Papers

Let $R$ be a Noetherian standard $\mathbb{N}$-graded ring, and let $I_1,\ldots,I_r$ be homogeneous ideals of $R$. Let $L,M$, and $N$ be finitely generated $\mathbb{Z}$-graded $R$-modules with $N\subseteq M$. For $\underline{n} :=(n_1,\dots,n_r)\in\mathbb N^r$, set ${\bf I}^{\underline{n}}:=I_1^{n_1}\cdots I_r^{n_r}$. We study the asymptotic behaviour of the (Castelnuovo-Mumford) regularity of two families of Ext and Tor modules: those obtained from the quotients ${\bf I}^{\underline{n}}M/{\bf I}^{\underline{n}}N$, and those obtained from the quotients $M/{\bf I}^{\underline{n}}N$, with $L$ as the first argument in both the families. For each fixed homological degree, we prove that the regularity of the former family is eventually given by the supremum of finitely many linear functions of $\underline{n}$, where the coefficient of $n_i$ belongs to the set of degrees of generators of $I_i$. For the second family, when at least one of the ideals $I_i$ is generated by homogeneous elements of positive degree, we establish, under suitable conditions, a dichotomy: the regularity is either eventually equal to the regularity of the Ext or Tor module of $(L,M)$, or exhibits the same asymptotic behaviour as in the first family. These results follow from a theorem describing the asymptotic behaviour of the regularity of $(U+{\bf I}^{\underline{n}}V)/{\bf I}^{\underline{n}}W$, under suitable conditions, where $U$, $V$, and $W$ are graded submodules of a finitely generated graded $R$-module with $W\subseteq V$.

Let $R$ be a $\phi$-ring and put $N=Nil(R)$. We determine the $\phi$-weak global dimension of every $\phi$-Prüfer ring. Its only possible values are $0$, $1$, and $\infty$: the value is $0$ when $R/N$ is a field, it is $1$ when $R$ is strong and $R/N$ is a nonfield Prüfer domain, and it is infinite when $R$ is not strong. No nilpotence assumption on $N$ is needed. The proof is based on the canonical kernel $K=ker(R\to\phi(R))$. In the local non-strong case, $K$ is a nonzero torsion divisible module over the valuation domain $R/N$, and its self-Tor groups satisfy \[ Tor^R_{2n}(K,K)=0\quad(n\geq 1), \qquad Tor^R_{2n+1}(K,K)\neq 0\quad(n\geq 0). \] It follows that $fd_RK=\infty$, and a filtered-colimit argument transfers this obstruction to cyclic quotients by nonnil principal ideals. We also prove the parallel trichotomy for the $\phi$-global dimension of $\phi$-Dedekind rings. For strong $\phi$-rings we obtain a change-of-rings theorem without assuming that the quotient map $R\to R/N$ splits. If $D=R/N$, $Q=Frac(D)$ and $T=T(R)$ is the total quotient ring, then \[ R\cong T\times_Q D. \] Using Ferrand's flat patching theorem and Milnor's projective patching theorem, we prove that every torsion $D$-module $M$, regarded as an $R$-module through $R\to D$, satisfies \[ fd_R(M)=fd_D(M), \qquad pd_R(M)=pd_D(M). \] Consequently, for every strong $\phi$-ring, \[\phi-wgd(R)=wgld(R/N), \qquad \phi-gld(R)=gld(R/N). \]

Let $(R,\m)$ be a one-dimensional Cohen–Macaulay local ring with a canonical fractional ideal $R\subseteq K\subseteq\overline R$, and let $v$ denote its embedding dimension. Put $\sigma(R)=\sup\{j\ge0\mid\tr(K^j)\supseteq\m\}$. By definition, $R$ is nearly Gorenstein if and only if $\sigma(R)>0$. We prove that $R$ is almost Gorenstein if and only if $\sigma(R)\ge v$, equivalently, $\sigma(R)=\infty$. Thus every finite value of $\sigma(R)$ is at most $v-1$. We then determine the structure of rings for which $\sigma(R)$ attains its largest finite value $v-1$, just below the threshold for the almost Gorenstein property. This occurs if and only if $K^{v-1}\cong\m$ and $K^v\not\cong\m$. When these conditions hold, $R$ has type two, with multiplicity $e$ greater than $v$ and $e \equiv 3 \pmod{v-1}$. As an application, we characterize the numerical semigroup rings $R$ with $\sigma(R)=v-1$ as those generated by an arithmetic progression, of type two and multiplicity greater than $v$.

