Standard Graded Ring Research Articles

Hyperplane sections of non-standard graded rings

The purpose of this paper is to explain a method on the generalization of the Bertini-type theorem on standard graded rings to the non-standard graded case of certain type.

Comparing generalized Gorenstein properties in semi-standard graded rings

Semi-standard graded rings are a generalized notion of standard graded rings. In this paper, we compare generalized notions of the Gorenstein property in semi-standard graded rings. We discuss the commonalities between standard graded rings and semi-standard graded rings, as well as elucidate distinctive phenomena present in semi-standard graded rings that are absent in standard graded rings.

Lim Ulrich sequences and Lech’s conjecture

The long standing Lech’s conjecture in commutative algebra states that for a flat local extension \((R,\mathfrak {m})\rightarrow (S,\mathfrak {n})\) of Noetherian local rings, we have an inequality on the Hilbert–Samuel multiplicities: \(e(R)\le e(S)\). In general the conjecture is wide open when \(\dim R>3\), even in equal characteristic. In this paper, we prove Lech’s conjecture in all dimensions, provided \((R,\mathfrak {m})\) is a standard graded ring over a perfect field localized at the homogeneous maximal ideal. We introduce the notions of lim Ulrich and weakly lim Ulrich sequences. Roughly speaking these are sequences of finitely generated modules that are not necessarily Cohen–Macaulay, but asymptotically behave like Ulrich modules. We prove that the existence of these sequences imply Lech’s conjecture. Though the existence of Ulrich modules is known in very limited cases, we construct weakly lim Ulrich sequences for all standard graded domains over perfect fields of positive characteristic.

Hilbert-Kunz density functions and F-thresholds

The first author had shown earlier that for a standard graded ring R and a graded ideal I in characteristic p>0, with ℓ(R/I)<∞, there exists a compactly supported continuous function fR,I whose Riemann integral is the HK multiplicity eHK(R,I). We explore further some other invariants, namely the shape of the graph of fR,m (where m is the graded maximal ideal of R) and the maximum support (denoted as α(R,I)) of fR,I.In case R is a domain of dimension d≥2, we prove that (R,m) is a regular ring if and only if fR,m has a symmetry fR,m(x)=fR,m(d−x), for all x.If R is strongly F-regular on the punctured spectrum then we prove that the F-threshold cI(m) coincides with α(R,I).As a consequence, if R is a two dimensional domain and I is generated by homogeneous elements of the same degree, then we have (1) a formula for the F-threshold cI(m) in terms of the minimum strong Harder-Narasimhan slope of the syzygy bundle and (2) a well defined notion of the F-threshold cI(m) in characteristic 0.This characterisation readily computes cI(n)(m), for the set of all irreducible plane trinomials k[x,y,z]/(h), where m=(x,y,z) and I(n)=(xn,yn,zn).

Asymptotic Hilbert–Kunz multiplicity

For a pair (M,I), where M is finitely generated graded module over a standard graded ring R of dimension d≥2, and I is a graded ideal with ℓ(R/I)<∞ and generated by elements of the same degree, we prove that limq→∞⁡e1(M,I[q])/qd exists, where e1(M,I[q]) denotes the first coefficient of the Hilbert–Samuel polynomial of (M,I[q]).We use this to get an expression for limk→∞⁡[eHK(M,Ik)−e0(M,Ik)/d!]/kd−1, where eHK denotes the Hilbert–Kunz multiplicity. In particular, if dim⁡ M=d then we deduce that the difference eHK(M,Ik)−e(M,Ik)/d! grows at least as a fixed positive multiple of kd−1 as k→∞.This is proved using ‘renormalized’ HK density functions.

Hilbert-Kunz density function and Hilbert-Kunz multiplicity

For a pair $(M, I)$, where $M$ is finitely generated graded module over a standard graded ring $R$ of dimension $d$, and $I$ is a graded ideal with $\ell(R/I) < \infty$, we introduce a new invariant $HKd(M, I)$ called the {\em Hilbert-Kunz density function}. In Theorem 1.1, we relate this to the Hilbert-Kunz multiplicity $e_{HK}(M,I)$ by an integral formula. We prove that the Hilbert-Kunz density function is additive. Moreover it satisfies a multiplicative formula for a Segre product of rings. This gives a formula for $e_{HK}$ of the Segre product of rings in terms of the HKd of the rings involved. As a corollary, $e_{HK}$ of the Segre product of any finite number of Projective curves is a rational number. As an another application we see that $e_{HK}(R, {\bf m}^k) - e(R, {\bf m}^k)/d!$ grows at least as a fixed positive multiple of $k^{d-1}$ as $k\to \infty$.

Tameness and Artinianness of Graded Generalized Local Cohomology Modules

Let R=⊕n ≥ 0 Rn be a standard graded ring, 𝔞 ⊇ ⊕n &gt; 0 Rn an ideal of R, and M, N two finitely generated graded R-modules. This paper studies the homogeneous components of graded generalized local cohomology modules. We show that for any i ≥ 0, the n-th graded component [Formula: see text] of the i-th generalized local cohomology module of M and N with respect to 𝔞 vanishes for all n ≫ 0. Some sufficient conditions are proposed to satisfy the equality [Formula: see text]. Also, some sufficient conditions are proposed for the tameness of [Formula: see text] such that [Formula: see text] or i= cd 𝔞(M,N), where [Formula: see text] and cd 𝔞(M,N) denote the R+-finiteness dimension and the cohomological dimension of M and N with respect to 𝔞, respectively. Finally, we consider the Artinian property of some submodules and quotient modules of [Formula: see text], where j is the first or last non-minimax level of [Formula: see text].

