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A counterexample to the rigidity conjecture for rings
math.AC
An example is constructed of a local ring and a module of finite type and finite projective dimension over that ring such that the module is not rigid. This shows that the rigidity conjecture is false.
math
1
Ideals associated to two sequences and a matrix
math.AC
Let $\u_{1\times n}$, $\X_{n\times n}$, and $\v_{n\times 1}$ be matrices of indeterminates, $\Adj \X$ be the classical adjoint of $\X$, and $H(n)$ be the ideal $I_1(\u\X)+I_1(\X\v)+I_1(\v\u-\Adj \X)$. Vasconcelos has conjectured that $H(n)$ is a perfect Gorenstein ideal of grade $2n$. In this paper, we obtain the minimal free resolution of $H(n)$; and thereby establish Vasconcelos' conjecture.
math
2
On the Betti numbers of some Gorenstein ideals
math.AC
Assume $R$ is a polynomial ring over a field and $I$ is a homogeneous Gorenstein ideal of codimension $g\ge3$ and initial degree $p\ge2$. We prove that the number of minimal generators $\nu(I_p)$ of $I$ that are in degree $p$ is bounded above by $\nu_0={p+g-1\choose g-1}-{p+g-3\choose g-1}$, which is the number of minimal generators of the defining ideal of the extremal Gorenstein algebra of codimension $g$ and initial degree $p$. Further, $I$ is itself extremal if $\nu(I_p)=\nu_0$.
math
3
Laurent coefficients and Ext of finite graded modules
math.AC
Let $R=\bigoplus_{n\ges0}R_n$ be a graded commutative ring generated over a field $K=R_0$ by homogeneous elements $x_1,\dots,x_e$ of positive degrees $d_1,\dots,d_e$. The Hilbert-Serre Theorem shows that for each finite graded $R$--module $M=\bigoplus_{n\in\BZ}M_n$ the {\it Hilbert series\/} $\sum_{n\in\BZ}(\rank_K M_n)t^n$ is the Laurent expansion around $0$ of a rational function $$ H_M(t)=\frac{q_M(t)}{\prod_{i=1}^e(1-t^{d_i})} $$ with $q_M(t)\in\BZ[t,\ti]$. We demonstrate that Laurent expansions $\left[M\right]_z$ of $H_M(t)$ around other points $z$ of the extended complex plane $\overline\BC$ also carry important structural information.
math
4
Analogs of Gröbner Bases in Polynomial Rings over a Ring
math.AC
In this paper we will define analogs of Gr\"obner bases for $R$-subalgebras and their ideals in a polynomial ring $R[x_1,\ldots,x_n]$ where $R$ is a noetherian integral domain with multiplicative identity and in which we can determine ideal membership and compute syzygies. The main goal is to present and verify algorithms for constructing these Gr\"obner basis counterparts. As an application, we will produce a method for computing generators for the first syzygy module of a subset of an $R$-subalgebra of $R[x_1,\ldots,x_n]$ where each coordinate of each syzygy must be an element of the subalgebra.
math
5
Links of prime ideals and their Rees algebras
math.AC
In a previous paper we exhibited the somewhat surprising property that most direct links of prime ideals in Gorenstein rings are equimultiple ideals with reduction number $1$. This led to the construction of large families of Cohen--Macaulay Rees algebras. The first goal of this paper is to extend this result to arbitrary Cohen--Macaulay rings. The means of the proof are changed since one cannot depend so heavily on linkage theory. We then study the structure of the Rees algebra of these links, more specifically we describe their canonical module in sufficient detail to be able to characterize self--linked prime ideals. In the last section multiplicity estimates for classes of such ideals are established.
math
6
Hilbert functions of graded algebras over Artinian rings
math.AC
In this paper we give an effective characterization of Hilbert functions and polynomials of standard algebras over an Artinian equicharacteristic local ring; the cohomological properties of such algebras are also studied. We describe algorithms to check the admissibility of a given function or polynomial as a Hilbert function or polynomial, and to produce a standard algebra with a given Hilbert function.
math
7
Extremal Betti Numbers and Applications to Monomial Ideals
math.AC
In this short note we introduce a notion of extremality for Betti numbers of a minimal free resolution, which can be seen as a refinement of the notion of Mumford-Castelnuovo regularity. We show that extremal Betti numbers of an arbitrary submodule of a free S-module are preserved when taking the generic initial module. We relate extremal multigraded Betti numbers in the minimal resolution of a square free monomial ideal with those of the monomial ideal corresponding to the Alexander dual simplicial complex and generalize theorems of Eagon-Reiner and Terai. As an application we give easy (alternative) proofs of classical criteria due to Hochster, Reisner, and Stanley.
math
8
Permanental Ideals
math.AC
The principal result is a primary decomposition of ideals generated by the (2x2)-subpermanents of a generic matrix. These permanental ideals almost always have embedded components and their minimal primes are of three distinct heights. Thus the permanental ideals are almost never Cohen-Macaulay, in contrast with determinantal ideals.
math
9
Multiplicative Invariants and Semigroup Algebras
math.AC
Let G be a finite group acting by automorphism on a lattice A, and hence on the group algebra S=k[A]. The algebra of G-invariants in S is called an algebra of multiplicative invariants. We investigate when algebras of multiplicative invariants are semigroup algebras. In particular, we present an explicit version of a result of Farkas stating that multiplicative invariants of finite reflection groups are indeed semigroup algebras. On the other hand, multiplicative invariants arising from fixed point free actions are shown to never be semigroup algebras. In particular, this holds whenever G has odd prime order.
math
10
Truncations of the ring of number-theoretic functions
math.AC
We study the ring of all functions from the positive integers to some field. This ring, which we call \emph{the ring of number-theoretic functions}, is an inverse limit of the ``truncations'' \Gamma_n consisting of all functions f for which f(m)=0 whenever m > n. Each \Gamma_n is a zero-dimensional, finitely generated (K)-algebra, which may be expressed as the quotient of a finitely generated polynomial ring with a \emph{reversely stable} monomial ideal. Using the description of the free minimal resolution of stable ideals, given by Eliahou-Kervaire, and some additional arguments by Aramova-Herzog and Peeva, we give the Poincar\'e-Betti series for \Gamma_n.
