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We introduce the notion of residual intersections of modules and prove their existence. We show that projective dimension one modules have Cohen-Macaulay residual intersections, namely they satisfy the relevant Artin-Nagata property. We then establish a formula for the core of orientable modules satisfying certain homological conditions, extending previous results of Corso, Polini, and Ulrich on the core of projective one modules. Finally, we provide examples of classes of modules that satisfy our assumptions.

A splinter is a notion of singularity that has seen numerous recent applications, especially in connection with the direct summand theorem, the mixed characteristic minimal model program, Cohen-Macaulayness of absolute integral closures and cohomology vanishing theorems. Nevertheless, many basic questions about these singularities remain elusive. One outstanding problem is whether the splinter property spreads from a point to an open neighborhood of a noetherian scheme. Our paper addresses this problem in prime characteristic, where we show that a locally noetherian scheme that has finite Frobenius or that is locally essentially of finite type over a quasi-excellent local ring has an open splinter locus. In particular, all varieties over fields of positive characteristic have open splinter loci. Intimate connections are established between the openness of splinter loci and $F$-compatible ideals, which are prime characteristic analogues of log canonical centers. We prove the surprising fact that for a large class of noetherian rings with pure (aka universally injective) Frobenius, the splinter condition is detected by the splitting of a single generically \'etale finite extension. We also show that for a noetherian $\textbf{N}$-graded ring over a field, the homogeneous maximal ideal detects the splinter property., Comment: Comments welcome, minor updates in v2

Codimension two Artinian algebras $A$ have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three AG algebras - the most promising results so far have concerned the weak Lefschetz property for such algebras. We here show that every standard-graded codimension three Artinian Gorenstein algebra $A$ having low maximum value of the Hilbert function - at most six - has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is greater than the socle degree of $A$, we show that $A$ is almost strong Lefschetz. This quite modest result is nevertheless arguably the most encompassing so far concerning the strong Lefschetz property for graded codimension three AG algebras.

Let $\mathbb{K}$ be a field and $I$ be a square-free monomial ideal in the polynomial ring $\mathbb{K}[x_1, \ldots, x_n]$. The Green-Lazarsfeld index, $\mathrm{index}(I)$, counts the number of steps to reach to a syzygy minimally generated by a nonlinear form in a graded minimal free resolution of $I$. In this paper, we study this invariant for $I$ and its powers from a combinatorial point of view. We characterize all square-free monomial ideals $I$ generated in degree $3$ such that $\mathrm{index}(I)>1$. Utilizing this result, we also characterize all square-free monomial ideals generated in degree $3$ such that $\mathrm{index}(I)>1$ and $\mathrm{index}(I^2)=1$. In case $n\leq5$, it is shown that $\mathrm{index}(I^k)>1$ for all $k$ if $I$ is any square-free monomial ideal with $\mathrm{index}(I)>1$., 14 pages, 4 figures

Given a simple undirected graph $G$ there is a simplicial complex $\mathrm{Ind}(G)$, called the independence complex, whose faces correspond to the independent sets of $G$. This is a well studied concept because it provides a fertile ground for interactions between commutative algebra, graph theory and algebraic topology. One of the line of research pursued by many authors is to determine the graph classes for which the associated independence complex is Cohen-Macaulay. For example, it is known that when $G$ is a chordal graph the complex $\mathrm{Ind}(G)$ is in fact vertex decomposable, the strongest condition in the Cohen-Macaulay ladder. In this article we consider a generalization of independence complex. Given $r\geq 1$, a subset of the vertex set is called $r$-independent if the connected components of the induced subgraph have cardinality at most $r$. The collection of all $r$-independent subsets of $G$ form a simplicial complex called the $r$-independence complex and is denoted by $\mathrm{Ind}_r(G)$. It is known that when $G$ is a chordal graph the complex $\mathrm{Ind}_r(G)$ has the homotopy type of a wedge of spheres. Hence it is natural to ask which of these complexes are shellable or even vertex decomposable. We prove, using Woodroofe's chordal hypergraph notion, that these complexes are always shellable when the underlying chordal graph is a tree. Further, using the notion of vertex splittable ideals we show that for caterpillar graphs the associated $r$-independence complex is vertex decomposable for all values of $r$. We also construct chordal graphs on $2r+2$ vertices such that their $r$-independence complexes are not sequentially Cohen-Macaulay for any $r \ge 2$., Final version. To appear in International Journal of Algebra and Computation

