The Riemann-Roch theorem on a graph G is closely related to Alexander duality
in combinatorial commutive algebra. We study the lattice ideal given by chip
firing on G and the initial ideal whose standard monomials are the G-parking
functions. When G is a saturated graph, these ideals are generic and the Scarf
complex is a minimal free resolution. Otherwise, syzygies are obtained by
degeneration. We also develop a self-contained Riemann-Roch theory for artinian
monomial ideals.

We study Koszul homology over Gorenstein rings. If an ideal is strongly
Cohen-Macaulay, the Koszul homology algebra satisfies Poincar\'e duality. We
prove a version of this duality which holds for all ideals and allows us to
give two criteria for an ideal to be strongly Cohen-Macaulay. The first can be
compared to a result of Hartshorne and Ogus; the second is a generalization of
a result of Herzog, Simis, and Vasconcelos using sliding depth.

We compute the Betti numbers for all the powers of initial and final
lexsegment edge ideals. For the powers of the edge ideal of an anti-$d-$path,
we prove that they have linear quotients and we characterize the normally
torsion-free ideals. We determine a class of non-squarefree ideals, arising
from some particular graphs, which are normally torsion-free.

We construct several pairwise-incomparable bounds on the projective
dimensions of edge ideals, each of which is sharp for certain classes of
graphs. Our bounds use combinatorial properties of the associated graphs; in
particular we draw heavily from the topic of dominating sets. Through
Hochster's Formula, these bounds recover and strengthen existing results on the
homological connectivity of graph independence complexes.

We study notions of robustness of Markov kernels and probability distribution
of a system that is described by $n$ input random variables and one output
random variable. Markov kernels can be expanded in a series of potentials that
allow to describe the system's behaviour after knockouts. Robustness imposes
structural constraints on these potentials. Robustness of probability
distributions is defined via conditional independence statements. These
statements can be studied algebraically. The corresponding conditional
independence ideals are related to binary edge ideals.

Let $f:A\rightarrow B$ be a ring homomorphism and let $J$ be an ideal of $B$.
In this paper, we characterize $R\bowtie^fJ$ to be Von Neumann regular ring and
SFT ring, respectively.

Let $G$ be a finite group and $k$ be a field. Let $G$ act on the rational
function field $k(x_g:g\in G)$ by $k$-automorphisms defined by $g\cdot
x_h=x_{gh}$ for any $g,h\in G$. Noether's problem asks whether the fixed field
$k(G)=k(x_g:g\in G)^G$ is rational (i.e. purely transcendental) over $k$.
Theorem 1. If $G$ is a group of order $2^n$ ($n\ge 4$) and of exponent $2^e$
such that (i) $e\ge n-2$ and (ii) $\zeta_{2^{e-1}} \in k$, then $k(G)$ is
$k$-rational. Theorem 2. Let $G$ be a group of order $4n$ where $n$ is any
positive integer (it is unnecessary to assume that $n$ is a power of 2).

We demonstrate how primary decomposition of commutative monoid congruences
fails to capture the essence of primary decomposition in commutative rings by
exhibiting a more sensitive theory of mesoprimary decomposition of congruences,
complete with witnesses, associated prime objects, and an analogue of
irreducible decomposition called coprincipal decomposition. We lift the
combinatorial theory of mesoprimary decomposition to binomial ideals in monoid
algebras.

We establish restrictions on the Hilbert function of standard graded
Gorenstein algebras with only quadratic relations. Furthermore, we pose some
intriguing conjectures and provide evidence for them by proving them in some
cases using a number of different techniques, including liaison theory and
generic initial ideals.

We use a result of Hellus about generalized local duality to describe some
generalized Matlis duals for certain quasi-\F-modules. Furthermore, we apply
this description to obtain examples for non-artinian local cohomology modules
by the theory of \F-modules. In particular, we get a new view on Hartshorne's
counterexample for a conjecture by Grothendieck about the finiteness of
$Hom_R(R/I,H^i_I(R))$ for a noetherian local Ring $R$ and an ideal $I \subseteq
R$.

Let $R$ be a polynomial ring in $N$ variables over an arbitrary field $K$ and
let $I$ be an ideal of $R$ generated by $n$ polynomials of degree at most 2. We
show that there is a bound on the projective dimension of $R/I$ that depends
only on $n$, and not on $N$.

Given two determinantal rings over a field, we consider the diagonal ideal,
the kernel of the multiplication map. The defining equations of the special
fiber ring of the diagonal ideal are known. The special fiber ring of the
diagonal ideal is the homogeneous coordinate ring of join variety. When the
join variety is the whole space, we study the blowup along the diagonal. We
prove that the Rees algebra and the symmetric algebra of the diagonal ideal
coincide for some cases.

For a local ring $(R, \mf{m})$, $R$ has exactly two (respectively, three)
nontrivial ideals if and only if its maximal ideal $\mf{m}$ is cyclic with
nilpotency index three (respectively, four). In this paper, we determine the
structure of finite local rings which have at most three nontrivial ideals.

We continue the program started in \cite{M1} to understand the commutative
algebra of the projective coordinate rings of line bundles on the moduli
$\mathcal{M}_{C, \vec{p}}(SL_2(\C))$ of quasi-parabolic principal bundles on a
marked projective curve. We prove a general theorem about presentations of
these rings, which implies that for generic marked curves $(C, \vec{p})$ the
square of any effective line bundle has projective coordinate ring generated in
degree 1 with a presenting ideal generated in degree 3.

We complement two papers on supertropical valuation theory ([IKR1],[IKR2]) by
providing natural examples of m-valuations (= monoid valuations), after that of
supervaluations and transmissions between them. The supervaluations discussed
have values in totally ordered supertropical semirings, and the transmissions
discussed respect the orderings. Basics of a theory of such semirings and
transmissions are developed as far as needed.

This paper concerns the question of whether a more direct limit can be used
to obtain the limit Hilbert Kunz multiplicity, a possible candidate for a
characteristic zero Hilbert-Kunz multiplicity. The main goal is to establish an
affirmative answer for one of the main cases for which the limit Hilbert Kunz
multiplicity is even known to exist, namely that of graded ideals in the
homogeneous coordinate ring of smooth projective curves.