math
11
Taylor and minimal resolutions of homogeneous polynomial ideals
math.AC
We give a necessary and sufficient condition on a homogeneous polynomial ideal for its Taylor complex to be exact. Then we give a combinatorial construction of a minimal resolution for ideals satisfying the above condition (in particular for monomial ideals).
math
12
Generalised Hilbert Numerators II
math.AC
We associate to each $r$-multigraded, locally finitely generated ideal in the "large polynomial ring" on countably many indeterminates a power series in $r$ variables; this power series is the limit in the adic topology of the numerators of the rational functions which give the Hilbert series of the truncations of the ideal. We characterise the set of all power series so obtained. Our main technical tools are an approximation result which asserts that truncation and the forming of initial ideals commute in a filtered sense, and standard inclusion/exclusion, M\"obius inversion, and LCM-lattice homology methods generalised to monomial ideals in countably many variables.
math
13
Some conjectures about the Hilbert series of generic ideals in the exterior algebra
math.AC
We give conjectures on the "asymptotic" behaviour of the Hilbert series of (quotients by) generic ideals in the exterior algebra, as the number of variables tend to infinity. Our conjectures are supported by extensive computer calculations.
math
14
Bounds for Betti numbers
math.AC
In this paper we prove parts of a conjecture of Herzog giving lower bounds on the rank of the free modules appearing in the linear strand of a graded $k$-th syzygy module over the polynomial ring. If in addition the module is $\mathbb{Z}^n$-graded we show that the conjecture holds in full generality. Furthermore, we give lower and upper bounds for the graded Betti numbers of graded ideals with a linear resolution and a fixed number of generators.
math
15
Lifting Grobner bases from the exterior algebra
math.AC
In the article "Non-commutative Grobner bases for commutative algebras", Eisenbud-Peeva-Sturmfels proved a number of results regarding Grobner bases and initial ideals of those ideals in the free associative algebra which contain the commutator ideal. We prove similar results for ideals which contains the anti-commutator ideal (the defining ideal of the exterior algebra). We define one notion of generic initial ideals in the free assoicative algebra, and show that gin's of ideals containing the commutator ideal, or the anti-commutator ideal, are finitely generated.
math
16
Rank one discrete valuations of $k((X_1,...X_n))$
math.AC
In this paper we study the rank one discrete valuations of $k((X_1,... ,X_n))$ whose center in $k\lcor\X\rcor$ is the maximal ideal $(\X)$. In sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote section 2 to finding an element of value 1, that is, a parameter. The field of coefficients is the residue field of the valuation, and it is given in section 5. The constructions given in these sections are not effective in the general case, because we need either to use the Zorn's lemma or to know explicitly a section $\sigma$ of the natural homomorphism $R_v\to\d$ between the ring and the residue field of the valuation $v$. However, as a consequence of this construction, in section 7, we prove that $k((\X))$ can be embedded into a field $L((\Y))$, where the {\em ``extended valuation'' is as close as possible to the usual order function}.
math
17
On the dimension of discrete valuations of k((X1,...,Xn))
math.AC
Let $v$ be a rank-one discrete valuation of the field $k((\X))$. We know, after \cite{Bri2}, that if $n=2$ then the dimension of $v$ is 1 and if $v$ is the usual order function over $k((\X))$ its dimension is $n-1$. In this paper we prove that, in the general case, the dimension of a rank-one discrete valuation can be any number between 1 and $n-1$.
math
18
Homological properties of bigraded algebras
math.AC
We study the x- and y-regularity of a bigraded K-algebra R. These notions are used to study asymptotic properties of certain finitely generated bigraded modules. As an application we get for any equigenerated graded ideal I upper bounds for the number j_0 for which reg(I^j) is a linear function for j >= j_0. Finally we give upper bounds for the x- and y-regularity of generalized Veronese algebras.
math
19
Subalgebras of bigraded Koszul algebras
math.AC
We show that diagonal subalgebras and generalized Veronese subrings of a bigraded Koszul algebra are Koszul. We give upper bounds for the regularity of sidediagonal and relative Veronese modules and apply the results to symmetric algebras and Rees rings.
math
20
On Cohen-Macaulay rings of invariants
math.AC
We investigate the transfer of the Cohen-Macaulay property from a commutative ring to a subring of invariants under the action of a finite group. Our point of view is ring theoretic and not a priori tailored to a particular type of group action. As an illustration, we briefly discuss the special case of multiplicative actions, that is, actions on group algebras $k[\bbZ^n]$ via an action on $\bbZ^n$.
math
21
Resolutions by mapping cones
math.AC
In this paper we study resolutions which arise as iterated mapping cones.
math
22
Sequentially Cohen-Macaulay modules and local cohomology
math.AC
The main result of the paper states that for a graded ideal I in a polynomial ring R over a field of characteristic 0, the Hilbert functions of the local cohomology modules of R/I and of R/Gin(I) coincide if and only if R/I is sequentially Cohen-Macaulay.
math
23
Asymptotic linear bounds for the Castelnuovo-Mumford regularity
math.AC
We prove asymptotic linear bounds for the Castelnuovo-Mumford regularity of certain filtrations of homogeneous ideals whose Rees algebras need not to be Noetherian.
math
24
Groebner bases and regularity of Rees algebras
math.AC
In this paper we study homological properties of the Rees ring R of the graded maximal ideal of a standard graded k-algebra A. In particular we are interested the comparison of the depth and regularity of A and R.
math
25
Conservation of the noetherianity by perfect transcendental field extensions
math.AC
Let $k$ be a perfect field of characteristic $p>0$, $k(t)_{per}$ the perfect closure of $k(t)$ and $A$ a $k$-algebra. We characterize whether the ring $A\otimes_k k(t)_{per}$ is noetherian or not. As a consequence, we prove that the ring $A\otimes_k k(t)_{per}$ is noetherian when $A$ is the ring of formal power series in $n$ indeterminates over $k$.
