direct sums of E9R(j)(g) with p £ V.
(2) M* belongs to Ker(RrV) if and only if M* is quasi-isomorphic to a gr-injective complex whose components are direct sums of E9R(j)(g) with p € Specgr(R) — V.
(3) RF is an abstract local cohomology functor.
Proof. By [22, Theorem 5.4] or [11, Proposition B.2], every complex of graded R-module has a K-injective resolution. For any gr-injective complex J * € K(I), Rrv (J*) = rv (J *) is the subcom-plex of J * consisting of gr-injective modules supported in V. Hence every object of Im(RrV) (resp. Ker(RrV)) is a gr-injective complex whose components are direct sums
of er(§XsO with P G V (resp.
p € Specgr(R) - V). In particular, if p € V (resp. p € Specgr{R) - V), then E9R{f){g) € /m(RTv) (resp. EfUf^g) € Ker(KTv)). Since Hom,Gr{R)(E%(^)(g)),E%(^)(h)) = 0 for p € F and q € Specgr(R) — V, we can see that HomK(i)(J1', J*) = HomK(i)(J1', rV(J*)) for any J* € Im(RrV) and J* € K(I). Hence it follows from the above equivalence that RrV is a right adjoint of the natural embedding i : Im(RrV) A D(Gr(R)). □
Proof of Theorem 3.1. (1) ^ (2), (2) ^ (3), (3) ^ (1), and (3) ^ (4) have already been proved, respectively, in Lemmas 3.2(2), 3.3, 3.1 (3), and 3.4(3).
(4) ^ (1). Assume that RF is an abstract local cohomology functor. We have to show that F(M/F(M)) = 0 for any graded module M. It is enough to show that F(E/F(E)) = 0 for any gr-injective module E. In fact, for any graded module M, taking the gr-injective hull Egr (M) of M, we have F(M/F(M)) c F(Egr(M)/F(Egr(M))) by Lemma 1.1.
Note that the natural inclusion F c 1 of functors on Gr(R) induces a natural morphism 0 : RF a 1 of functors on D+(Gr(R)). Since (Im(RF), Ker(RF)) is a stable t-structure on D+(Gr(R)), it follows from Theorem 1.2 that every gr-injective module E is embedded in a triangle
RF(E) ——A E A N A RF(E)[1],
with RF(E) € Im(RF) and N € Ker(RF). Since E is a gr-injective module and since RF is the right derived functor of a left-exact functor, RF(E) = F(E) is a submodule of E via the morphism 0(E). Therefore, we have N ^ E/F(E) in D+(Gr(R)). In particular, H0(RF(E/F(E))) ^ H0(RF(N)) = 0. Since F is a left exact functor, it is concluded that F(E/F(E)) = 0 as desired. This completes the proof of Theorem 3.1. □
Funding. This work was supported by the Chinese Scholarship Council.
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Received 05.04.2019
Lu Li, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991, Russia. E-mail: [email protected]
Л.Лу
Структурная теорема для gr-инъективных модулей над gr-нётеровыми G-градуированными коммутативными кольцами и локальные когомологические функторы
Цитата: Известия Института математики и информатики Удмуртского государственного университета. 2019. Т. 53. С. 127-137.
Ключевые слова: градуированные коммутативные кольца, gr-бассовы числа, локальные когомологические функторы, производные категории, радикальные функторы.
УДК: 512.7
DOI: 10.20537/2226-3594-2019-53-11
Хорошо известно, что разложение инъективных модулей над нётеровыми кольцами является одним из наиболее эстетичных и важных результатов в коммутативной алгебре. Наша цель — доказать аналогичные результаты для градуированных нётеровых колец. В этой статье мы изучаем структурную теорему для gr-инъективных модулей над gr-нётеровыми G-градуированными коммутативными кольцами, даем определение gr-бассовых чисел и изучаем их свойства. Мы покажем, что каждый gr-инъективный модуль имеет неразложимое разложение. Пусть R — gr-нётерово градуированное кольцо, а M — gr-конечно порожденный R-модуль. Мы дадим формулу для выражения чисел Басса с помощью функтора Ext. Мы определяем функтор сечения rV с носителем в замкнутом по специализации подмножестве V из Specgr (R) и абстрактный локальный когомологический функтор. В заключение мы покажем, что левый точный радикальный функтор F имеет вид rV для замкнутого по специализации подмножества V.
Финансирование. Работа выполнена при поддержке Китайского стипендиального совета.
Поступила в редакцию 05.04.2019
Лу Ли, механико-математический факультет, Московский государственный университет им. М. В. Ломоносова, 119991, Россия, г. Москва, ГСП-1, Ленинские горы. E-mail: [email protected]