Let G be a graph and let I be its edge ideal. Our main result shows that the
sets of associated primes of the powers of I form an ascending chain. It is
known that the sets of associated primes of I(i) and intcl(I(i)) stabilize for
large i, where "intcl" denotes integral closure and I(i) denotes the i-th power
of I. We show that for edge ideals their corresponding stable sets are equal.
To show our main result we use a classical result of Berge from matching theory
and certain notions from combinatorial optimization.

In this paper we investigate the class of rigid monomial ideals. We give a
characterization of the minimal free resolutions of certain classes of these
ideals. Specifically, we show that the ideals in a particular subclass of rigid
monomial ideals are lattice-linear and thus their minimal resolution can be
constructed as a poset resolution. We then use this result to give a
description of the minimal free resolution of a larger class of rigid monomial
ideals by using $\mathcal{L}(n)$, the lattice of all lcm-lattices of monomial
ideals with $n$ generators.

Let $G$ be a finite graph on the vertex set $[d] = \{1, ..., d \}$ with the
edges $e_1, ..., e_n$ and $K[\tb] = K[t_1, ..., t_d]$ the polynomial ring in
$d$ variables over a field $K$. The edge ring of $G$ is the semigroup ring
$K[G]$ which is generated by those monomials $\tb^e = t_it_j$ such that $e =
\{i, j\}$ is an edge of $G$. Let $K[\xb] = K[x_1, ..., x_n]$ be the polynomial
ring in $n$ variables over $K$ and define the surjective homomorphism $\pi :
K[\xb] \to K[G]$ by setting $\pi(x_i) = \tb^{e_i}$ for $i = 1, ..., n$. The
toric ideal $I_G$ of $G$ is the kernel of $\pi$.

We define a family of homogeneous ideals with large projective dimension and
regularity relative to the number of generators and their common degree. This
family subsumes and improves upon constructions given in [Cav04] and [McC]. In
particular, we describe a family of three-generated homogeneous ideals in
arbitrary characteristic whose projective dimension grows asymptotically as
sqrt{d}^(sqrt(d) - 1).

Let $R$ be a standard graded Noetherian algebra over an Artinian local ring.
Motivated by the work of Achilles and Manaresi in intersection theory, we first
express the multiplicity of $R$ by means of local $j$-multiplicities of various
hyperplane sections. When applied to a homogeneous inclusion $A\subseteq B$ of
standard graded Noetherian algebras over an Artinian local ring, this formula
yields the multiplicity of $A$ in terms of that of $B$ and of local
$j$-multiplicities of hyperplane sections along ${\rm Proj}\,(B)$.

We study the degeneration problem for maximal Cohen-Macaulay modules and give
several examples of such degenerations. It is proved that such degenerations
over an even-dimensional simple hypersurface singularity of type $(A_n)$ are
given by extensions. We also prove that all extended degenerations of maximal
Cohen-Macaulay modules over a Cohen-Macaulay complete local algebra of finite
representation type are obtained by iteration of extended degenerations of
Auslander-Reiten sequences.

We present an axiomatic framework for the residue structures induced by
Prufer extensions with a stress upon the intimate connection between their
arithmetic and arboreal theoretic properties. The main result of the paper
provides an adjunction relationship between two naturally defined functors
relating Prufer extensions and superrigid directed commutative regular
quasi-semirings.

It is proved that when R is a local ring of positive characteristic, $\phi$
is its Frobenius endomorphism, and some non-zero finite R-module has finite
flat dimension or finite injective dimension for the R-module structure induced
through $\phi$, then R is regular.

We consider finite dimensional representations of the dihedral group $D_{2p}$
over an algebraically closed field of characteristic two where $p$ is an odd
integer and study the degrees of generating and separating polynomials in the
corresponding ring of invariants. We give an upper bound for the degrees of the
polynomials in a minimal generating set that does not depend on $p$ when the
dimension of the representation is sufficiently large. We also show that $p+1$
is the minimal number such that the invariants up to that degree always form a
separating set.

Inspired by the works in linkage theory of ideals, the concept of sliding
depth of extension modules is defined to prove the Cohen-Macaulyness of linked
module if the base ring is merely Cohen-Macaulay. Some relations between this
new condition and other module-theory conditions such as G-dimension and
sequentially Cohen-Macaulay are established. By the way several already known
theorems in linkage theory are improved or recovered by new approaches.

We generalize a Brian\c{c}on-Skoda type theorem first studied by Aberbach and
Huneke. With some conditions on a regular local ring $(R,\m)$ containing a
field, and an ideal $I$ of $R$ with analytic spread $\ell$ and a minimal
reduction $J$, we prove that for all $w \geq -1$, $ \bar{I^{\ell+w}} \subseteq
J^{w+1} \mathfrak{a} (I,J),$ where $\mathfrak{a}(I,J)$ is the coefficient ideal
of $I$ relative to $J$, i.e. the largest ideal $\mathfrak{b}$ such that
$I\mathfrak{b}=J\mathfrak{b}$. Previously, this result was known only for
$\m$-primary ideals.

For a standard graded algebra $R$, we consider embeddings of the the poset of
Hilbert functions of quotients of $R$ into the poset of ideals of $R$, as a way
of classification of Hilbert functions. There are examples of rings for which
such embeddings do not exist. We describe how the embedding can be lifted to
certain ring extensions, which is then used in the case of polarization and
distraction. A version of a theorem of Clements--Lindstr\"om is proved.

We study the dependence of graded Betti numbers of monomial ideals on the
characteristic of the base field. The examples we describe include bipartite
ideals, Stanley--Reisner ideals of vertex-decomposable complexes and ideals
with componentwise linear resolutions. We give a description of bipartite
graphs and, using discrete Morse theory, provide a way of looking at the
homology of arbitrary simplicial complexes through bipartite ideals.

This survey of methods surrounding lattice point methods for binomial ideals
begins with a leisurely treatment of the geometric combinatorics of binomial
primary decomposition. It then proceeds to three independent applications whose
motivations come from outside of commutative algebra: hypergeometric systems,
combinatorial game theory, and chemical dynamics. The exposition is aimed at
students and researchers in algebra; it includes many examples, open problems,
and elementary introductions to the motivations and background from outside of
algebra.