math
26
Intersections of symbolic powers of prime ideals
math.AC
Let (R,m) be a local ring with prime ideals p and q such that p+q is an m-primary ideal. If R is regular and contains a field, and dim(R/p)+dim(R/q)=dim(R), we prove that p^{(r)}\cap q^{(n)}\subseteq m^{m+n} for all positive integers r and s. This is proved using a generalization of Serre's Intersection Theorem which we apply to a hypersurface R/fR. The generalization gives conditions that guarantee that Serre's bound on the intersection dimension dim(R/p)+dim(R/q) \leq dim(R) holds when R is nonregular.
math
27
Frobenius powers of non-complete intersections
math.AC
For a commutative ring $R$ of characteristic $p$, let $\phi : R \to R$ be the Frobenius homomorphism and let $^{\phi^r}R$ denote the $R$-module structure on $R$ defined via the $r$-th power of the Frobenius. We show that the Tor functor against the Frobenius module, $\Tor^R_*(-, {^{\phi^r}}R)$, is rigid for a certain class of depth zero rings which includes rings that are not complete intersection. We also show that $\Tor^R_*(-, {^{\phi^r}}R)$ is not rigid (non-vacuously) when $\depth (R) >0$ and $r$ is large enough. This answers a question of Avramov and Miller: does rigidity of $\Tor^R_*(-, {^{\phi^r}}R)$ hold for non-complete intersections?
math
28
On Kummer extensions of the power series field
math.AC
In this paper we study the Kummer extensions of the power series field $K=k((X_1,...,X_n)$, where $k$ is an algebraically closed field of arbitrary characteristic.
math
29
Algebraic Generalized Power Series and Automata
math.AC
A theorem of Christol states that a power series over a finite field is algebraic over the polynomial ring if and only if its coefficients can be generated by a finite automaton. Using Christol's result, we prove that the same assertion holds for generalized power series (whose index sets may be arbitrary well-ordered sets of nonnegative rationals).
math
30
The ring of arithmetical functions with unitary convolution: Divisorial and topological properties
math.AC
We study the ring of arithmetical functions with unitary convolution, giving an isomorphism to a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell-Everett between the ring of arithmetical functions with Dirichlet convolution and the power series ring on countably many variables. We topologize it with respect to a natural norm, and shove that all ideals are quasi-finite. Some elementary results on factorization into atoms are obtained. We prove the existence of an abundance of non-associate regular non-units.
math
31
Local rings of countable Cohen--Macaulay type
math.AC
We prove (the excellent case of) Schreyer's conjecture that a local ring with countable Cohen--Macaulay type has at most a one-dimensional singular locus. Furthermore we prove that the localization of a Cohen-Macaulay local ring of countable CM type is again of countable CM type.
math
32
The ring of arithmetical functions with unitary convolution: General Truncations
math.AC
Let A denote the ring of arithmetical functions with unitary convolution, and let V be a finite subset of the positive integers having the property that for every v in V, all unitary divisors of v lie in V. We study the truncation A_V, an artinian monomial quotient of a polynomial ring in finitely many indeterminates, isomorphic to the ``Artinified'' Stanley-Reisner ring C[\bar{\Delta(V)}] of a certain simplicial complex \Delta(V).
math
33
Grothendieck-Serre formula and bigraded Cohen-Macaulay Rees algebras
math.AC
The Grothendieck-Serre formula for the difference between the Hilbert function and Hilbert polynomial of a graded algebra is generalized for bigraded standard algebras. This is used to get a similar formula for the difference between the Bhattacharya function and Bhattacharya polynomial of two m-primary ideals I and J in a local ring (A,m) in terms of local cohomology modules of Rees algebras of I and J. The cohomology of a variation of the Kirby-Mehran complex for bigraded Rees algebras is studied which is used to characterize the Cohen-Macaulay property of bigraded Rees algebra of I and J for two dimensional Cohen-Macaulay local rings.
math
34
Hilbert coefficients and depths of form rings
math.AC
We present short and elementary proofs of two theorems of Huckaba and Marley, while generalizing them at the same time to the case of a module. The theorems concern a characterization of the depth of the associated graded ring of a Cohen-Macaulay module, with respect to a Hilbert filtration, in terms of the Hilbert coefficient e_1. As an application, we derive bounds on the higher Hilbert coefficient e_i in terms of e_0.
math
35
Hasse-Schmidt Derivations and Coefficient Fields in Positive Characteristics
math.AC
We show how to express any Hasse-Schmidt derivation of an algebra in terms of a finite number of them under natural hypothesis. As an application, we obtain coefficient fields of the completion of a regular local ring of positive characteristic in terms of Hasse-Schmidt derivations
math
36
Hilbert coefficients and depth of fiber cones
math.AC
Criteria are given in terms of certain Hilbert coefficients for the fiber cone F(I) of an m-primary ideal I in a Cohen-Macaulay local ring (R,m) so that it is Cohen-Macaulay or has depth at least dim(R)-1. A version of Huneke's fundamental lemma is proved for fiber cones. S. Goto's results concerning Cohen-Macaulay fiber cones of ideals with minimal multiplicity are obtained as consequences.
math
37
Free resolutions fo rmultigraded modules: a generalization of Taylor's construction
math.AC
Let $Q=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ with the standard $N^n$-grading. Let $\phi$ be a morphism of finite free $N^n$-graded $Q$-modules. We translate to this setting several notions and constructions that appear originally in the context of monomial ideals. First, using a modification of the Buchsbaum-Rim complex, we construct a canonical complex $T_\bullet(\phi)$ of finite free $N^n$-graded $Q$-modules that generalizes Taylor's resolution. This complex provides a free resolution for the cokernel $M$ of $\phi$ when $\phi$ satisfies certain rank criteria. We also introduce the Scarf complex of $\phi$, and a notion of ``generic'' morphism. Our main result is that the Scarf complex of $\phi$ is a minimal free resolution of $M$ when $\phi$ is minimal and generic. Finally, we introduce the LCM-lattice for $\phi$ and establish its significance in determining the minimal resolution of $M$.