Let $Q$ be a Gorenstein local ring, let $I$ be a proper ideal of $Q$, and set $R=Q/I$. We investigate when finite injective dimension of the conormal module $I/I^2$ forces $I$ to be generated by a regular sequence. We prove this implication for every ideal in the linkage class of a complete intersection. When $Q$ is regular, we also establish it under several numerical conditions in terms of the number of generators of $I$ and the type of $R$. For normal domain quotients, we show that finite injective dimension of $I/I^2$ forces $(\mathrm{ht}(I)+1)[\omega_R]=0$ in $\operatorname{Cl}(R)$, yielding the desired conclusion when the divisor class group is torsionfree. In the graded setting, we further prove the implication for all monomial ideals in polynomial rings. We also obtain a converse to Kunz's description of the first Koszul homology of an almost complete intersection.

We study ramification invariants of finite separable extensions of valued fields that are pure on a primitive element. In the immediate unibranched case, we prove that the extended trace ideal is generated by minimal-polynomial derivatives of integral primitive elements and that the Dedekind different is its divisorial closure. For Galois extensions, we express this derivative ideal as a product of subgroup ramification ideals. These formulas extend prime-degree calculations to pure extensions of higher degree. We compute the differential module, traces, differents, ramification ideals, and character Swan ideals of a pure immediate elementary abelian extension of degree $p^2$. A totally ramified example of the same degree gives a comparison with the defectless case.

We prove that the general version—the one that includes the dimension of the tensor product—of Peskine and Szpiro's Dimension Inequality holds over noetherian local rings of depth at most one.

The Eisenbud-Goto conjecture stated that for a homogeneous prime ideal in a polynomial ring over an algebraically closed field, the Castelnuovo-Mumford regularity is bounded above by the multiplicity minus height plus one. In 2017, McCullough and Peeva showed that the regularity of an arbitrary homogeneous prime cannot be bounded by any polynomial function of the multiplicity. For ideals generated by polynomials with at most $c$ terms, we present a bound on the regularity which is less than triple exponential in $c$ and the multiplicity. Furthermore, for toric ideals we get a bound that is smaller than double exponential in terms of multiplicity.

Building on work of Turner, we propose a proof in characteristic two of Quillen's conjecture on finiteness properties of the cotangent complex for homomorphisms between noetherian rings of possibly infinite flat dimension.

We prove that any counter-example to the two dimensional Jacobian Conjecture with integral coefficients and determinant of its Jacobian matrix equals one, has only paths of asymptotic values that can contain only finitely many points of the planar integral lattice. The second part of our paper gives some nontrivial applications of the theory of certain Theta functions to the two dimensional Jacobian Conjecture over the real field, and over the complex field.

We prove that a Cohen-Macaulay standard graded integral domain over a field is almost Gorenstein as a graded ring if and only if the completion of its localization at the graded maximal ideal is almost Gorenstein as a local ring.