Let $ e_1, ..., e_c $ be positive integers and let $ Y \subseteq \mathbb{P}^n$ be the monomial complete intersection defined by the vanishing of $x_1^{e_1}, ..., x_c^{e_c}$. In this paper we study sharp upper bounds on the number of equations and syzygies of subschemes parametrized by the Hilbert scheme of points $Hilb^d(Y)$, and discuss applications to the Hilbert scheme of points $Hilb^d(X)$ of arbitrary complete intersections $X \subseteq \mathbb{P}^n$., Final version

We study exceptional minuscule Schubert varieties and provide the defining equations of the defining ideals of their intersection with the big open cell. We also provide the resolutions of these ideals and characterize some of them in terms of fundamental examples of ideals in the theory of Gorenstein ideals., Comment: 21 pages, comments welcome

We investigate the Castelnuovo-Mumford regularity and the multiplicity of the toric ring associated with a three-dimensional Ferrers diagram. In particular, in the rectangular case, we provide direct formulas for these two important invariants. Then, we compare these invariants for an accompanying pair of Ferrers diagrams under some mild conditions and bound the Castelnuovo-Mumford regularity for more general cases., Comment: Revised following the reviewer's suggestions. 27 pages and 6 figures. Comments are welcome

We give degree formulas for Grothendieck polynomials indexed by vexillary permutations and $1432$-avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which appeared in previous joint work of the authors with Y. Ren and A. St. Dizier. We apply our formulas to compute Castelnuovo-Mumford regularity of classes of generalized determinantal ideals. In particular, we give combinatorial formulas for the regularities of all one-sided mixed ladder determinantal ideals. We also derive formulas for the regularities of certain Kazhdan-Lusztig ideals, including those coming from open patches of Schubert varieties in Grassmannians. This provides a correction to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri (2015)., 26 pages, revised version corrects compilation errors affecting Examples 3.1, 3.3, and 7.1

Over a $d$-dimensional Cohen-Macaulay local ring admitting a canonical module the definable closure of the class of balanced big Cohen-Macaulay modules is $d$-cotilting and is the smallest such class containing the maximal Cohen-Macaulay modules. We describe its cotilting structure and contrast it to the largest $d$-cotilting class containing the maximal Cohen-Macaulay modules. The enables an implicit classification of all $d$-cotilting classes containing the maximal Cohen-Macaulay modules., Comment: 13 pages, typos corrected, slightly expanded

Given a generically finite local extension of valuation rings $V \subset W$, the question of whether $W$ is the localization of a finitely generated $V$-algebra is significant for approaches to the problem of local uniformization of valuations using ramification theory. Hagen Knaf proposed a characterization of when $W$ is essentially of finite type over $V$ in terms of classical invariants of the extension of associated valuations. Knaf's conjecture has been verified in important special cases by Cutkosky and Novacoski using local uniformization of Abhyankar valuations and resolution of singularities of excellent surfaces in arbitrary characteristic, and by Cutkosky for valuation rings of function fields of characteristic $0$ using embedded resolution of singularities. In this paper we prove Knaf's conjecture in full generality., Comments very welcome!