math
38
On symbolic powers of prime ideals
math.AC
Let (R,m) be a regular local ring with prime ideals p and q such that p+q is m-primary and dim(R/p)+dim(R/q)=dim(R). It has been conjectured by Kurano and Roberts that p^{(n)} \cap q \subseteq m^{n+1} for all positive integers n. We discuss this conjecture and related conjectures. In particular, we prove that this conjecture holds for all regular local rings if and only if it holds for all localizations of polynomial algebras over complete discrete valuation rings. In addition, we give examples showing that certain generalizations to nonregular rings do not hold.
math
39
Test ideals in diagonal hypersurface rings II
math.AC
Let $R=k[x_1, ..., x_n]/(x_1^d + ... + x_n^d)$, where $k$ is a field of characteristic $p$, $p$ does not divide $d$ and $n \geq 3$. We describe a method for computing the test ideal for these diagonal hypersurface rings. This method involves using a characterization of test ideals in Gorenstein rings as well as developing a way to compute tight closures of certain ideals despite the lack of a general algorithm. In addition, we compute examples of test ideals in diagonal hypersurface rings of small characteristic (relative to $d$) including several that are not integrally closed. These examples provide a negative answer to Smith's (2000, Comm. in Alg.) question of whether the test id eal in general is always integrally closed.
math
40
The F-signature and strong F-regularity
math.AC
We show that the F-signature of a local ring of characteristic p, defined by Huneke and Leuschke, is positive if and only if the ring is strongly F-regular.
math
41
Hypersurfaces of bounded Cohen--Macaulay type
math.AC
Let R = k[[x_0,...,x_d]]/(f), where k is a field and f is a non-zero non-unit of the formal power series ring k[[x_0,...,x_d]]. We investigate the question of which rings of this form have bounded Cohen--Macaulay type, that is, have a bound on the multiplicities of the indecomposable maximal Cohen--Macaulay modules. As with finite Cohen--Macaulay type, if the characteristic is different from two, the question reduces to the one-dimensional case: The ring R has bounded Cohen--Macaulay type if and only if R is isomorphic to k[[x_0,...,x_d]]/(g+x_2^2+...+x_d^2), where g is an element of k[[x_0,x_1]] and k[[x_0,x_1]]/(g) has bounded Cohen--Macaulay type. We determine which rings of the form k[[x_0,x_1]]/(g) have bounded Cohen--Macaulay type.
math
42
Extensions of a Dualizing Complex by its Ring: Commutative Versions of a Conjecture of Tachikawa
math.AC
Let $(R,\fm,k)$ be a commutative noetherian local ring with dualizing complex $\dua R$, normalized by $\Ext^{\depth(R)}_R(k,\dua R)\cong k$. Partly motivated by a long standing conjecture of Tachikawa on (not necessarily commutative) $k$-algebras of finite rank, we conjecture that if $\Ext^n_R(\dua R,R)=0$ for all $n>0$, then $R$ is Gorenstein, and prove this in several significant cases.
math
43
The ring of arithmetical functions with unitary convolution: the [n]-truncation
math.AC
We study a certain truncation of the ring of arithmetical functions with unitary convolution, consisting of functions vanishing on arguments >n. The truncations are artinian monomial quotients of a polynomial ring in finitely many indeterminates, and are isomorphic to the ``artinified'' Stanley-Reisner rings of certain simplicial complexes.
math
44
The Graph of Monomial Ideals
math.AC
There is a natural infinite graph whose vertices are the monomial ideals in a polynomial ring. The definition involves Gr\"obner bases or the action of an algebraic torus. We present algorithms for computing the (affine schemes representing) edges in this graph. We study the induced subgraphs on multigraded Hilbert schemes and on square-free monomial ideals. In the latter case, the edges correspond to generalized bistellar flips.
math
45
The first Mayr-Meyer ideal
math.AC
This paper gives a complete primary decomposition of the first, that is, the smallest, Mayr-Meyer ideal, its radical, and the intersection of its minimal components. The particular membership problem which makes the Mayr-Meyer ideals' complexity doubly exponential in the number of variables is here examined also for the radical and the intersection of the minimal components. It is proved that for the first Mayr-Meyer ideal the complexity of this membership problem is the same as for its radical. This problem was motivated by a question of Bayer, Huneke and Stillman.
math
46
The minimal components of the Mayr-Meyer ideals
math.AC
Mayr and Meyer found ideals $J(n,d)$ (in a polynomial ring in $10n+2$ variables over a field $k$ and generators of degree at most $d+2$) with ideal membership property which is doubly exponential in $n$. This paper is a first step in understanding the primary decomposition of these ideals: it is proved here that $J(n,d)$ has $nd^2 + 20$ minimal prime ideals. Also, all the minimal components are computed, and the intersection of the minimal components as well.
math
47
Tight closure commutes with localization in binomial rings
math.AC
It is proved that tight closure commutes with localization in any domain which has a module finite extension in which tight closure is known to commute with localization. It follows that tight closure commutes with localization in binomial rings, in particular in semigroup or toric rings.
math
48
Local cohomology modules with infinite dimensional socles
math.AC
Let T be a commutative Noetherian local ring of dimension at least two and R=T[x_1,...,x_n] a polynomial ring in n variables over T. Consider R as a graded ring with deg T = 0 and deg x_i = 1 for all i. Let I=R_+ and f a homogeneous polynomial whose coefficients form a system of parameters for T. We show that the socle of H^n_I(R/fR) is infinite dimensional, generalizing an example due to Hartshorne.
math
49
A cancellation theorem for ideals
math.AC
We prove cancellation theorems for special ideals in Gorenstein local rings. These theorems take the form that if KI is contained in JI, then K is contained in J.