Let $k$ be a field of characteristic zero and $D_N(a_n)=a_{n+1}a_{n+2}$ the adjacent-quadratic locally nilpotent derivation on $A_N=k[a_0,\dots,a_N]$. We determine the invariant ring at $N=5$, where deleting $a_0$ gives the polynomial base kernel $B^{D_5}=K=k[p,q,J_1,J_2]$, so localization gives the polynomial-line invariant algebra $(A_5^{D_5})_{pq}=K_{pq}[W]$ with $W=F_0/(p^2q^4)$, but this does not by itself determine the global kernel. An invariant $H$ of weight $24$ and $a_0$-degree $2$ lies outside $K[F_0]$, and adjoining it gives the six-generator hypersurface $C_5$; the obstruction is explained by algebraic residue behavior defeating naive denominator descent. An invariant $F_{46}^{int}$ of weight $46$ and $a_0$-degree $4$ lies outside $C_5$ and satisfies $p^2F_{46}^{int}\in C_5$. To pass from this candidate to the full invariant ring, a second slice chart arising from $f_{13}$, together with exact $q$- and $p$-saturation, yields $A_5^{D_5}=C_5[F_{46}^{int}]$, hence finite generation, with a codimension-two complete-intersection presentation and explicit Hilbert series. At $N=6$ the transported $B$-level ring is exact, while the $A_6$ analysis is established through total degree $13$ and the full $A_6$ invariant ring remains open.

The off-diagonal Gram map for h vectors in an M-dimensional quadratic space is flat with reduced fibers of rational singularities (FRS) in characteristic zero if and only if M > max_{2 <= r <= h} (2h - r + 1 - 2h/r). Equivalently, this gives the exact FRS convolution threshold of x |-> (x_i x_j){i < j}. For its zero-fiber ideal I{M,h}, we also prove lct(I_{M,h}) = min{ binom(h,2), min_{0 <= k <= h-2} ( binom(k+1,2) + M(h-k)/2 ) }. In the strict range, all truncated coefficient maps have geometrically integral complete-intersection fibers and are faithfully flat over Z[1/2], uniformly over graphs on h vertices. The proof uses a sharp weighted estimate for hollow symmetric matrices over finite local principal ideal rings, paired with isotropic lower bounds. Within the established point-counting approach to rational singularities, this estimate also gives an explicit modulus of continuity for Gram-map densities over every local field of odd residue characteristic.

Let \[ S_h=\langle a,ah+d,ah+2d,\ldots,ah+kd\rangle, \qquad \gcd(a,d)=1,\quad 1<k<a,\quad h> 1, \] be a numerical monoid generated by a generalized arithmetic sequence. In this paper, we determine the delta set \[\Delta(S_h),\] resolving an open problem posed by Christopher O'Neill and Roberto Pelayo in 2014 \cite{ONeill}. Writing $e=h-1$ and letting $E(e,d)$ denote the set of values arising from the subtractive Euclidean algorithm applied to $e$ and $d$, we prove that \[ \Delta(S_h) = E(h-1,d)\cup \{M-r(h-1):M-r(h-1)>0,\ r\geq0\}, \] where \[ M=\left\lceil\frac{a}{k}\right\rceil(h-1)+d \] is the maximal element of $\Delta(S_h)$. In particular, $d$ and $h-1$ always belong to $\Delta(S_h)$. We further establish a reduction theorem showing that $\Delta(S_h)$ is a multiple of the delta set of a primitive monoid satisfying $\gcd(d,h-1)=1$, and deduce that the minimum delta is $\gcd(d,h-1)$. We investigate the relationship between delta sets and catenary degrees for these monoids, obtaining an explicit connection between their maximal values and formulating several conjectures concerning the catenary degree set. We also find an interesting way to generate metallic fibonacci sequences through Delta Sets. These results provide a structural description of factorization invariants for generalized arithmetic numerical monoids and suggest several directions for further study.

Let $F$ be a projective variety, let $H$ be a normally generated line bundle on $F$, and let $X\in |H^{\otimes d}|$ be a general projective variety. We establish a uniform criterion, formulated in terms of the graded Betti numbers of the section ring $R(F,H)$, that provides sufficient conditions for every automorphism of $X$ to extend to an automorphism of $F$. This extends the classical results of Matsumura-Monsky on automorphisms of hypersurfaces in projective spaces to a broader class of polarized varieties $(F,H)$, including rational homogeneous varieties of Picard number one, smooth Fano threefolds with their anticanonical polarization, and examples of hyperkähler and Calabi-Yau varieties.