A set of polynomials $M$ is called a {\it submodule} of $\mathbb{C} [x_1, \dots, x_n ]$ if $M$ is a translation invariant linear subspace of $\mathbb{C} [x_1, \dots, x_n ]$. We present a description of the submodules of $\mathbb{C} [x,y]$ in terms of a special type of submodules. We say that the submodule $M$ of $\mathbb{C} [x,y]$ is an {\it L-module of order} $s$ if, whenever $F(x,y)=\sum_{n=0}^N f_n (x) \cdot y^n \in M$ is such that $f_0 =\ldots = f_{s-1}=0$, then $F=0$. We show that the proper submodules of $\mathbb{C} [x,y]$ are the sums $M_d +M$, where $M_d =\{ F\in \mathbb{C} [x,y] \colon \textit{deg}_x F, Comment: 22 pages

We introduce and study the set of radical stable operations of an integral domain $D$. We show that their set is a complete lattice that is the join-completion of the set of spectral semistar operations, and we characterize when every radical operation is spectral (under the hypothesis that $D$ is rad-colon coherent). When $D$ is a Pr\"ufer domain such that every set of minimal prime ideals is scattered, we completely classify stable semistar operations.

We show that the second power of the cover ideal of a path graph has linear quotients. To prove our result we construct a recursively defined order on the generators of the ideal which yields linear quotients. Our construction has a natural generalization to the larger class of chordal graphs. This generalization allows us to raise some questions that are related to some open problems about powers of cover ideals of chordal graphs., revised, 19 pages, 3 figures

In this article we investigate a pair of surjective local ring maps $S_1\leftarrow R\to S_2$ and their relation to the canonical projection $R\to S_1\otimes_R S_2$, where $S_1,S_2$ are Tor-independent over $R$. Our main result asserts a structural connection between the homotopy Lie algebra of $S:=S_1\otimes_R S_2$, denoted $\pi(S)$, in terms of those of $R,S_1$ and $S_2$. Namely, $\pi(S)$ is the pullback of (adjusted) Lie algebras along the maps $\pi(S_i)\to \pi(R)$ in various cases, including when the maps above have residual characteristic zero. Consequences to the main theorem include structural results on Andr\'{e}-Quillen cohomology, stable cohomology, and Tor algebras, as well as an equality relating the Poincar\'{e} series of the common residue field of $R,S_1,S_2$ and $S$., Comment: 20 pages. Corrected a mistake in 1.7; simplified and reorganized Sections 4 and 5

Let $\mathcal{P}$ be a simple thin polyomino and $\Bbbk$ a field. Let $R$ be the toric $\Bbbk$-algebra associated to $\mathcal{P}$. Write the Hilbert series of $R$ as $h_{R}(t)/(1-t)^{\dim(R)}$. We show that $$(-1)^{\left\lfloor{\frac{\mathrm{deg} h_R(t)}{2}}\right\rfloor}h_{R}(-1) \geq 0$$ if $R$ is Gorenstein. This shows that the Gorenstein rings associated to simple thin polyominoes satisfy the Charney-Davis conjecture., Comment: To appear in Communications in Algebra

We describe new classes of noetherian local rings $R$ whose finitely generated modules $M$ have the property that $Tor_i^R(M,M)=0$ for $i\gg 0$ implies that $M$ has finite projective dimension, or $Ext^i_R(M,M)=0$ for $i\gg 0$ implies that $M$ has finite projective dimension or finite injective dimension., 23 pages

Let $\mathbb{K}$ be a field and $R = \mathbb{K}[x_1, \ldots, x_n]$. We obtain an improved upper bound for asymptotic resurgence of squarefree monomial ideals in $R$. We study the effect on the resurgence when sum, product and intersection of ideals are taken. We obtain sharp upper and lower bounds for the resurgence and asymptotic resurgence of cover ideals of finite simple graphs in terms of associated combinatorial invariants. We also explicitly compute the resurgence and asymptotic resurgence of cover ideals of several classes of graphs. We characterize a graph being bipartite in terms of the resurgence and asymptotic resurgence of edge and cover ideals. We also compute explicitly the resurgence and asymptotic resurgence of edge ideals of some classes of graphs., Accepted for publication in Mathematische Zeitschrift(Math Z.)