math
50
Cofiniteness and associated primes of local cohomology modules
math.AC
Let R be a regular local ring of dimension d, I an ideal of R, and M a finitely generated R-module of dimension n. We prove that the set of associated primes of Ext^i_R(R/I,H^j_I(M)) is finite for all i and j in the following cases: (1) dim M\le 3; (2) dim R\le 4; (3) dim M/IM \le 2 and M satisfies Serre's condition S_{n-3}; (4) dim M/IM\le 3, R is unramified, and M is faithful and satisfies S_{n-3}. We also prove that if dim R/I\ge 2 and the punctured spectrum of R/I is disconnected then H^{d-1}_I(R) is not I-cofinite. This generalizes a result due to Huneke and Koh.
math
51
Tight closure in non-equidimensional rings
math.AC
An equidimensional local ring is F-rational if and only if one ideal generated by a system of parameters is tightly closed. The question of whether a non-equidimensional local ring can have a tightly closed ideal generated by a system of parameters has been a long-standing open problem, and for certain classes of non-equidimensional rings we prove that this is not possible. A key point is that tight closure has a colon capturing property in equidimensional rings that it does not have in non-equidimensional rings. We define a new closure operation, one that rectifies the absence of the colon capturing property of tight closure in non-equidimensional rings. This closure operation agrees with tight closure when the ring is equidimensional, and we prove that the F-rationality of a local ring is equivalent to a single system of parameters being closed with respect to this new closure operation.
math
52
Failure of F-purity and F-regularity in certain rings of invariants
math.AC
We demonstrate that the ring of invariants for the natural action of a subgroup G of GL_n(F_q) on a polynomial ring R=K[X_1,...,X_n] need not be F-pure. In these examples G is the symplectic group over a finite field, and the invariant subrings are always complete intersections by the work of Carlisle and Kropholler. These examples are of special interest from the point of view of studying the Frobenius closures and tight closures of ideals as contractions from certain extension rings: they provide instances when the socle element modulo an ideal generated by a system of parameters is forced into the expansion of the ideal to a module-finite extension ring which is a separable (in fact, Galois) extension. This element is also forced into the expanded ideal in a linearly disjoint purely inseparable extension since it is in the Frobenius closure of the ideal. The second part of this paper studies the alternating group A_n acting on the polynomial ring R by permuting the variables. We determine when the ring of invariants for this action is F-regular.
math
53
A computation of tight closure in diagonal hypersurfaces
math.AC
In the ring R=K[X,Y,Z]/(X^3+Y^3+Z^3), where K is a field of prime characteristic p other than 3, determining the tight closure of the ideal (X^2, Y^2, Z^2)R had existed as a classic example of the difficulty involved in tight closure computations. We settle this question, compute the Frobenius closure of this ideal, and generalize these results to the diagonal hypersurfaces K[X_1,...,X_n]/(X_1^n + ... + X_n^n).
math
54
Deformation of F-purity and F-regularity
math.AC
Hochster and Huneke showed that the property of F-regularity deforms for Gorenstein rings, i.e., if (R,m) is a Gorenstein local ring such that R/tR is F-regular for some nonzerodivisor t in m, then R is F-regular. This result was later extended to the case of Q-Gorenstein rings by Smith (for rings of characteristic zero) and Aberbach, Katzman, and MacCrimmon (for rings of positive characteristic). We investigate the deformation of strong F-regularity using an anti-canonical cover of R, i.e., a symbolic Rees algebra S = R + It + I^(2)t^2 + ..., where I is an inverse for the canonical module in the divisor class group of the ring R. We show that strong F-regularity deforms in the case that the symbolic powers I^(i) satisfy the Serre condition S_3 for all i > 0, and the ring S is Noetherian. We also construct examples which show that the property of F-purity does not deform.
math
55
F-regularity does not deform
math.AC
We show that the property of F-regularity does not deform, and thereby settle this longstanding open question in the theory of tight closure. Specifically, we construct a three dimensional domain R which is not F-regular (or even F-pure), but has a quotient R/tR which is F-regular. Similar examples are also constructed over fields of characteristic zero. Our work has an immediate application to questions related to flat base change: Hochster and Huneke showed that if (A,m) -> (R,n) is a flat map with regular generic and closed fibers, R is excellent, and A is weakly F-regular, then R is weakly F-regular. They asked if the condition on the fibers may be relaxed by requiring instead that the two fibers are F-regular. Our result shows that this condition is not enough, even if the base ring A is a discrete valuation ring: taking A=K[[t]] as a subring of the ring R above, the closed fiber R/tR and the generic fiber R[1/t] are F-regular, whereas R is not.
math
56
Extension of weakly and strongly F-regular rings by flat maps
math.AC
Let (R,m) -> (S,n) be a flat local homomorphism of excellent local rings. We investigate the conditions under which the weak or strong F-regularity of R passes to S. We show that is suffices that the closed fiber S/mS be Gorenstein and either F-finite (if R and S have a common test element), or F-rational (otherwise).
math
57
Primary Decomposition: Compatibility, Independence and Linear Growth
math.AC
For finitely generated modules $N \subsetneq M$ over a Noetherian ring $R$, we study the following properties about primary decomposition: (1) The Compatibility property, which says that if $\ass (M/N)=\{P_1, P_2, ..., P_s\}$ and $Q_i$ is a $P_i$-primary component of $N \subsetneq M$ for each $i=1,2,...,s$, then $N =Q_1 \cap Q_2 \cap ... \cap Q_s$; (2) For a given subset $X=\{P_1, P_2, ..., P_r \} \subseteq \ass(M/N)$, $X$ is an open subset of $\ass(M/N)$ if and only if the intersections $Q_1 \cap Q_2\cap ... \cap Q_r= Q_1' \cap Q_2' \cap ... \cap Q_r'$ for all possible $P_i$-primary components $Q_i$ and $Q_i'$ of $N\subsetneq M$; (3) A new proof of the `Linear Growth' property, which says that for any fixed ideals $I_1, I_2, ..., I_t$ of $R$, there exists a $k \in \mathbb N$ such that for any $n_1, n_2, ..., n_t \in \mathbb N$ there exists a primary decomposition of $I_1^{n_1}I_2^{n_2}... I_t^{n_t}M \subset M$ such that every $P$-primary component $Q$ of that primary decomposition contains $P^{k(n_1+n_2+...+n_t)}M$.