Raedschelders, Špenko and Van den Bergh proved that the Plücker coordinate ring $C_n$ of the Grassmannian $\operatorname{Gr}(2,n)$ has finite Frobenius representation type when the characteristic $p$ satisfies $p\ge\max\{n-2,3\}$, and they determined the indecomposable summands of its Frobenius pushforwards up to nonzero multiplicity. We give nonrecursive finite formulas for every entry of the Frobenius multiplicity matrix and prove that each entry is a polynomial in $p$ of degree at most $2n-3$. Consequently the $F$-signature of $C_n$, and the asymptotic proportion of every indecomposable summand, is a single rational function of $p$ throughout the characteristic range. Its limit as $p\to\infty$ is the leading coefficient of the first free multiplicity, which we evaluate in closed form through Bernoulli numbers. For $\operatorname{Gr}(2,4)$ we find $s_p(C_4)=(13p^2+8)/(5(3p^2+2))$, and we calculate the analogous rational functions for $\operatorname{Gr}(2,5)$ and $\operatorname{Gr}(2,6)$ with computer assistance. We also construct the full characteristic polynomial from the Riemann-Roch weights and two residual matrices whose entries have degree at most $n-1$ in $p$. In the sequel, we use this work to show rank-two determinantal rings have FFRT, and to calculate their $F$-signatures.

Let $H=\langle a_1,\ldots,a_n\rangle$ be a numerical semigroup. A conjecture on numerical semigroup rings predicts that the defining ideal $I_H$ has a determinantal presentation if and only if the set of pseudo-Frobenius numbers of $H$ has form $\{h+\alpha,h+2\alpha,\ldots,h+(n-1)\alpha\}$ for $h\ge 0, \alpha>0$. We prove this conjecture under the condition that $h$ has a unique factorization in $H$. We also give characterization this unique-factorization condition. As an application, we consider the family of affine-orbit numerical semigroups determined by $m=cA_{n-1}+1$, $1\le c\le a$. We prove that the relevant element $h$ has a unique factorization and the defining ideal is generated by the 2-minors of a $2\times n$-matrix. In particular, the Eagon–Northcott complex of the matrix mentioned above gives the graded minimal free resolution of $k[H]$.

In this paper, we investigate the componentwise linear property of support-two monomial ideals. Our first main result shows that if $I$ is a support-two monomial ideal, then its underlying simple graph $G_I$ is co-chordal; equivalently, by Fröberg's theorem, $\sqrt{I}$ admits a linear resolution. This phenomenon is quite rare for general monomial ideals. In fact, there exist monomial ideals with linear resolutions whose radicals fail to have linear resolutions, even when the radical is the edge ideal of a graph. For any monomial ideal $I$, one always has $\mu(I)\geq \mu(\sqrt{I})$. In the literature, support-two monomial ideals satisfying $\mu(I)=\mu(\sqrt{I})$ are of special interest, as they include edge ideals of simple graphs, weighted oriented graphs, edge-weighted graphs, and vertex-weighted graphs. We refer to such ideals as minimal support-two monomial ideals. We explicitly characterize all minimal support-two monomial ideals, as well as their powers, that admit linear resolutions. Next, we classify the linearity of non-minimal ones. Consequently, we obtain a complete classification of linear support-two monomial ideals, which shows that the property of being linear does not depend on the characteristics of the base field for support-two monomial ideals.

In this article, we compute the limit Hilbert–Kunz multiplicities and limit F-signatures of all the simple singularities and the Hilbert–Kunz multiplicities of the Fermat hypersurfaces, generalizing the well-known sec z + tan z formula by Gessel and Monsky for the A1 singularities, using the theory of h-functions. We also obtain a functional equation relating the phi-function of a hypersurface to its reflection. Using limit F-invariants, we generalize both a characterization of regular local rings through F-invariants and the known cases of the Watanabe–Yoshida conjecture to characteristic zero.