Let $G$ be a finite graph and $K[G]$ the edge ring of $G$. Based on the
technique of Gr\"obner bases and initial ideals, it will be proved that, given
integers $f$ and $d$ with $7 \leq f \leq d$, there exists a finite graph $G$ on
$[d]={1,...,d}$ with $\depth K[G] = f$ and with $\Krull-dim K[G] = d$.

In this paper we primarily study monomial ideals and their minimal free
resolutions by studying their associated LCM lattices. In particular, we
formally define the notion of coordinatizing a finite atomic lattice P to
produce a monomial ideal whose LCM lattice is P, and we give a complete
characterization of all such coordinatizations. We prove that all relations in
the lattice L(n) of all finite atomic lattices with n ordered atoms can be
realized as deformations of exponents of monomial ideals. We also give
structural results for L(n).

We prove that for m > 2, the m-th symbolic power of a Stanley-Reisner ideal
is Cohen-Macaulay if and only if the simplicial complex is a matroid.
Similarly, the m-th ordinary power is Cohen-Macaulay for some m > 2 if and only
if the complex is a complete intersection. These results solve several open
questions on the Cohen-Macaulayness of ordinary and symbolic powers of
Stanley-Reisner ideals. Moreover, they have interesting consequences on the
Cohen-Macaulayness of symbolic powers of facet ideals and cover ideals.

Let K be a field and let m_0,...,m_{n} be an almost arithmetic sequence of
positive integers. Let C be a toric variety in the affine (n+1)-space, defined
parametrically by x_0=t^{m_0},...,x_{n}=t^{m_{n}}. In this paper we produce a
minimal Gr\"obner basis for the toric ideal which is the defining ideal of C
and give sufficient and necessary conditions for this basis to be the reduced
Gr\"obner basis of C, correcting a previous work of \cite{Sen} and giving a
much simpler proof than that of \cite{Ayy}.

Starting from \cite{Ayy2} we compute the Groebner basis for the defining
ideal, P, of the monomial curves that correspond to arithmetic sequences, and
then give an elegant description of the generators of powers of the initial
ideal of P, inP. The first result of this paper introduces a procedure for
generating infinite families of Ratliff-Rush ideals, in polynomial rings with
multivariables, from a Ratliff-Rush ideal in polynomial rings with two
variables. The second result is to prove that all powers of inP are
Ratliff-Rush.

Let $R$ be a commutative Noetherian ring of Krull dimension $d$ admitting a
dualizing complex $D$ and let $\frak a$ be any ideal of $R$, we prove that
$\Gamma_{\frak a}(G)$ is Gorenstein injective for any Gorenstein injective
$R$-module $G$. Let $(R,\frak m)$ be a local ring and $M$ be a finitely
generated $R$-module.

The objective of this paper is to lay out the algebraic theory of
supertropical vector spaces and linear algebra, utilizing the key antisymmetric
relation of ``ghost surpasses.''Special attention is paid to the various
notions of ``base,'' which include d-base and s-base, and these are compared to
other treatments in the tropical theory. Whereas the number of elements in a
d-base may vary according to the d-base, it is shown that when an s-base
exists, it is unique up to permutation and multiplication by scalars, and can
be identified with a set of ``critical'' elements.

Let $G$ be a simple graph on the vertex set $V(G) = [n] = \{1,\ldots,n\}$ and
edge ideal $E(G)$. We consider the class of closed graphs. A closed graph is a
simple graph satisfying the following property: for all edges $\{i, j\}$ and
$\{k, \ell\}$ with $i < j$ and $k < \ell$ one has $\{j, \ell\}\in E(G)$ if $i =
k$, and $\{i, k\}\in E(G)$ if $j = \ell$. We state some criteria for the
closedness of a graph $G$ that do not depend necessarily from the labelling of
its vertex set.

In this paper we study monomial ideals attached to posets, introduce
generalized Hibi rings and investigate their algebraic and homological
properties. The main tools to study these objects are Groebner basis theory,
the concept of sortability due to Sturmfels and the theory of weakly
polymatroidal ideals.

Let $(R,\fm)$ be a local ring and $(-)^{\vee}$ denote the Matlis duality
functor. We investigate the relationship between Foxby equivalence and local
duality through generalized local cohomology. Assume that $R$ possesses a
normalized dualizing complex $D$ and $X$ and $Y$ are two homologically bounded
complexes of $R$-modules with finitely generated homology modules.

Stanley-Reisner rings of Buchsbaum* complexes are studied by means of their
quotients modulo a linear system of parameters. The socle of these quotients is
computed. Extending a recent result by Novik and Swartz for orientable homology
manifolds without boundary, it is shown that modulo a part of their socle these
quotients are level algebras. This provides new restrictions on the face
vectors of Buchsbaum* complexes.

We show in this paper that the principal component of the first order jet
scheme over the classical determinantal variety of m x n matrices of rank at
most 1 is arithmetically Cohen-Macaulay, by showing that an associated
Stanley-Reisner simplicial complex is shellable.

In this paper, we prove the vanishing theorem of Dual Bass numbers (Theorem
5.10).

The functional decomposition of polynomials has been a topic of great
interest and importance in pure and computer algebra and their applications.
The structure of compositions of (suitably normalized) polynomials f=g(h) over
finite fields is well understood in many cases, but quite poorly when the
degrees of both components are divisible by the characteristic p. This work
investigates the decomposition of polynomials whose degree is a power of p.

Let $R=\Bbbk[x_1,\..., x_n]$ and $M=R^s/I$ a multigraded squarefree module.
We discuss the construction of cochain complexes associated to $M$ and we show
how to interpret homological invariants of $M$ in terms of topological
computations. This is a generalization of the well studied case of squarefree
monomial ideals.

The algebra of basic covers of a graph G, denoted by \A(G), was introduced by
Juergen Herzog as a suitable quotient of the vertex cover algebra. In this
paper we show that if the graph is bipartite then \A(G) is a homogeneous
algebra with straightening laws and thus is Koszul. Furthermore, we compute the
Krull dimension of \A(G) in terms of the combinatorics of G. As a consequence
we get new upper bounds on the arithmetical rank of monomial ideals of pure
codimension 2.