math
58
Unmixed local rings with minimal Hilbert-Kunz multiplicity are regular
math.AC
We give a new and simple proof that unmixed local rings having Hilbert-Kunz multiplicity equal to 1 must be regular.
math
59
A numerical characterization of the S_2-ification of a Rees algebra
math.AC
Let A be a local ring with maximal ideal m. For an arbitrary ideal I of A, we define the generalized Hilbert coefficients j_k(I) \in Z^{k+1} (k=0,1,...,dim A). When the ideal I is m-primary, j_k(I)=(0,...,0,(-1)^k e_k(I)), where e_k(I) is the classical k-th Hilbert coefficient of I. Using these coefficients, we give a numerical characterization of the homogeneous components of the S_2-ification of S=A[It,t^{-1}].
math
60
Lifting chains of prime ideals
math.AC
We give an elementary proof that for a ring homomorphism A -> B, satisfying the property that every ideal in A is contracted from B, the following property holds: for every chain of prime ideals p_0 \subset ... \subset p_r in A there exists a chain of prime ideals q_0 \subset ... \subset q_r in B such that q_i \cap A = p_i.
math
61
How to rescue solid closure
math.AC
We define a closure operation for ideals in a commutative ring which has all the good properties of solid closure (at least in the case of equal characteristic) but such that also every ideal in a regular ring is closed. This gives in particular a kind of tight closure theory in characteristic zero without referring to positive characteristic.
math
62
Notes on the behavior of the Ratliff-Rush filtration
math.AC
We establish new classes of Ratliff-Rush closed ideals and some pathological behavior of the Ratliff-Rush closure. In particular, Ratliff-Rush closure does not behave well under passage modulo superficial elements, taking powers of ideals, associated primes, leading term ideals, and the minimal number of generators. In contrast, the reduction number of the Ratliff-Rush filtration behaves better: it preserves some information on the reduction number of the ideal.
math
63
Linear bounds on growth of associated primes
math.AC
We find explicit bounds on the primary components and on the Castelnuovo-Mumford regularity of powers of monomial ideals. We also analyze the primary decompositions of Katzman's example.
math
64
Cohen--Macaulayness of tensor products
math.AC
Let $(R,\fm)$ be a commutative Noetherian local ring. Suppose that $M$ and $N$ are finitely generated modules over $R$ such that $M$ has finite projective dimension and such that $\Tor^R_i(M,N)=0$ for all $i>0$. The main result of this note gives a condition on $M$ which is necessary and sufficient for the tensor product of $M$ and $N$ to be a Cohen--Macaulay module over $R$, provided $N$ is itself a Cohen--Macaulay module.
math
65
On the embedded primes of the Mayr-Meyer ideals
math.AC
This paper investigates the doubly exponential ideal membership property of the Mayr-Meyer ideals from the point of view of their associated primes. A doubly-exponential upper bound on the set of associated primes is proved. In the paper a new family of ideals emerges which also has the doubly exponential ideal membership property. More on the new family can be found in author's paper "A new family of ideals with the doubly exponential ideal membership property".
math
66
A new family of ideals with the doubly exponential ideal membership property
math.AC
Mayr and Meyer found ideals with the doubly exponential ideal membership property. In the analysis of the associated primes of these ideals (in math.AC/0209344), a new family of ideals arose. This new family is presented and analyzed in this paper. It is proved that this new family also satisfies the doubly exponential ideal membership property. Furthermore, the set of associated primes of this family can be computed inductively.
math
67
Finiteness of $\bold{\bigcup_e \Ass F^e(M)}$ and its connections to tight closure
math.AC
The paper shows that if the set of associated primes of Frobenius powers of ideals or a closely related set of primes is finite then if tight closure does not commute with localisation one can find a counter-example where $R$ is complete local and we are localizing at a prime ideal $P \subset R$ with $\dim (R/P)=1$. If one assumes further that for any local ring $(R,m)$ of prime characteristic $p$ and every finitely generated $R$-module $\bar M$ the set $ \bigcup_e \Ass G^e (\bar M) $ has finitely many maximal elements and, in addition, for every $R$-module $\bar M$ there exists a positive integer $B>0$ such that $m^{qB}$ kills $\H_m^0(F^e(\bar M))$ (or $\H_m^0(G^e(\bar M))$) then it is shown tight closure commutes with localization. The author then produces an example of an ideal in an hypersurface whose union of sets associated primes of all its Frobenius powers form an infinite set.
math
68
Residues for Akizuki's one-dimensional local domain
math.AC
For a one-dimensional local domain $C_M$ constructed by Akizuki, we find residue maps which give rise to a local duality. The completion of $C_M$ is described using these residue maps.
math
69
Normal ideals of graded rings
math.AC
For a graded domain $R=k[X_0,...,X_m]/J$ over an arbitrary domain $k$, it is shown that the ideal generated by elements of degree $\geq mA$, where $A$ is the least common multiple of the weights of the $X_i$, is a normal ideal.
math
70
Tight closure and linkage classes in Gorenstein rings
math.AC
We study the relationship between the tight closure of an ideal and the sum of all ideals in its linkage class.
math
71
Vanishing of cohomology over Gorenstein rings of small codimension
math.AC
We prove that if M, N are finite modules over a Gorenstein local ring R of codimension at most 4, then the vanishing of Ext^n_R(M,N) for n\gg 0 is equivalent to the vanishing of Ext^n_R(N,M) for n\gg 0. Furthermore, if the completion of $R$ has no embedded deformation, then such vanishing occurs if and only if M or N has finite projective dimension.