We discuss a class of mixed characteristic rings defined analogously to Stanley-Reisner rings by replacing one variable with a uniformizing parameter for a discrete valuation ring. We adapt Hochster's formula for Tor and Ext modules, Hochster's formula for local cohomology, and Terai's criterion for Serre conditions to this new setting; one of the tools needed is cellular sheaf cohomology, for which we give a brief treatment.

In this paper, we investigate two related questions concerning the Boolean polynomial semiring in two variables. First, we study prime ideals in $\mathbb{B}[x,y]$, beginning with a complete classification of those generated by linear polynomials. Second, motivated by the limitations of ideals in capturing geometric phenomena over semirings, we introduce a notion of projective closure via congruences. We prove that this projective closure coincides with the topological closure in the projective space of prime congruences.

We prove that every commutative Gaussian ring has finitistic weak dimension at most two. The proof works with a fixed module of finite flat dimension and does not assume that the weak global dimension of the ring is finite. For a local Gaussian ring $A$, with nilradical $N$, the kernel $I$ of $A\to A_N$ satisfies $NI=I^2=0$. A hypothetical module $M$ of flat dimension three gives a nonzero torsion module $H=\Tor_2^A(A/N,M)$ over the valuation domain $A/N$, with $H\cong\Tor_1^A(I,M)$ and $I\otimes_{A/N}H=0$. Two carefully controlled localizations, a propagation argument in the chain ring $A/D$, where $D=\{x:x^2=0\}$, and an annihilator factorization in a filtered system of cyclic modules yield a contradiction.

We classify the deep points of cluster varieties of cluster algebras of punctured marked surfaces with boundary.

Let $Q$ be a local ring, $f$ a nonzerodivisor, and $R=Q/(f)$. We characterize weak and Serre liftability of cyclic $R$-modules under suitable hypotheses, obtaining complete-intersection criteria for Serre liftability in small height. We prove that the tensor product of two weakly liftable cyclic modules remains weakly liftable when their first Tor module vanishes, and give sufficient conditions for tensor products to preserve Serre liftability. We construct an Artinian Gorenstein quotient $Q/I$, with $Q$ regular, that is perfect over $R$ and Serre liftable but not weakly liftable to $Q$, showing that the negative answer to a question of Jorgensen persists in this more restrictive setting. Finally, we characterize when a lift induces a lift of the Auslander transpose and obtain a freeness criterion related to the Auslander-Reiten conjecture, recovering a theorem of Ghosh and Samanta.

We implement a Macaulay2 package for computations involving rings of truncated Witt vectors $W_n(R)$ of a finitely generated $\F_p$-algebra $R$. This package includes ring operations (addition, multiplication, Frobenius, Verschiebung, etc.) on elements of $W_n(R)$, conversion between tuple representatives and ghost map representatives of elements of truncated Witt vectors over polynomial rings, and explicit computation of $W_n(R)$ as a finite-type algebra over $\Z/p^n$. This functionality is based on an algorithm we developed for performing arithmetic operations in rings of finite length $p$-typical Witt vectors over finitely generated $\F_p$-algebras. In addition, we implement an algorithm to calculate lifts of Frobenius from $\F_p$-algebras to flat lifts over $\Z/p^2\Z$, and an algorithm to find the quasi-$F$-splitting height of a local or graded complete intersection ring.

We show that the Knörrer functor induces an equivalence ${\underline{\operatorname{MCM}}}(R) \simeq {\underline{\operatorname{MCM}}}(A)$ of stable categories of maximal Cohen-Macaulay modules, where $R = S/(f)$ is a complete hypersurface ring and $A = S[\![u,v]\!]/(f+uv)$ is the hyperbolic extension of $R$. No hypothesis is placed on the residue field or on its characteristic, $f$ need not define an isolated singularity, and $S$ need not contain a field. This is in contrast to the iterated double branched cover $R^{\sharp\sharp} = S[\![z,w]\!]/(f+z^2+w^2)$, which is stably equivalent to $R$ only in characteristic not equal to $2$. We also give a direct construction of a free resolution identifying $\operatorname{syz}_2^A(M)$ with the image of $M \oplus \operatorname{syz}_1^R M$ under the functor, and an example in characteristic two in which the corresponding statement for the double branched cover $S[\![z]\!]/(f+z^2)$ fails.