Associated to any toric ideal are two special generating sets: the universal
Gr\"obner basis and the Graver basis. While the former is a subset of the
typically much larger Graver basis, there are cases for which the two sets
coincide. The most prominent examples among them are toric ideals of unimodular
matrices. Yet, a general classification of all matrices for which both sets
agree is far from known.

It is known that any torsion element in a lambda-ring is nilpotent. In this
note we deduce a sharp estimate for the nilpotence degree of such an element.

Let (R,m) be a local ring and U_R=Spec(R) -{m} be the punctured spectrum of
R. Gabber conjectured that if R is a complete intersection of dimension 3, then
the abelian group Pic(U_R) is torsion-free. In this note we prove Gabber's
statement for the hypersurface case. We also point out certain connections
between Gabber's Conjecture, Van den Bergh's notion of non-commutative crepant
resolutions and some well-studied questions in homological algebra over local
rings.

For valued fields $K$ of rank higher than 1, we describe how elements in the
henselization $K^h$ of $K$ can be approximated from within $K$; our result is a
handy generalization of the well-known fact that in rank 1, all of these
elements lie in the completion of $K$. We apply the result to show that if an
element $z$ algebraic over $K$ can be approximated from within $K$ in the same
way as an element in $K^h$, then $K(z)$ is not linearly disjoint from $K^h$
over $K$.

We give a criterion for maps on ultrametric spaces to be surjective and to
preserve spherical completeness. We show how Hensel's Lemma and the
multi-dimensional Hensel's Lemma follow from our result. We give an easy proof
that the latter holds in every henselian field. We also prove a basic
infinite-dimensional Implicit Function Theorem.

We discuss the role of additive polynomials and $p$-polynomials in the theory
of valued fields of positive characteristic and in their model theory. We
outline the basic properties of rings of additive polynomials and discuss
properties of valued fields of positive characteristic as modules over such
rings. We prove the existence of Frobenius-closed bases of algebraic function
fields $F|K$ in one variable and deduce that $F/K$ is a free module over the
ring of additive polynomials with coefficients in $K$.

We consider the Zariski space of all places of an algebraic function field
$F|K$ of arbitrary characteristic and investigate its structure by means of its
patch topology. We show that certain sets of places with nice properties (e.g.,
prime divisors, places of maximal rank, zero-dimensional discrete places) lie
dense in this topology. Further, we give several equivalent characterizations
of fields that are large, in the sense of F. Pop's Annals paper {\it Embedding
problems over large fields}.

This paper gives a survey on a valuation theoretical approach to local
uniformization in positive characteristic, the model theory of valued fields in
positive characteristic, and their connection with the valuation theoretical
phenomenon of defect.

We apply Miller's theory on multigraded modules over a polynomial ring to the
study of the Stanley depth of these modules. Several tools for Stanley's
conjecture are developed, and a few partial answers are given. For example, we
show that taking the Alexander duality twice (but with different "centers") is
useful for this subject. Generalizing a result of Apel, we prove that Stanley's
conjecture holds for the quotient by a cogeneric monomial ideals.

This paper concerned with the $w$-Jaffard domains and study this class of
domains in pullback constructions. We give new examples of $w$-Jaffard domains.
In particular we give an example of a $w$-Jaffard non-Jaffard domain. As
another application we established that for each pair of positive integers
$(n,m)$ with $n+1\leq m\leq 2n+1$, there is an (integrally closed) integral
domain $R$ such that $w$-$\dim(R)=n$ and $w[X]$-$\dim(R[X])=m$.

In this article, we find the equations defining the Rees algebra for certain
Monomial Curves explicitly and use them to prove that the blowup scheme is not
smooth. This proves a conjecture of Francia in affirmative, which says that a
dimension one prime in a regular local ring is a complete intersection if it
has a smooth blowup.

In this article, we prove that the Buchsbaum-Rim function
$\ell_A(\S_{\nu+1}(F)/N^{\nu+1})$ of a parameter module $N$ in $F$ is bounded
above by $e(F/N) \binom{\nu+d+r-1}{d+r-1}$ for every integer $\nu \geq 0$.
Moreover, it turns out that the base ring $A$ is Cohen-Macaulay once the
equality holds for some integer $\nu$. As a direct consequence, we observe that
the first Buchsbaum-Rim coefficient $e_1(F/N)$ of a parameter module $N$ is
always non-positive.

Let $R$ be a commutative ring with identity. We define a graph
$\Gamma_{\aut}(R)$ on $ R$, with vertices elements of $R$, such that any two
distinct vertices $x, y$ are adjacent if and only if there exists $\sigma \in
\aut$ such that $\sigma(x)=y$. The idea is to apply graph theory to study orbit
spaces of rings under automorphisms. In this article, we define the notion of a
ring of type $n$ for $n\geq 0$ and characterize all rings of type zero.

In this paper we study the Hilbert coefficients the fiber cone. We also
compare the depths of the fiber cone and the associated graded ring.

We show that if $(R, \m)$ is a Cohen-Macaulay local ring and $I$ is an ideal
of minimal mixed multiplicity, then $\depth G(I) \geq d- 1$ implies that
$\depth F(I) \geq d-1$. We use this to show that if $I$ is a contracted ideal
in a two dimensional regular local ring then $\depth R[It]-1= \depth G(I) =
\depth F(I)$. We also give an infinite class of ideals where $R[It]$ is
Cohen-Macaulay but $F(I)$ is not.

For a prime p>2 and q=p^n, we compute a finite generating set for the
SL_2(F_q)-invariants of the second symmetric power representation, showing the
invariants are a hypersurface and the field of fractions is a purely
transcendental extension of the coefficient field. As an intermediate result,
we show the invariants of the Sylow p-subgroups are also hypersurfaces.

We provide an axiomatic framework for working with a wide variety of closure
operations on ideals and submodules in commutative algebra, including notions
of reduction, independence, spread, and special parts of closures. This
framework is applied to tight, Frobenius, and integral closures. Applications
are given to evolutions and special Brian\c{c}on-Skoda theorems.

