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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 minim... | 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 mini... | 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... | 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 algorit... | 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 arbitr... | 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 f... | 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... | 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 determ... | 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... | 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 generate... | 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 ... | 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 gi... | 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-commutat... | 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... | 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 ... | 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... | 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 ... | 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 ... | 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 w... | 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 c... | 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 ... | 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 va... | 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, ... | 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 ... | 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 coe... | 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 fib... | 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 Buc... | 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 conj... | 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 develop... | 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... | 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... | 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... | 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 sch... | 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 th... | 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... | 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... | 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_... | 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-e... | 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. T... | 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, an... | 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 zer... | 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... | 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_... | 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 ... | 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 =... | 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... | 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... | 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 produ... | 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 membe... | 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... | 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 $\dim
(R/P)=1$.
... | 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 h... | 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 $\... | 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... | 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 ... | 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... | 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 clos... | 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-Northcot... | 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 no... | 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... | 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 e... | 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 gener... | 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... | 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 interse... | 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, w... | 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
characterist... | 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.... | 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 te... | 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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