Local Base Ring Research Articles

Some Properties of Serre Subcategories in the Graded Local Cohomology Modules

LetR=⊕n≥0Rnbe a standard homogeneous Noetherian ring with local base ring(R0,m0)and letMbe a finitely generated gradedR-module. LetHR+i(M)be theith local cohomology module ofMwith respect toR+=⊕n>0Rn. LetSbe a Serre subcategory of the category ofR-modules and letibe a nonnegative integer. In this paper, ifdim⁡R0≤1,then we investigate some conditions under which theR-modulesR0/m0 ⊗R0 HR+i(M),Γm0R(HR+i(M))andHm0R1(HR+i(M))are inSfor alli≥0. Also, we prove that ifdim⁡R0≤2, then the gradedR-moduleHm01(HR+i(M))is inSfor alli≥0. Finally, we prove that ifnis the biggest integer such thatHai(M)∉S, thenHR+i(M)/m0HR+i(M)∈Sfor alli≥n.

Artinianness of Composed Graded Local Cohomology Modules

AbstractLet be a graded Noetherian ring with local base ring (R0 ,m0) and let . Let M and N be finitely generated graded R-modules and let a = a0 + R+ an ideal of R. We show that and are Artinian for some i s and j s with a specified property, where bo is an ideal of R0 such that a0 + b0 is an m0-primary ideal.

Finiteness of graded generalized local cohomology modules

We consider two finitely generated graded modules over a homogeneous Noetherian ring \(R = \oplus _{n \in \mathbb{N}_0 } R_n\) with a local base ring (R 0, m0) and irrelevant ideal R + of R. We study the generalized local cohomology modules H b i (M,N) with respect to the ideal b = b0 + R +, where b0 is an ideal of R 0. We prove that if dimR 0/b0 ≤ 1, then the following cases hold: for all i ≥ 0, the R-module H b i (M,N)/a0 H b i (M,N) is Artinian, where \(\sqrt {\mathfrak{a}_0 + \mathfrak{b}_0 } = \mathfrak{m}_0\); for all i ≥ 0, the set \(Ass_{R_0 } \left( {H_\mathfrak{b}^i \left( {M,N} \right)_n } \right)\) is asymptotically stable as n→−∞. Moreover, if H b i (M,N) n is a finitely generated R 0-module for all n ≤ n 0 and all j < i, where n 0 ∈ ℤ and i ∈ ℕ0, then for all n ≤ n 0, the set \(Ass_{R_0 } \left( {H_\mathfrak{b}^i \left( {M,N} \right)_n } \right)\) is finite.

Artinianness of Certain Graded Local Cohomology Modules

AbstractWe show that ifR= ⴲn∈ℕ0Rnis a Noetherian homogeneous ring with local base ring (R0, m0), irrelevant idealR+, andMa finitely generated gradedR-module, thenis Artinian forj= 0, 1 wheret= inf﹛i∈ ℕ0:is not finitely generated﹜. Also, we prove that if cd(R+,M) = 2, then for eachi∈ ℕ0,) is Artinian if and only ifis Artinian, where cd(R+,M) is the cohomological dimension ofMwith respect toR+. This improves some results of R. Sazeedeh.

On the Artinianness of Graded Local Cohomology Modules

Let R = ⨁n≥ 0 Rn be a homogeneous noetherian ring with local base ring [Formula: see text], and N a finitely generated graded R-module. Let [Formula: see text] be the i-th local cohomology module of N with respect to R+ := ⨁n &gt; 0 Rn. Let t be the largest integer such that [Formula: see text] is not minimax. We prove that [Formula: see text] is [Formula: see text]-coartinian for any i &gt; t, and [Formula: see text] is artinian. Let s be the first integer such that [Formula: see text] is not minimax. We show that for any i ≤ s, the graded module [Formula: see text] is artinian.