Let $I_G$ be the toric ideal of a graph $G$. We characterize in graph
theoretical terms the primitive, the minimal, the indispensable and the
fundamental binomials of the toric ideal $I_G$.

We define a notion of Gorenstein flat dimension for unbounded complexes over
left GF-closed rings. Over Gorenstein rings we introduce a notion of Gorenstein
cohomology for complexes; we also define a generalized Tate cohomology for
complexes over Gorenstein rings, and we show that there is a close connection
between the absolute, the Gorenstein and the generalized Tate cohomology.

Contravariantly finite resolving subcategories of the category of finitely
generated modules have been playing an important role in the representation
theory of algebras. In this paper we study contravariantly finite resolving
subcategories over commutative rings. The main purpose of this paper is to
classify contravariantly finite resolving subcategories over a henselian
Gorenstein local ring; in fact there exist only three ones.

The conjecture of Wolmer Vasconcelos on the vanishing of the first Hilbert
coefficient $e_1(Q)$ is solved affirmatively, where $Q$ is a parameter ideal in
a Noetherian local ring. Basic properties of the rings for which $e_1(Q)$
vanishes are derived. The invariance of $e_1(Q)$ for parameter ideals $Q$ and
its relationship to Buchsbaum rings are studied.

This paper takes a new look at ideals generated by 2x2 minors of 2x3 matrices
whose entries are powers of three elements not necessarily forming a regular
sequence. A special case of this are the ideals determining monomial curves in
three dimensional space, which were already studied by Herzog. In the broader
context studied here, these ideals are identified as Northcott ideals in the
sense of Vasconcelos, and so their liaison properties are displayed. It is
shown that they are set-theoretically complete intersections, revisiting the
work of Bresinsky and of Valla.

We show that, for the edge ideals of forests, the arithmetical rank equals
the projective dimension of the corresponding quotient ring.

Let $A$ be an integral domain. We study new conditions on families of
integral ideals of $A$ in order to get that $A$ is of $t$-finite character
(i.e., each nonzero element of $A$ is contained in finitely many $t$-maximal
ideals). We also investigate problems connected with the local invertibility of
ideals.

We study square-free S-modules with support on a simplicial graph, and
investigate an analogy between Cohen--Macaulay modules, locally of rank 1,
supported on a connected graph and line bundles on a curve. We use the
combinatorial structure of the graph to prove a corresponding Riemann-Roch
theorem, we study the Jacobian for a connected graph, and we study
Brill-Noether theory for 2-connected graphs.

In this paper we show that if the Stanley-Reisner ring of the simplicial
complex of independent sets of a bipartite graph $G$ satisfies Serre's
condition $S_2$, then $G$ is Cohen-Macaulay. As a consequence, the
characterization of Cohen-Macaulay bipartite graphs due to Herzog and Hibi
carries over this family of bipartite graphs. We check that the equivalence of
Cohen-Macaulay property and the condition $S_2$ is also true for chordal graphs
and we classify cyclic graphs with respect to the condition $S_2$.

We give an especially simple proof of a theorem in graph theory that forms
the key part of the solution to a problem in commutative algebra, on how to
characterize the integral closure of a polynomial ring generated by quadratic
monomials.

Let $X$ be a projective scheme over a field $k$ and let $A$ be the local ring
at the vertex of the affine cone of $X$ under some embedding
$X\hookrightarrow\mathbb{P}^n_k$. We prove that, when $\ch(k)>0$, the Lyubeznik
numbers $\lambda_{i,j}(A)$ are intrinsic numerical invariants of $X$, i.e.,
$\lambda_{i,j}(A)$ depend only on $X$, but not on the embedding.

Let $R=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ of
characteristic $p>0,$ and let $I=(f_1,...,f_s)$ be an ideal of $R.$ We prove
that every associated prime $P$ of $H^i_I(R)$ satisfies $\text{dim}R/P\geqslant
n-\sum\text{deg}f_i.$ In characteristic 0 the question is open.

We study the linear space generated by the multigraded Betti diagrams of
Z^n-graded artinian modules of codimension n whose resolutions become pure of a
given type when taking total degrees. We show that the multigraded Betti
diagram of the equivariant resolution constructed by D.Eisenbud, J.Weyman, and
the author, and all its twists, form a basis for this linear space.

We introduce and study a deformation of commutative polynomial algebras in
even numbers of variables. We also discuss some connections and applications of
this deformation to the generalized Laguerre orthogonal polynomials and the
interchanges of right and left total symbols of differential operators of
polynomial algebras. Furthermore, a more conceptual re-formulation for the
image conjecture [Z3] is also given in terms of the deformed algebras.
Consequently, the well-known Jacobian conjecture [Ke] is reduced to an open
problem on this deformation of polynomial algebras.

We define the symmetric Auslander category A^s(R) to consist of complexes of
projective modules whose left- and right-tails are equal to the left- and right
tails of totally acyclic complexes of projective modules.

The symmetric Auslander category contains A(R), the ordinary Auslander
category. It is well known that A(R) is intimately related to Gorenstein
projective modules, and our main result is that A^s(R) is similarly related to
what can reasonably be called Gorenstein projective homomorphisms. Namely,
there is an equivalence of triangulated categories:

The face ring of a simplicial complex modulo m generic linear forms is shown
to have finite local cohomology if and only if the link of every face of
dimension m or more is `nonsingular', i.e., has the homology of a wedge of
spheres of the expected dimension. This is derived from an enumerative result
for local cohomology of face rings modulo generic linear forms, as compared
with local cohomology of the face ring itself. The enumerative result is
generalized in slightly weaker form to squarefree modules.

Let $B$ be a Noetherian regular local ring with algebraically closed residue
field $k$, and $G\subset\Aut(B)$ a cyclic group of local automorphisms of prime
order acting trivially on $k$. Let $A$ be the ring of $G$-invariants of $B$,
assume that $A$ is Noetherian. We study conditions under which $A$ is again
regular; in particular, we prove that $B$ is a monogenous $A$-algebra if and
only if $G$ is a generalized pseudo-reflection.