This paper considers the problems of finite determinacy and approximation of flat analytic maps from germs of real or complex analytic spaces. It is shown that the flatness of analytic maps from germs of real or complex analytic spaces whose local rings are Cohen-Macaulay is finitely determined. Further, it is shown that flat maps from complete intersection and Cohen-Macaulay analytic germs can be arbitrarily closely approximated by algebraic and Nash maps respectively in such a way that the Hilbert-Samuel function of the special fibre is preserved. It is also proved that in the complex case the preservation of the Hilbert-Samuel function implies the preservation of Whitney's tangent cone., Comment: 13 pages. Substantial improvements in writing, existing theorems strengthened, several results added. arXiv admin note: text overlap with arXiv:1910.11498

This paper investigates when local cohomology modules have an annihilator that does not depend on the choice of an ideal. Takahashi classified the dominant resolving subcategories of the category of finitely generated modules over a commutative Noetherian ring. We show that his classification theorem describes annihilation results of local cohomology modules over a finite-dimensional ring with certain assumptions or a Cohen-Macaulay ring., Comment: 10 pages; In the second version, minor revisions were made, including Remark 5.5

Let $M$ and $N$ be finitely generated graded modules over a graded complete intersection $R$ such that $\operatorname{Ext}_R^i(M,N)$ has finite length for all $i\gg 0$. We show that the even and odd Hilbert polynomials, which give the lengths of $\operatorname{Ext}^i_R(M,N)$ for all large even $i$ and all large odd $i$, have the same degree and leading coefficient whenever the highest degree of these polynomials is at least the dimension of $M$ or $N$. Refinements of this result are given when $R$ is regular in small codimensions., Minor edits. Final version, to appear in Mathematische Zeitschrift; 23 pages

The Auslander-Reiten conjecture is a notorious open problem about the vanishing of Ext modules. In a Cohen-Macaulay complete local ring $R$ with a parameter ideal $Q$, the Auslander-Reiten conjecture holds for $R$ if and only if it holds for the residue ring $R/Q$. In the former part of this paper, we study the Auslander-Reiten conjecture for the ring $R/Q^\ell$ in connection with that for $R$, and prove the equivalence of them for the case where $R$ is Gorenstein and $\ell\le \dim R$. In the latter part, we generalize the result of the minimal multiplicity by J. Sally. Due to these two of our results, we see that the Auslander-Reiten conjecture holds if there exists an Ulrich ideal whose residue ring is a complete intersection. We also explore the Auslander-Reiten conjecture for determinantal rings., Comment: 19 pages. Errors are corrected

We show that the number of rational points of a subgroup inside a toric variety over a finite field defined by a homogeneous lattice ideal can be computed via Smith normal form of the matrix whose columns constitute a basis of the lattice. This generalizes and yields a concise toric geometric proof of the same fact proven purely algebraically by Lopez and Villarreal for the case of a projective space and a standard homogeneous lattice ideal of dimension one. We also prove a Nullstellensatz type theorem over a finite field establishing a one to one correspondence between subgroups of the dense split torus and certain homogeneous lattice ideals. As application, we compute the main parameters of generalized toric codes on subgroups of the torus of Hirzebruch surfaces, generalizing the existing literature., Comment: 26 pages, 5 figures, supported by TUBITAK 119F177, to appear in Finite Fields and their Applications

For any commutative ring $A$ we introduce a generalization of $S$--noetherian rings using a here\-ditary torsion theory $\sigma$ instead of a multiplicatively closed subset $S\subseteq{A}$. It is proved that totally noetherian w.r.t. $\sigma$ is a local property, and if $A$ is a totally noetherian ring w.r.t $\sigma$, then $\sigma$ is of finite type.