math
72
A conjecture of Herzog and Conca on counting of paths
math.AC
A formula concerning counting of paths was conjectured by Herzog and Conca few years ago. Recently, Krattenthaler and Prohaska gave an affirmative answer to this conjecture. In this paper we generalize this formula.
math
73
Bounds for numbers of generators for a class of submodules of a finitely generated module
math.AC
The aim of this paper is to obtain a uniform bound for a certain class of submodules from the following theorem: Let $(R,\frak m)$ be a local ring, let $M$ be a finite $R$--module of dimension $d\ge 1$ and let $\frak q$ be an ideal of $R$ generated by a system of parameters on $M$. Let $N$ be a submodule of $M$ with $\depth M/N\ge d-1$. Then $\ell(N/\frak qN)\le\ell(M/\frak qM)$.
math
74
Links of prime ideals
math.AC
We exhibit the elementary but somewhat surprising property that most direct links of prime ideals in Gorenstein rings are equimultiple ideals. It leads to the construction of a bountiful set of Cohen--Macaulay Rees algebras.
math
75
Strongly Cohen-Macaulay ideals of small second analytic deviation
math.AC
We characterize the strongly Cohen-Macaulay ideals of second analytic deviation one in terms of depth properties of the powers of the ideal in the `standard range.' This provides an explanation of the behaviour of certain ideals that have appeared in the literature.
math
76
On residually S_2 ideals and projective dimension one modules
math.AC
We prove that certain modules are faithful. This enables us to draw consequences about the reduction number and the integral closure of some classes of ideals.
math
77
Reduction numbers and initial ideals
math.AC
The reduction number r(A) of a standard graded algebra A is the least integer k such that there exists a minimal reduction J of the homogeneous maximal ideal m of A such that Jm^k=m^{k+1}. Vasconcelos conjectured that the reduction number of A=R/I can only increase by passing to the initial ideal, i.e r(R/I)\leq r(R/in(I)). The goal of this note is to prove the conjecture.
math
78
Castelnuovo-Mumford regularity of products of ideals
math.AC
We discuss the behavior of the Castelnuovo-Mumford regularity under certain operations on ideals and modules, like products or powers. In particular, we show that reg(IM) can be larger than reg(M)+reg(I) even when I is an ideal of linear forms and M is a module with a linear resolution. On the other hand, we show that any product of ideals of linear forms has a linear resolution. We also discuss the case of polymatroidal ideals and show that any product of determinantal ideals of a generic Hankel matrix has a linear resolution.
math
79
A note on cancellation of reflexive modules
math.AC
We prove that cancellation of reflexive modules over affine rings holds under some restrictions. We construct examples to show that this is false even over polynomial rings without the extra assumptions.
math
80
The structure of the core of ideals
math.AC
The core of an $R$-ideal $I$ is the intersection of all reductions of $I$. This object was introduced by D. Rees and J. Sally and later studied by C. Huneke and I. Swanson, who showed in particular its connection to J. Lipman's notion of adjoint of an ideal. Being an a priori infinite intersection of ideals, the core is difficult to describe explicitly. We prove in a broad setting that: ${\rm core}(I)$ is a finite intersection of minimal reductions; ${\rm core}(I)$ is a finite intersection of general minimal reductions; ${\rm core}(I)$ is the contraction to $R$ of a `universal' ideal; ${\rm core}(I)$ behaves well under flat extensions. The proofs are based on general multiplicity estimates for certain modules.
math
81
Core and residual intersections of ideals
math.AC
D. Rees and J. Sally defined the core of an $R$-ideal $I$ as the intersection of all $($minimal$)$ reductions of $I$. However, it is not easy to give an explicit characterization of it in terms of data attached to the ideal. Until recently, the only case in which a closed formula was known is the one of integrally closed ideals in a two-dimensional regular local ring, due to C. Huneke and I. Swanson. The main result of this paper explicitly describes the core of a broad class of ideals with good residual properties in an arbitrary local Cohen-Macaulay ring. We also find sharp bounds on the number of minimal reductions that one needs to intersect to get the core.
math
82
Reduction numbers of links of irreducible varieties
math.AC
The reductions of an ideal $I$ give a natural pathway to the properties of $I$, with the advantage of having fewer generators. In this paper we primarily focus on a conjecture about the reduction exponent of links of a broad class of primary ideals. The existence of an algebra structure on the Koszul and Eagon-Northcott resolutions is the main tool for detailing the known cases of the conjecture. In the last section we relate the conjecture to a formula involving the length of the first Koszul homology modules of these ideals.
math
83
Core of projective dimension onemodules
math.AC
The core of a projective dimension one module is computed explicitly in terms of Fitting ideals. In particular, our formula recovers previous work by R. Mohan on integrally closed torsionfree modules over a two-dimensional regular local ring.
math
84
Generic Gaussian ideals
math.AC
The content of a polynomial $f(t)$ is the ideal generated by its coefficients. Our aim here is to consider a beautiful formula of Dedekind-Mertens on the content of the product of two polynomials, to explain some of its features from the point of view of Cohen-Macaulay algebras and to apply it to obtain some Noether normalizations of certain toric rings. Furthermore, the structure of the primary decomposition of generic products is given and some extensions to joins of toric rings are considered.
math
85
Q-Gorenstein splinter rings of characteristic p are F-regular
math.AC
An integral domain R is said to be a splinter if it is a direct summand, as an R-module, of every module-finite extension ring. Hochster's direct summand conjecture is precisely the conjecture that every regular local ring is a splinter. An integral domain containing the rational numbers is a splinter if and only if it is a normal ring, but the notion is more subtle for rings of positive characteristic: F-regular rings are splinters, and Hochster and Huneke proved that the converse is true for Gorenstein rings. We extend their result by showing that Q-Gorenstein splinters of positive characteristic are F-regular.