Let K be a finite field with q elements and let X be a subset of a projective
space of dimension s-1, over the field K, which is parameterized by Laurent
monomials. We introduce the class of parameterized linear codes arising from X
and present algebraic methods to compute their dimensions and lengths. Using
tools from commutative algebra, along with the theory of lattices and finite
fields, we study the structure of the graded ideal I(X) generated by the
homogeneous polynomials of K[t1,...,ts] that vanish on X. It is shown that I(X)
is a lattice ideal.

As is well known, the geometry of the interpolation site of a multivariate
polynomial interpolation problem constitutes a dominant factor for the
structures of the interpolation polynomials. Solving interpolation problems on
interpolation sites with special geometries in theory may be a key step to the
development of general multivariate interpolation theory. In this paper, we
introduce a new type of 2-dimensional interpolation sites, tower interpolation
sites, whose associated degree reducing Lagrange interpolation monomial and
Newton bases w.r.t.

Let R be a two-dimensional regular local ring having an algebraically closed
residue field and let a be a complete ideal of finite colength in R. In this
article we investigate the jumping numbers of a by means of the dual graph of
the minimal log resolution of the pair (X,a). Our main result is a
combinatorial criterium for a positive rational number to be a jumping number.
In particular, we associate to each jumping number certain ordered tree
structures on the dual graph.

Exploiting symmetry in Groebner basis computations is difficult when the
symmetry takes the form of a group acting by automorphisms on monomials in
finitely many variables. This is largely due to the fact that the group
elements, being invertible, cannot preserve a term order. By contrast, inspired
by work of Aschenbrenner and Hillar, we introduce the concept of equivariant
Groebner basis in a setting where a_monoid_ acts by_homomorphisms_ on monomials
in potentially infinitely many variables.

A recent result of Boij-Soederberg and Eisenbud-Schreyer proves that the
Betti diagram of any graded module decomposes as a positive rational linear
combination of pure diagrams. We consider the follow-up question of whether
this numerical decomposition ever corresponds to an actual filtration of the
minimal free resolution itself. Our main result is an affirmative answer to
this question in many surprising cases. As applications of our technique, we
also obtain new results about the semigroup of Betti diagrams and about very
singular spaces of matrices.

Let $f:A \to B$ be a ring homomorphism and let $J$ be an ideal of $B$. In
this paper, we study the amalgamation of $A$ with $B$ along $J$ with respect to
$f$ (denoted by ${A\Join^fJ}$), a construction that provides a general frame
for studying the amalgamated duplication of a ring along an ideal, introduced
and studied by D'Anna and Fontana in 2007, and other classical constructions
(such as the $A+ XB[X]$, the $A+ XB[[X]]$ and the $D+M$ constructions). In
particular, we completely describe the prime spectrum of the amalgamated
duplication and we give bounds for its Krull dimension.

We derive a set of polynomial and quasipolynomial identities for degrees of
syzygies in the Hilbert series H(d^m;z) of nonsymmetric numerical semigroups
S(d^m) of arbitrary generating set of positive integers d^m={d_1,...,d_m},
m\geq 3. These identities were obtained by studying together the rational
representation of the Hilbert series H(d^m;z) and the quasipolynomial
representation of the Sylvester waves in the restricted partition function
W(s,d^m). In the cases of symmetric semigroups and complete intersections these
identities become more compact.

The F-threshold $c^J(\a)$ of an ideal $\a$ with respect to an ideal $J$ is a
positive characteristic invariant obtained by comparing the powers of $\a$ with
the Frobenius powers of $J$. We study a conjecture formulated in an earlier
paper by the same authors together with M. Musta\c{t}\u{a}, which bounds
$c^J(\a)$ in terms of the multiplicities $e(\a)$ and $e(J)$, when $\a$ and $J$
are zero-dimensional ideals, and $J$ is generated by a system of parameters. We
prove the conjecture when $\a$ and $J$ are generated by homogeneous systems of
parameters in a Noetherian graded $k$-algebra.

We give a complete factorization of the invariant factors of resultant
matrices built from birational parameterizations of rational plane curves in
terms of the singular points of the curve and their multiplicity graph. This
allows us to prove the validity of some conjectures about these invariants
stated by Chen, Wang and Liu in [J. Symbolic Comput. 43(2):92-117, 2008]. As a
byproduct, we also give a complete factorization of the D-resultant for
rational functions in terms of the similar data extracted from the
multiplicities.

We study the Hilbert function and the Hilbert series of the vertex cover
algebra $A(G)$, where $G$ is a Cohen-Macaulay bipartite graph.

We study the structure of Stanley-Reisner rings associated to cyclic
polytopes, using ideas from unprojection theory. Consider the boundary
simplicial complex Delta(d,m) of the d-dimensional cyclic polytope with m
vertices. We show how to express the Stanley-Reisner ring of Delta(d,m+1) in
terms of the Stanley-Reisner rings of Delta(d,m) and Delta(d-2,m-1). As an
application, we use the Kustin-Miller complex construction to identify the
minimal graded free resolutions of these rings. In particular, we recover
results of Schenzel, Terai and Hibi about their graded Betti numbers.

In this paper, we prove that a squarefree monomial ideal of height 2 whose
quotient ring is Cohen-Macaulay is set-theoretic complete intersection.

Let $R$ be a commutative ring with identity. For an $R$-module $M$, the
notion of strongly prime submodule of $M$ is defined. It is shown that this
notion of prime submodule inherits most of the essential properties of the
usual notion of prime ideal. In particular, the Generalized Principal Ideal
Theorem is extended to modules.

Border bases arise as a canonical generalization of Gr\"obner bases. We
extend our previous work [arXiv:0911.0859] to arbitrary order ideals: we devise
a polyhedral characterization of order ideals and border bases: order ideals
that support a border basis correspond one-to-one to integral points of the
order ideal polytope. In particular, we establish a crucial connection between
the ideal and its combinatorial structure.

The lemma given by Schmitt and Vogel is an important tool in the study of
arithmetical rank of squarefree monomial ideals. In this paper, we give a
Schmitt-Vogel type lemma for reductions as an analogous result.