Upper and Lower Bounds for Finiteness of Graded Local Cohomology Modules

Let R = ⨁ n ∈ ℕ0 R n be a noetherian homogeneous ring with local base ring (R 0, 𝔪0) and irrelevant ideal R +, let M be a finitely generated graded R-module. We shall define the upper finiteness dimension (lower finiteness dimension) of M with respect to R + and 𝔪0, denoted by u R +, 𝔪0 (M)(l R +, 𝔪0 (M)) and we prove that are artinian for all i ≤ u R +, 𝔪0 (M) (i ≥ l R +, 𝔪0 (M)). These results yield us to get some similar results for new finiteness dimensions induced by minimax and cofinite graded modules.

Bounds for the Castelnuovo-Mumford Regularity

We extend the “linearly exponential” bound for the Castelnuovo-Mumford regularity of a graded ideal in a polynomial ring K[x1, . . . , xr] over a field (established by Galligo and Giusti in characteristic 0 and recently, by Caviglia-Sbarra for abitrary K) to graded submodules of a graded module over a homogeneous Cohen-Macaulay ring R = ⊕n≥0Rn with artinian local base ring R0. As an application we get a “linearly exponential” bound for the Castelnuovo-Mumford regularity of a graded R-module M in terms of the degrees which occur in a minimal free presentation of M.

Results on Graded Generalized Local Cohomology

Let R = ⨁ i≥0 R i be a positively graded Noetherian ring with irrelevant ideal R + = ⨁ i≥1 R i , and let M, N be two finitely generated graded R-modules such that pd(M) is finite. We show that the least integer i for which is not asymptotically zero is equal to the least integer i such that . Also, whenever R is homogeneous with local base ring (R 0, 𝔪0), we prove some Artinian and tameness property for .

Bounds for regularity and coregularity of graded modules

Let M be a finitely generated graded module over a Noetherian homogeneous ring R with local base ring (R0, m0) .I fR0 is of dimension one, then we show that reg i+1 (M) and coreg i+1 (M) are bounded for all i ∈ N0. We improve these bounds, if in addition, R0 is either regular or analytically irreducible of unequal characteristic.

Finiteness of graded local cohomology modules

Let R = ⨁ n ∈ N 0 R n be a Noetherian homogeneous ring with local base ring ( R 0 , m 0 ) and irrelevant ideal R + , let M be a finitely generated graded R -module. In this paper we show that H m 0 R 1 ( H R + 1 ( M ) ) is Artinian and H m 0 R i ( H R + 1 ( M ) ) is Artinian for each i in the case where R + is principal. Moreover, for the case where ara ( R + ) = 2 , we prove that, for each i ∈ N 0 , H m 0 R i ( H R + 2 ( M ) ) is Artinian if and only if H m 0 R i + 2 ( H R + 1 ( M ) ) is Artinian. We also prove that H m 0 d ( H R + c ( M ) ) is Artinian, where d = dim ( R 0 ) and c is the cohomological dimension of M with respect to R + . Finally we present some examples which show that H m 0 R 2 ( H R + 1 ( M ) ) and H m 0 R 3 ( H R + 1 ( M ) ) need not be Artinian.

Artinianess of graded local cohomology modules

Let R = ⊕ neN R n be a Noetherian homogeneous ring with local base ring (R 0 , m 0 ) and let M be a finitely generated graded R-module. Let a be the largest integer such that H a R+ (M) is not Artinian. We will prove that H i R+ (M) /m 0 H i R+ (M) are Artinian for all i ≥ a and there exists a polynomial P ∈ Q[x] of degree less than a such that length R0 (H a R+ (M) n /m 0 H a R+ (M) n ) = P(n) for all n << 0. Let s be the first integer such that the local cohomology module H s R (M) is not R + -cofinite. We will show that for all i < s the graded module r m0 (H i R+ (M)) is Artinian.

An avoidance principle with an application to the asymptotic behaviour of graded local cohomology

We present an avoidance principle for certain graded rings. As an application we fill a gap in the proof of a result of Brodmann, Rohrer and Sazeedeh about the antipolynomiality of the Hilbert–Samuel multiplicity of the graded components of the local cohomology modules of a finitely generated module over a Noetherian homogeneous ring with two-dimensional local base ring.