Let $I$ be an equidimensional ideal of a ring polynomial $R$ over $\mathbb{C}$ and let $J$ be its generic linkage. We prove that there is a uniform bound of the difference between the F-pure thresholds of $I_p$ and $J_p$ via the generalized Frobenius powers of ideals. This provides evidence that the F-pure threshold of an equidimensional ideal $I$ is less than that of its generic linkage. As a corollary we recover a result on log canonical thresholds of generic linkage by Niu., Minor revisions

Since Hochster's work, spectral spaces have attracted increasing interest. Through this note we intend to show that the set of proper ideals of a ring endowed with coarse lower topology is a spectral space., 4 pages

The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the singularities of the vanishing locus. Work of Musta\c{t}\u{a}, later extended by Bitoun and the third author, provides an analogous Bernstein-Sato theory for regular rings of positive characteristic. In this paper, we extend this theory to singular ambient rings in positive characteristic. We establish finiteness and rationality results for Bernstein-Sato roots for large classes of singular rings, and relate these roots to other classes of numerical invariants defined via the Frobenius map. We also obtain a number of new results and simplified arguments in the regular case., Comment: v2: minor fixes after referee report; 58 pages; comments appreciated

We introduce the notion of Burch submodules and weakly $\mathfrak m$-full submodules of modules over local rings and study their properties. One of our main results shows that Burch submodules satisfy 2-Tor rigid and test property. We also show that over a local ring $(R,\mathfrak m)$ a submodule $M$ of a finitely generated $R$-module $X$, such that either $M=\mathfrak m X$ or $M(\subseteq \mathfrak m X$) is weakly $\mathfrak m$-full in $X$, is 1-Tor rigid and a test module provided that $X$ is faithful (and $X/M$ has finite length when $M$ is weakly $\mathfrak m$-full). As an application, we give a new class of rings such that a conjecture of Huneke and Wiegand is affirmative over them., Revised version; 31 pages. Key word Ulrich module and some new results added. Accepted at Illinois Journal of Mathematics

Let $D$ be an integral domain and $\Gamma$ be a torsion-free commutative cancellative (additive) semigroup with identity element and quotient group $G$. In this paper, we show that if char$(D)=0$ (resp., char$(D)=p>0$), then $D[\Gamma]$ is a weakly Krull domain if and only if $D$ is a weakly Krull UMT-domain, $\Gamma$ is a weakly Krull UMT-monoid, and $G$ is of type $(0,0,0, \dots )$ (resp., type $(0,0,0, \dots )$ except $p$). Moreover, we give arithmetical applications of this result.

We study the upper bound of the colength of trace of the canonical module in one-dimensional Cohen-Macaulay rings. We answer the two questions posed by Herzog-Hibi-Stamate and Kobayashi., 9 pages

Let $R=\mathbb{K}[X_1, \ldots , X_n ]$ be a polynomial ring over a field $\mathbb{K}$. We introduce an endomorphism $\mathcal{F}^{[m]}: R \rightarrow R $ and denote the image of an ideal $I$ of $R$ via this endomorphism as $I^{[m]}$ and call it to be the $m$ \textit{-th square power} of $I$. In this article, we study some homological invariants of $I^{[m]}$ such as regularity, projective dimension, associated primes and depth for some families of ideals e.g. monomial ideals., 9 pages

In this article we study the defining ideal of Rees algebras of ideals of star configurations. We characterize when these ideals are of linear type and provide sufficient conditions for them to be of fiber type. In the case of star configurations of height two, we give a full description of the defining ideal of the Rees algebra, by explicitly identifying a minimal generating set., 32 pages. In this updated version, we fixed typos and made a few changes in the exposition of Sections 2, 3 and 6