Let $\fa$ be an ideal of a Noetherian local ring $(R,\fm)$ and $M$ a finitely
generated $R$-module. In this paper we introduce some criterions on
Artinianness of formal local cohomology, in particular vanishing and finiteness
of local cohomology modules. We find out the lower and upper bound for
Artinianness of formal local cohomology

We study $h$-vectors of simplicial complexes which satisfy Serre's condition
($S_r$). We say that a simplicial complex $\Delta$ satisfies Serre's condition
($S_r$) if $\tilde H_i(\lk_\Delta(F);K)=0$ for all faces $F \in \Delta$ and for
all $i < \min \{r-1,\dim \lk_\Delta(F)\}$, where $\lk_\Delta(F)$ is the link of
$\Delta$ with respect to $F$ and where $\tilde H_i(\Delta;K)$ is the reduced
homology groups of $\Delta$ over a field $K$.

We show that, under mild conditions, the (normalized) Frobenius splitting
numbers of a local ring of prime characteristic are lower semicontinuous.

In this paper, we partially confirm a conjecture, proposed by Cimpoeas,
Keller, Shen, Streib and Young, on the Stanley depth of squarefree Veronese
ideals $I_{n,d}$. They conjecture that, for positive integers $1 \le d \le n$,
$\sdepth (I_{n,d})= \lfloor \binom{n}{d+1}/\binom{n}{d} t\rfloor+d$. Herzog,
Vladoiu and Zheng established a connection between the Stanley depths of
quotients of monomial ideals and interval partitions of certain associated
partially ordered sets.

This paper studies Frobenius maps on injective hulls of residue fields of
complete local rings with a view toward providing constructive descriptions of
objects originating from the theory of tight closure. Specifically, the paper
describes algorithms for computing parameter test ideals, and tight closure of
certain submodules of the injective hull of residue fields of a class of
well-behaved rings which includes all quasi-Gorenstein complete local rings.

We study obstructions to existence of non-commutative crepant resolutions, in
the sense of Van den Bergh, over local complete intersections.

Let (T,m) be a complete local (Notherian) ring, C a finite set of pairwise
incomparable nonmaximal prime ideals of T, and p a nonzero element. We provide
necessary and sufficient conditions for T to be the completion of an integral
domain A containing the prime ideal pA whose formal fiber is semilocal with
maximal ideals the elements of C.

In this paper, it is proved that, if a toric ideal possesses a fundamental
binomial none of whose monomials is squarefree, then the corresponding
semigroup ring is not very ample. Moreover, very ample semigroup rings of
Lawrence type are discussed. As an application, we study very ampleness of
configurations arising from contingency tables.

In this paper we study the isomorphism classes of Artinian Gorenstein local
rings with socle degree three by means of Macaulay's inverse system. We prove
that their classification is equivalent to the projective classification of the
hypersurfaces of $\mathbb P ^{n }$ of degree three. This is an unexpected
result because it reduces the study of this class of local rings to the
homogeneous case. The result has applications in problems concerning the
punctual Hilbert scheme $Hilb_d (\mathbb P^n)$ and in relation to the problem
of the rationality of the Poincar\'e series of local rings.

We give a cohomological interpretation of orbit sets of unimodular rows of
length d+1 over smooth algebras of Krull dimension d.

Let $I\subset S=K[x_1,...,x_n]$ be a lexsegment edge ideal or the Alexander
dual of such an ideal. In both cases it turns out that the arithmetical rank of
$I$ is equal to the projective dimension of $S/I.$

We show that determinantal varieties defined by maximal minors of a generic
matrix have a non-commutative desingularization, in that we construct a maximal
Cohen-Macaulay module over such a variety whose endomorphism ring is
Cohen-Macaulay and has finite global dimension. In the case of the determinant
of a square matrix, this gives a non-commutative crepant resolution.

A new approach is established to computing the image of a rational map,
whereby the use of approximation complexes is complemented with a detailed
analysis of the torsion of the symmetric algebra in certain degrees. In the
case the map is everywhere defined this analysis provides free resolutions of
graded parts of the Rees algebra of the base ideal in degrees where it does not
coincide with the corresponding symmetric algebra. A surprising fact is that
the torsion in those degrees only contributes to the first free module in the
resolution of the symmetric algebra modulo torsion.

Let $G$ be a simple undirected graph on $n$ vertices. Francisco and Van Tuyl
have shown that if $G$ is chordal, then $\bigcap_{\{x_i,x_j\}\in E_G} <
x_i,x_j>$ is componentwise linear. A natural question that arises is for which
$t_{ij}>1$ the ideal $\bigcap_{\{x_i,x_j\}\in E_G}< x_i, x_j>^{t_{ij}}$ is
componentwise linear, if $G$ is chordal. In this report we show that
$\bigcap_{\{x_i,x_j\}\in E_G} < x_i, x_j>^{t}$ is componentwise linear for all
$n\geq 3$ and positive $t$, if $G$ is a complete graph.

We introduce a conjecture about constructing critically (s+1)-chromatic
graphs from critically s-chromatic graphs. We then show how this conjecture
implies that any unmixed height two square-free monomial ideal I, i.e., the
cover ideal of a finite simple graph, has the persistence property, that is,
Ass(R/I^s) \subseteq Ass(R/I^{s+1}) for all s >= 1. To support our conjecture,
we prove that the statement is true if we also assume that \chi_f(G), the
fractional chromatic number of the graph G, satisfies \chi(G) -1 < \chi_f(G) <=
\chi(G).

In this paper, we study the pair $(\GP(R),\GP(R)^{\perp})$ where $\GP(R)$ is
the class of all Gorenstein projective modules. We prove that it is complete
hereditary cotorsion theory provided $l.\Ggldim(R)<\infty$. We discuss also,
when every Gorenstein projective module is Gorenstein flat.

Numerical invariants of a minimal free resolution of a module $M$ over a
regular local ring $(R,\n)$ can be studied by taking advantage of the rich
literature on the graded case. The key is to fix suitable $\n$-stable
filtrations ${\mathbb M} $ of $M $ and to compare the Betti numbers of $M$ with
those of the associated graded module $ gr_{\mathbb M}(M). $ This approach has
the advantage that the same module $M$ can be detected by using different
filtrations on it.