On graded generalized local cohomology

Let \( R = {\mathop \oplus \limits_{i\underline{\underline > } 0} }R_{i} \) be a homogeneous Noetherian ring with local base ring (R0,m0) and let M,N be two finitely generated graded R-modules. Let \( H^{i}_{{R + }} {\left( {M,N} \right)} \) denote the i-th graded generalized local cohomology of N relative to M with support in \( R = {\mathop \oplus \limits_{i\underline{\underline > } 0} }R_{i} \) . We study the vanishing, tameness and asymptotical stability of the homogeneous components of \( H^{i}_{{R + }} {\left( {M,N} \right)} \).

Hilbert-Samuel coefficients and postulation numbers of graded components of certain local cohomology modules

Let R = ⨁ n ≥ 0 R n R = \bigoplus _{n \geq 0} R_n be a Noetherian homogeneous ring with one-dimensional local base ring ( R 0 , m 0 ) (R_0, {\mathfrak m}_0) . Let q 0 ⊆ R 0 {\mathfrak q}_0 \subseteq R_0 be an m 0 {\mathfrak m}_0 -primary ideal, let M M be a finitely generated graded R R -module and let i ∈ N 0 i \in {\mathbb N}_0 . Let H R + i ( M ) H^i_{R_+}(M) denote the i i -th local cohomology module of M M with respect to the irrelevant ideal R + := ⨁ n &gt; 0 R n R_+:= \bigoplus _{n &gt; 0} R_n of R R . We show that the first Hilbert-Samuel coefficient e 1 ( q 0 , H R + i ( M ) n ) e_1 \big ( {\mathfrak q}_0, H^i_{R_+}(M)_n \big ) of the n n -th graded component of H R + i ( M ) H^i_{R_+}(M) with respect to q 0 {\mathfrak q}_0 is antipolynomial of degree &gt; i &gt; i in n n . In addition, we prove that the postulation numbers of the components H R + i ( M ) n H^i_{R_+} (M)_n with respect to q 0 {\mathfrak q}_0 have a common upper bound.

Multiplicities of graded components of local cohomology modules

Let M be a finitely generated graded module over a Noetherian homogeneous ring R with local base ring ( R 0 , m 0 ) . Then, the nth graded component H R + i ( M ) n of the ith local cohomology module of M with respect to the irrelevant ideal R + of R is a finitely generated R 0 -module which vanishes for all n ≫ 0 . In various situations we show that, for an m 0 -primary ideal q 0 ⊆ R 0 , the multiplicity e q 0 ( H R + i ( M ) n ) of H R + i ( M ) n is antipolynomial in n of degree less than i. In particular we consider the following three cases: (a) i < g ( M ) , where g ( M ) is the so-called cohomological finite length dimension of M; (b) i = g ( M ) ; (c) dim ( R 0 ) = 2 . In cases (a) and (b) we express the degree and the leading coefficient of the representing polynomial in terms of local cohomological data of M (e.g. the sheaf induced by M) on Proj ( R ) . We also show that the lengths of the graded components of various graded submodules of H R + i ( M ) are antipolynomial of degree less than i and prove invariance results on these degrees.

Local cohomology over homogeneous rings with one-dimensional local base ring

Let R = ○+ n≥0 R n be a homogeneous Noetherian ring with local base ring (R 0 , m 0 ) and let M be a finitely generated graded R-module. Let H i R+ (M) be the i-th local cohomology module of M with respect to R + := ○+ n>0 R n . If dim R 0 ≤ 1, the R-modules Γ m0R (H ι Ρ+ (M)), (0:H ι Ρ+ (M) m 0 ) and H i R+(M)/m 0 H i R+(M) are Artinian for all i ∈ No. As a consequence, much can be said on the asymptotic behaviour of the Ro-modules H i R+(M) n for n → -∞.