math
86
A generalized Dedekind-Mertens lemma and its converse
math.AC
We study content ideals of polynomials and their behavior under multiplication. We give a generalization of the Lemma of Dedekind-Mertens and prove the converse under suitable dimensionality restrictions.
math
87
Separable integral extensions and plus closure
math.AC
Let R be an excellent local domain of positive characteristic, and R^+ denote the integral closure of R in an algebraic closure of its fraction field. Hochster and Huneke proved that R^+ is a big Cohen-Macaulay algebra for R, and asked if there is a smaller R-algebra with the Cohen-Macaulay property. In this paper we establish the existence of a smaller big Cohen-Macaulay algebra which is, moreover, a separable extension.
math
88
Multi-symbolic Rees algebras and strong F-regularity
math.AC
Let I be a divisorial ideal of a strongly F-regular ring R. Watanabe asked if the symbolic Rees algebra R_s(I) is Cohen-Macaulay whenever it is Noetherian. We develop the notion of multi-symbolic Rees algebras, and use this to show that R_s(I) is indeed Cohen-Macaulay whenever a certain auxiliary ring is finitely generated over R.
math
89
Veronese subrings and tight closure
math.AC
We determine when graded rings have F-rational or F-regular Veronese subrings, and develop techniques of constructing F-rational rings which are not F-regular.
math
90
Gorenstein Dimensions under Base Change
math.AC
The so-called 'change-of-ring' results are well-known expressions which present several connections between projective, injective and flat dimensions over the various base rings. In this note we extend these results to the Gorenstein dimensions over Cohen-Macaulay local rings.
math
91
Intersection multiplicities over Gorenstein rings
math.AC
We construct a complex of free-modules over a Gorenstein ring R of dimension five, for which the Euler characteristic and Dutta multiplicity are different. This complex is the resolution of an R-module of finite length and finite projective dimension. As a consequence, the ring R has a nonzero Todd class tau_3(R) and a bounded free complex whose local Chern character does not vanish on this class. In the course of our work, we construct a module N of finite length and finite projective dimension over the hypersurface A=K[u,v,w,x,y,z]/(ux+vy+wz), such that the Serre intersection multiplicity of the modules N and A/(u,v,w)A is -2.
math
92
Todd classes of affine cones of Grassmannians
math.AC
A local ring R is said to be a Roberts ring if tau_R([R]) = [Spec R]_dim R, where tau_R is the Riemann-Roch map for Spec R. Such rings satisfy a vanishing theorem for the Serre intersection multiplicity, as was established by Paul Roberts in his proof of the Serre vanishing conjecture. It is known that complete intersections are Roberts rings, and the first author proved that a determinantal ring is a Roberts ring precisely if it is complete intersection. Let A_d(n) denote the affine cone of the Grassmannian G_d(n) under the Plucker embedding. We determine which of the rings A_d(n) are Roberts rings.
math
93
On a generalization of test ideals
math.AC
The test ideal $\tau(R)$ of a ring $R$ of prime characteristic is an important object in the theory of tight closure. In this paper, we study a generalization of the test ideal, which is the ideal $\tau(\a^t)$ associated to a given ideal $\a$ with rational exponent $t \ge 0$. We first prove a key lemma of this paper, which gives a characterization of the ideal $\tau(\a^t)$. As applications of this key lemma, we generalize the preceding results on the behavior of the test ideal $\tau(R)$. Moreover, we prove an analog of so-called Skoda's theorem, which is formulated algebraically via adjoint ideals by Lipman in his proof of the "modified Brian\c{c}on--Skoda theorem."
math
94
The Direct Summand Conjecture in Dimension Three
math.AC
The direct summand conjecture asserts that if R is a regular local ring and S is a module-finite R-algebra containing R, then R is a direct summand of S as an R-module. It was previously known to be true if R contains a field or if dim R is at most two. In this article, the result is demonstrated for mixed characteristic rings of dimension three. The proof is accomplished by showing that an extension of plus closure has the colon-capturing property in dimension three.
math
95
On the integral closure of ideals
math.AC
Among the several types of closures of an ideal $I$ that have been defined and studied in the past decades, the integral closure $\bar{I}$ has a central place being one of the earliest and most relevant. Despite this role, it is often a difficult challenge to describe it concretely once the generators of $I$ are known. Our aim in this note is to show that in a broad class of ideals their radicals play a fundamental role in testing for integral closedness, and in case $I\neq \bar{I}$, $\surd{I}$ is still helpful in finding some fresh new elements in $\bar{I}\setminus I$. Among the classes of ideals under consideration are: complete intersection ideals of codimension two, generic complete intersection ideals, and generically Gorenstein ideals.
math
96
Sally modules and associated graded rings
math.AC
We study the depth properties of the associated graded ring of an m-primary ideal I in terms of numerical data attached to the ideal I. We also find bounds on the Hilbert coefficients of I by means of the Sally module S_J(I) of I with respect to a minimal reduction J of I.
math
97
Test ideals and flat base change problems in tight closure theory
math.AC
Test ideals are an important concept in tight closure theory and their behavior via flat base change can be very difficult to understand. Our paper presents results regarding this behavior under flat maps with reasonably nice (but far from smooth) fibers. This involves analyzing, in depth, a special type of ideal of test elements, called the CS test ideal. Besides providing new results, the paper also contains extensions of a theorem by G. Lyubeznik and K. E. Smith on the completely stable test ideal and of theorems by F. Enescu and, independently, M. Hashimoto on the behavior of F-rationality under flat base change.
math
98
Tensor Products of Some Special Rings
math.AC
In this paper we solve a problem, originally raised by Grothendieck, on the properties, i.e. Complete intersection, Gorenstein, Cohen--Macaulay, that are conserved under tensor product of algebras over a field $k$.
math
99
Counting of paths and the multiplicity of determinantal rings
math.AC
In this paper, we derive several formulas of counting families of non-intersecting paths for two-sided ladder-shaped regions. As an application, we give a new proof to a combinatorial interpretation of Fibonacci numbers obtained by G. Andrews in 1974.
math