We show that an Artinian quotient of K[x, y, z] by an ideal I generated by
powers of linear forms has the Weak Lefschetz property. If the syzygy bundle of
I is semistable this follows from results of Brenner-Kaid; our proof works
without this hypothesis, which typically does not hold.

We compute the derived functors of the third symmetric-power functor and
their cross-effects for certain values. These calculations match predictions by
the first named author and largely prove them in general.

In this note we characterize the (resp., weak) Gorenstein global dimension
for an arbitrary ring. Also, we extend the well-known Hilbert's syzygy Theorem
to the weak Gorenstein global dimension and we study the weak Gorenstein
homological dimensions of direct product of rings, which gives examples of
non-coherent rings of finite Gorenstein dimensions $>0$ and infinite classical
weak dimension.

Let $K$ be a field and $S=K[x_1,...,x_n]$. In 1982, Stanley defined what is
now called the Stanley depth of an $S$-module $M$, denoted $\sdepth(M)$, and
conjectured that $\depth(M) \le \sdepth(M)$ for all finitely generated
$S$-modules $M$. This conjecture remains open for most cases. However, Herzog,
Vladoiu and Zheng recently proposed a method of attack in the case when $M = I
/ J$ with $J \subset I$ being monomial $S$-ideals. Specifically, their method
associates $M$ with a partially ordered set.

Given an arbitrary graph G, we study its basic covers algebra, which is the
symbolic fiber cone of the Alexander dual of the edge ideal of G. Extending
results of Villarreal and Benedetti-Constantinescu-Varbaro, valid only in the
case when G is bipartite, we characterize in a combinatorial fashion the
situations when: 1) the basic covers algebra is a domain, and 2) it is a domain
and in addition (the edge ideal of) G is unmixed.

Let R be a local ring with maximal ideal m admitting a non-zero element
a\in\fm for which the ideal (0:a) is isomorphic to R/aR.

We study minimal free resolutions of finitely generated R-modules M, with
particular attention to the case when m^4=0. Let e denote the minimal number of
generators of m. If R is Gorenstein with m^4=0 and e\ge 3, we show that \Poi
MRt is rational with denominator \HH R{-t} =1-et+et^2-t^3, for each finitely
generated R-module M. In particular, this conclusion applies to generic
Gorenstein algebras of socle degree 3.

We study almost reverse lexicographic ideals in a polynomial ring over a
field of arbitrary characteristic. We give a criterion for a given sequence of
nonnegative integers to be the Hilbert function of an almost reverse
lexicographic ideal in the polynomial ring. Then it will be shown that every
Fr\"{o}berg sequence satisfies this criterion. Using this, we show that
Fr\"{o}berg conjecture and Moreno-Socias conjecture are all true for the case
that the base field is of characteristic 0.

Normaliz is a program for solving linear systems of inequalities. In this
paper we present the algorithms implemented in the program, starting with
version 2.0.

Lie antialgebras is a class of supercommutative algebras recently appeared in
symplectic geometry. We define the notion of enveloping algebra of a Lie
antialgebra and study its properties. We show that every Lie antialgebra is
canonically related to a Lie superalgebra and prove that its enveloping algebra
is a quotient of the enveloping algebra of the corresponding Lie superalgebra.

There are two seemingly unrelated classical objects associated to a
simplicial complex: a hierarchical model and a Stanley-Reisner ring. A
hierarchical model gives rise to a toric ideal, a relationship that is a staple
of algebraic statistics. In this note, we explore the connection between
degrees of Markov bases elements of the model and the rows of the Betti diagram
of the Stanley-Reisner ideal. We propose a precise conjecture, which we
establish in several cases, most notably for decomposable and
vertex-decomposable complexes.

This paper introduces and studies a particular subclasses of the class of
commutative rings with finite Gorenstein global (resp., weak) dimensions.

A finite poset $P$ is called "simplicial", if it has the smallest element
$0^$, and every interval $[0^, x]$ is a boolean algebra. The face poset of a
simplicial complex is a typical example. Stanley assigned the graded ring $A_P$
to $P$ generalizing the Stanley-Reisner ring of a simplicial complex. This ring
has been studied from both combinatorial and topological perspective. In this
paper, we will give a concise description of a dualizing complex of $A_P$. As
an application, we will construct the squarefree module theory over $A_P$.

Let R be an isolated hypersurface singularity, and let M and N be finitely
generated R-modules. As R is a hypersurface, the torsion modules of M against N
are eventually periodic of period two (i.e., Tor_i^R(M,N) is isomorphic to
Tor_{i+2}^R(M,N) for i sufficiently large). Since R has only an isolated
singularity, these torsion modules are of finite length for i sufficiently
large. The theta invariant of the pair (M,N) is defined by Hochster to be
length(Tor_{2i}^R(M,N)) - length(Tor_{2i+1}^R(M,N)) for i sufficiently large.
H.

It is shown that the diameter $\diam (H^1_\mfr(R/I))$ of the first local
cohomology module of a tetrahedral curve $C= C(a_1,...,a_6)$ can be explicitly
expressed in terms of the $a_i$ and is the smallest non-negative integer $k$
such that $\mfr^k H^1_\mfr(R/I)=0$. From that one can describe all
arithmetically Cohen-Macaulay or Buchsbaum tetrahedral curves.

Bounds for the maximum degree of a minimal Gr\"obner basis of simplicial
toric ideals with respect to the reverse lexicographic order are given. These
bounds are close to the bound stated in Eisenbud-Goto's Conjecture on the
Castelnuovo-Mumford regularity.

The theory of "subalgebra basis" analogous to standard basis (the
generalization of Gr\"{o}bner bases to monomial ordering which are not
necessarily well ordering \cite{GP1}.) for ideals in polynomial rings over a
field is developed. We call these bases "SASBI Basis" for "Subalgebra Analogue
to Standard Basis for Ideals". The case of global orderings, here they are
called "SAGBI Basis" for "Subalgebra Analogue to Gr\"{o}bner Basis for Ideals",
is treated in \cite{RS1}. Sasbi bases may be infinite.

In this paper, we compare the Gorenstein homological dimension of a ring $R$
and of its trivial ring extension by an module $E$.