In this paper, we study the Hankel edge ideals of graphs. We determine the minimal prime ideals of the Hankel edge ideal of labeled Hamiltonian and semi-Hamiltonian graphs, and we investigate radicality, being a complete intersection, almost complete intersection and set theoretic complete intersection for such graphs. We also consider the Hankel edge ideal of trees with a natural labeling, called rooted labeling. We characterize such trees whose Hankel edge ideal is a complete intersection, and moreover, we determine those whose initial ideal with respect to the reverse lexicographic order satisfies this property., 15 pages, 6 figures

In this paper we define and explore the analytic spread $\ell(\mathcal I)$ of a filtration in a local ring. We show that, especially for divisorial and symbolic filtrations, some basic properties of the analytic spread of an ideal extend to filtrations, even when the filtration is non Noetherian. We also illustrate some significant differences between the analytic spread of a filtration and the analytic spread of an ideal with examples. In the case of an ideal $I$, we have the classical bounds $\mbox{ht}(I)\le\ell(I)\le \dim R$. The upper bound $\ell(\mathcal I)\le \dim R$ is true for filtrations $\mathcal I$, but the lower bound is not true for all filtrations. We show that for the filtration $\mathcal I$ of symbolic powers of a height two prime ideal $\mathfrak p$ in a regular local ring of dimension three (a space curve singularity), so that $\mbox{ht}(\mathcal I) =2$ and $\dim R=3$, we have that $0\le \ell(\mathcal I)\le 2$ and all values of 0,1 and 2 can occur. In the cases of analytic spread 0 and 1 the symbolic algebra is necessarily non-Noetherian. The symbolic algebra is non-Noetherian if and only if $\ell(\mathfrak p^{(n)})=3$ for all symbolic powers of $\mathfrak p$ and if and only if $\ell(\mathcal I_a)=3$ for all truncations $\mathcal I_a$ of $\mathcal I$., Comment: 23 pages. Final version. Theorem 4.6 is a little stronger

We study singularity categories of exact categories with a focus on those associated to a complete hereditary cotorsion pair. As an application we identify a non-affine analogue of the singularity category of a Gorenstein local ring; with this Buchweitz's classic equivalence of three categories over Gorenstein local rings has been generalized to schemes, a project started by Murfet and Salarian more than ten years ago. As another application we use the framework to characterize rings of finite finitistic dimension., Minor revisions. Final version, to appear in Q.J. Math; 24 pp

Given a square-free monomial ideal $I$ in a polynomial ring $R$ over a field $\mathbb{K}$, one can associate it with its LCM-lattice and its hypergraph. In this short note, we establish the connection between the LCM-lattice and the hypergraph, and in doing so we provide a sufficient condition for removing higher dimension edges of the hypergraph without impacting the projective dimension of the square-free monomial ideal. We also offer algorithms to compute the projective dimension of a class of square-free monomial ideals built using the new result and previous results of Lin-Mantero., The paper is split into two via the suggestion of the editor. The second part will be submitted soon

Let $R$ be a general ring. Duality pairs of $R$-modules were introduced by Holm-Jorgensen. Most examples satisfy further properties making them what we call semi-complete duality pairs in this paper. We attach a relative theory of Gorenstein homological algebra to any given semi-complete duality pair $\mathfrak{D} = (\mathcal{L},\mathcal{A})$. This generalizes the homological theory of the AC-Gorenstein modules defined by Bravo-Gillespie-Hovey, and we apply this to other semi-complete duality pairs. The main application is that the Ding injective modules are the right side of a complete (perfect) cotorsion pair, over any ring. Completeness of the Gorenstein flat cotorsion pair over any ring arises from the same duality pair., Updated references and Remarks

In this article we study a theory of support varieties over a skew complete intersection $R$, i.e. a skew polynomial ring modulo an ideal generated by a sequence of regular normal elements. We compute the derived braided Hochschild cohomology of $R$ relative to the skew polynomial ring and show its action on $\mathrm{Ext}_R(M,N)$ is noetherian for finitely generated $R$-modules $M$ and $N$ respecting the braiding of $R$. When the parameters defining the skew polynomial ring are roots of unity we use this action to define a support theory. In this setting applications include a proof of the Generalized Auslander-Reiten Conjecture and that $R$ possesses symmetric complexity.