Castelnuovo–Mumford Regularity and Cohomological Dimension

Let be a standard graded ring, be the irrelevant ideal of R and 𝔞0 be an ideal of R 0. In this paper, as a generalization of the concept of Castelnouvo–Mumford regularity reg(M) of a finitely generated graded R-module M, we define the regularity of M with respect to 𝔞0 + R +, say reg𝔞0 + R + (M). We also study some relations of this new invariant with the classic one. To this end, we need to consider the cohomological dimension of some finitely generated R 0-modules. Also, we will express reg𝔞0 + R + (M) in terms of some invariants of the minimal graded free resolution of M and see that in a special case this invariant is independent of the choice of 𝔞0.

Toric ideals for high Veronese subrings of toric algebras

We prove that the defining ideal of a sufficiently high Veronese subring of a toric algebra admits a quadratic Gröbner basis consisting of binomials. More generally, we prove that the defining ideal of a sufficiently high Veronese subring of a standard graded ring admits a quadratic Gröbner basis. We give a lower bound on $d$ such that the defining ideal of $d$th Veronese subring admits a quadratic Gröbner basis. Eisenbud–Reeves–Totaro stated the same theorem without a proof with some lower bound on $d$. In many cases, our lower bound is less than Eisenbud–Reeves–Totaro’s lower bound.

Sequences that Preserve the Homological Degree

The homological degree is a cohomological degree generalizing the classical degree function of a module. Given a finitely generated graded module M, over a standard graded ring A, this note is concerned with the question of when, for an element x in the augmentation ideal of A that is not in any associated prime of M other than the augmentation ideal itself, the homological degree of M is equal to the homological degree of M/xM. This question is answered when the dimension of M is one or two.

Generalized local cohomology and regularity of Ext modules

Let R be a Noetherian standard graded ring, and M and N two finitely generated graded R-modules. We introduce reg R ( M , N ) by using the notion of generalized local cohomology instead of local cohomology, in the definition of regularity. We prove that reg R ( M , N ) is finite in several cases. In the case that the base ring is a field, we show that reg R ( M , N ) = reg ( N ) − indeg ( M ) . This formula, together with a graded version of duality for generalized local cohomology, gives a formula for the minimum of the initial degrees of some Ext modules (in the case R is Cohen–Macaulay), of which the three usual definitions of regularity are special cases. Bounds for regularity of certain Ext modules are obtained, using the same circle of ideas.

Castelnuovo–Mumford regularity, postulation numbers, and reduction numbers

Suppose G is a standard graded ring over an infinite field. We obtain a sharp lower bound for the regularity of G in terms of the postulation number, the depth, and the dimension of G. We present a class of examples in dimension 1 where the postulation number is 0 and the regularity of G can take on any value between 1 and the embedding codimension of G. Suppose G = gr m ( R ) is the associated graded ring of a Cohen–Macaulay local ring ( R , m ) . We compute the regularity, the reduction number and the postulation number of G and consider the relationship among these invariants. In the case where dim G − grade G + ⩽ 1 , a precise description is known as to how these integers are related. We consider the case where dim G − grade G + ⩾ 2 , and prove that if dim G − grade G + = 2 , then reg G = max { p ( G ) + dim G − 1 , r ( m ) } , where p ( G ) is the postulation number of G and r ( m ) is the reduction number of m.

Tight closure of finite length modules in graded rings

In this article, we look at how the equivalence of tight closure and plus closure (or Frobenius closure) in the homogeneous m-coprimary case implies the same closure equivalence in the nonhomogeneous m-coprimary case in standard graded rings. Although our result does not depend upon dimension, the primary application is based on results known in dimension 2 due to the recent work of H. Brenner. We also demonstrate a connection between tight closure and the m-adic closure of modules extended to R+ or R∞. We finally show that unlike the Noetherian case, the injective hull of the residue field over R+ or R∞ contains elements that are not killed by any power of the maximal ideal of R. This fact presents an obstruction to one possible method of extending our main result to all modules.

On the asymptotic linearity of Castelnuovo–Mumford regularity

Let R be a standard graded ring over a commutative Noetherian ring with unity. Let I be an arbitrary graded ideal of R and M an arbitrary finitely generated graded R -module. We prove that there exist integers e and ρ M ( I ) such that reg ( I n M ) = ρ M ( I ) n + e for all large n . This generalizes earlier results in the case R is a polynomial ring. Note that standard techniques in the polynomial case cannot be used here.

Bounds on the a-invariant and reduction numbers of ideals

Let R be a d-dimensional standard graded ring over an Artinian local ring. Let M be the unique maximal homogeneous ideal of R. Let h i ( R) n denote the length of the nth graded component of the local cohomology module H i M (R) . Define the Eisenbud–Goto invariant EG( R) of R to be the number ∑ q=0 d−1 d−1 q h q M (R) 1−q. We prove that the a-invariant a( R) of the top local cohomology module H M d(R) satisfies the inequality: a( R)⩽ e( R)−ℓ( R 1)+( d−1)(ℓ( R 0)−1)+ EG( R). This bound is used to get upper bounds for the reduction number of an m -primary ideal I of a Cohen–Macaulay local ring (R, m) , when the associated graded ring of I has depth at least d−1.