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0 | 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 |
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