The first goal of the present paper is to study the class groups of the edge rings of complete multipartite graphs, denoted by $\Bbbk[K_{r_1,\ldots,r_n}]$, where $1 \leq r_1 \leq \cdots \leq r_n$. More concretely, we prove that the class group of $\Bbbk[K_{r_1,\ldots,r_n}]$ is isomorphic to $\mathbb{Z}^n$ if $n =3$ with $r_1 \geq 2$ or $n \geq 4$, while it turns out that the excluded cases can be deduced into Hibi rings. The second goal is to investigate the special class of divisorial ideals of $\Bbbk[K_{r_1,\ldots,r_n}]$, called conic divisorial ideals. We describe conic divisorial ideals for certain $K_{r_1,\ldots,r_n}$ including all cases where $\Bbbk[K_{r_1,\ldots,r_n}]$ is Gorenstein. Finally, we give a non-commutative crepant resolution (NCCR) of $\Bbbk[K_{r_1,\ldots,r_n}]$ in the case where it is Gorenstein., 21 pages

Let R be a commutative noetherian ring. The notion of n-wide subcategories of Mod R is introduced and studied in Matsui-Nam-Takahashi-Tri-Yen in relation to the cohomological dimension of a specialization-closed subset of Spec R. In this paper, we introduce the notions of n-coherent subsets of Spec R and n-uniform subcategories of D(Mod R), and explore their interactions with n-wide subcategories of Mod R. We obtain a commutative diagram which yields filtrations of subcategories of Mod R, D(Mod R) and subsets of Spec R and complements classification theorems of subcategories due to Gabriel, Krause, Neeman, Takahashi and Angeleri Hugel-Marks-Stovicek-Takahashi-Vitoria., Comment: 17 pages, to appear in IMRN

We study how Betti numbers of ideals in a local ring change under small perturbations. Given $p\in\mathbb N$ and given an ideal $I$ of a Noetherian local ring $(R,\mathfrak m)$, our main result states that there exists $N>0$ such that if $J$ is an ideal with $I\equiv J\bmod \mathfrak m^N$ and with the same Hilbert function as $I$, then the Betti numbers $\beta_i^R(R/I)$ and $\beta_i^R(R/J)$ coincide for $0\le i\le p$. Moreover, we present several cases in which an ideal $J$ such that $I \equiv J \bmod \mathfrak m^N$ is forced to have the same Hilbert function as $I$, and therefore the same Betti numbers., Comment: 13 pages

We construct Grassmannian categories of infinite rank, providing an infinite analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. Each Grassmannian category of infinite rank is given as the category of graded maximal Cohen-Macaulay modules over a certain hypersurface singularity. We show that generically free modules of rank $1$ in a Grassmannian category of infinite rank are in bijection with the Pl\"ucker coordinates in an appropriate Grassmannian cluster algebra of infinite rank. Moreover, this bijection is structure preserving, as it relates rigidity in the category to compatibility of Pl\"ucker coordinates. Along the way, we develop a combinatorial formula to compute the dimension of the $\mathrm{Ext}^1$-spaces between any two generically free modules of rank $1$ in the Grassmannian category of infinite rank., Comment: V2: revision, title change. Revised version to appear in IMRN

We introduce and study a new class of morphisms which includes morphisms represented by monomorphisms in the sense of Auslander and Bridger. As an application, we give not only an extension of Kato's theorem on morphisms represented by monomorphisms, but also a common generalization of several results due to Auslander and Bridger that describe relationships between torsionfreeness and the grades of Ext modules., 15 pages