Schreyer's algorithm is usually the fastest way to determine a (typically non-minimal) free resolution of a finitely presented module over a polynomial ring. We adapt a refined version of Schreyer's algorithm to compute free resolutions over the exterior algebra, relying on relative Groebner bases. We illustrate the use of our algorithm for the computation of cohomology of coherent sheaves over projective space, which by the BGG correspondence can be computed via free resolutions over the exterior algebra. Schreyer's method relies on a tree traversal and thus has the potential for parallel computations. We report on ongoing work on a massively parallel implementation, observing that Gnawali's parallel approach over polynomial rings can be carried over to our setting.

Let $\Delta_{m,n}$ be the chessboard complex, and let $J_{m,n}$ be the Stanley-Reisner ideal of its Alexander dual. We study the symbolic and ordinary powers of $J_{m,n}$, focusing on their homological and algebraic invariants. First, we characterize when $J_{m,n}^{(q)}$ has linear quotients. For the ordinary powers, we give sufficient conditions on $m$ and $n$ for $J_{m,n}^q$ to have linear quotients. We determine the maximal degree of a minimal generator and the Castelnuovo-Mumford regularity of $J_{m,n}^{(q)}$, obtaining explicit formulas for all $q\ge2$ and showing that both invariants have the same asymptotic slope. We also compute the initial degrees and Waldschmidt constant, proving that $\alpha(J_{m,n}^{(q)})$ is a degree-one quasi-polynomial of period two and $\widehat{\alpha}(J_{m,n})=mn/2$. Explicit formulas for the $\mathrm{v}$-numbers of symbolic and ordinary powers are obtained. Finally, for every $m\ge3$, we show that $J_{m,2m-1}$ is not weakly polymatroidal, disproving a conjecture of Lu and Wang in this setting.

We study the $\mathrm{v}$-numbers of ordinary and symbolic powers of the facet ideal and the Stanley–Reisner ideal of a chessboard complex. Exact formulas are established for the $\mathrm{v}$-numbers of ordinary and symbolic powers of the Stanley–Reisner ideal, as well as symbolic powers of the facet ideal. Furthermore, we show that the $\mathrm{v}$-number function of the ordinary powers is asymptotically linear.

Let $G$ be a finite simple graph and let $A_c(G)$ be the Artinian algebra associated with its cover ideal. We prove that $A_c(G)$ has the WLP when $\tau(G)>|V(G)|/2$, where $\tau(G)$ denotes the size of a minimum vertex cover of $G$. As a consequence, $A_c(G)$ has the WLP with high probability when the Erdős-Rényi random graph model is considered. Moreover, we study the borderline case $\tau(G)=|V(G)|/2$ and as a result, classify the WLP for paths, cycles, Ferrers graphs, and well-covered trees.

We explicitly investigate Gorenstein projective, injective and flat modules over the $n$-trivial extension $R\ltimes_{n}M$ of a ring $R$ by an $R$-bimodule $M$. Assume that $fd(M^{\otimes_{R}i}_{R})<\infty$ and $pd(_{R}M^{\otimes_{R}i})<\infty$ for any $1\leq i\leq n$, $fd(\textbf{Z}(R)_{R\ltimes_{n}M})<\infty$ and $pd(_{R\ltimes_{n}M}\textbf{Z}(R))<\infty$. It is proven that a left $R\ltimes_{n}M$-module $(X,f)$ is Gorenstein projective if and only if the sequence $M^{\otimes_{R}n+1}\otimes_{R} X\stackrel{(M\otimes f) \cdots(M^{\otimes_{R}n}\otimes f)}\longrightarrow M\otimes_{R} X\stackrel{f}\longrightarrow X$ is exact and coker$(f)$ is a Gorenstein projective left $R$-module. As a consequence, we characterize Gorenstein projective, injective and flat modules over tensor rings.