We show that the shortest nonzero polynomials vanishing on bounded-rank matrices and skew-symmetric matrices are the determinants and Pfaffians characterising the rank. Algebraically, this means that in the ideal generated by all $t$-minors or $t$-Pfaffians of a generic matrix or skew-symmetric matrix one cannot find any polynomial with fewer terms than those determinants or Pfaffians, respectively, and that those determinants and Pfaffians are essentially the only polynomials in the ideal with that many terms. As a key tool of independent interest, we show that the ideal of a sufficiently general $t$-dimensional subspace of an affine $n$-space does not contain polynomials with fewer than $t+1$ terms., 13 pages, comments welcome, v2: 15 pages, final version as in Bulletin LMS

We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We investigate far-flung Gorenstein rings in relation with the endomorphism algebras of the maximal ideals and numerical semigroup rings. We show that the solution of the Rohrbach problem in additive number theory provides an upper bound for the multiplicity of far-flung Gorenstein numerical semigroup rings. Reflexive modules over far-flung Gorenstein rings are also studied., 15 pages. Comments are welcome!

There are two motivation questions in \cite{MTS, MTS1} about Castelnuovo-Mumford regularity and vertex decomposable of simple graph $G$. In this paper, we disprove the questions by providing of two counterexamples., Comment: 6 pages. to appear in International Electronic Journal of Algebra

In this paper we explore the almost Cohen-Macaulayness of the associated graded ring of stretched $\mathfrak{m}$-primary ideals with small first Hilbert coefficient in a Cohen-Macaulay local ring $(A,\mathfrak{m})$. In particular, we explore the structure of stretched $\mathfrak{m}$-primary ideals satisfying the equality $\mathrm{e}_1(I)=\mathrm{e}_0(I)-\ell_A(A/I)+4$ where $\mathrm{e}_0(I)$ and $\mathrm{e}_1(I)$ denote the multiplicity and the first Hilbert coefficient respectively., 17 pages. arXiv admin note: substantial text overlap with arXiv:2109.12918

It is known that the numerical invariants Betti numbers and Bass numbers are worthwhile tools for decoding a large amount of information about modules over commutative rings. We highlight this fact, further, by establishing some criteria for certain semidualizing complexes via their Betti and Bass numbers. Two distinguished types of semidualizing complexes are the shifts of the underlying rings and dualizing complexes. Let $C$ be a semidualizing complex for an analytically irreducible local ring $R$ and set $n:=\sup C$ and $d:=\dim_RC$. We show that $C$ is quasi-isomorphic to a shift of $R$ if and only if the $n$th Betti number of $C$ is one. Also, we show that $C$ is a dualizing complex for $R$ if and only if the $d$th Bass number of $C$ is one., Comment: A slightly shorter version of this paper appears in Communications in Algebra

Given a commutative local ring $(R,\mathfrak m)$ and an ideal $I$ of $R$, a family of quotients of the Rees algebra $R[It]$ has been recently studied as a unified approach to the Nagata's idealization and the amalgamated duplication and as a way to construct interesting examples, especially integral domains. When $R$ is noetherian of prime characteristic, we compute the Hilbert-Kunz function of the members of this family and, provided that either $I$ is $\mathfrak{m}$-primary or $R$ is regular and F-finite, we also find their Hilbert-Kunz multiplicity. Some consequences and examples are explored., Comment: To appear in the Proceedings of the American Mathematical Society

A complex $X$ is called Ding injective if there exists an exact sequence of injective complexes $\ldots \rightarrow E_1 \rightarrow E_0 \rightarrow E_{-1} \rightarrow \ldots$ such that $X = Ker(E_0 \rightarrow E_{-1})$, and the sequence remains exact when the functor $Hom(A,-)$ is applied to it, for any $FP$-injective complex $A$. We prove that, over any ring $R$, a complex is Ding injective if and only if it is a complex of Ding injective modules. We use this to show that the class of Ding injective complexes is enveloping over any ring.