In 2004, Herzog, Hibi, and Zheng proved that a quadratic monomial ideal has a linear resolution if and only if all its powers have a linear resolution. We study a generalization of this result for square-free monomial ideals arising from facet ideals of a simplicial tree. We give a complete characterization of simplicial trees for which all powers of their facet ideal have a linear resolution. We compute the regularity of t-path ideals of rooted trees. In addition, we study the regularity of powers of t-path ideals of rooted trees. We pose a regularity upper bound conjecture for facet ideals of simplicial trees, which is as follows: if $\Delta $ is a d-dimensional simplicial tree connected in codimension one, then reg$(I(\Delta )^s) \leq (d+1)(s-1)~+$ reg$(I(\Delta ))$ for all $s \geq 1$. We prove this conjecture for some special classes of simplicial trees.

By extending some basic results about cohomological dimension of tensor products to non-positive DG-rings, the Intersection Theorem for DG-modules is examined over commutative noetherian local DG-rings with bounded cohomology. Some applications are provided. The first is to improve the DG-setting of the amplitude inequality in [Forum Math. 22 (2010) 941–948]. The second is to show Minamoto’s conjecture in [Israel J. Math. 242 (2021) 1–36]. The third is to obtain the DG-version of the Vasconcelos conjecture about Gorenstein rings.

Let X be a zero-dimensional reduced subscheme of a multiprojective space $\mathbb {V} $. Let $s_i$ be the length of the projection of X onto the ith component of $\mathbb {V}$. A result of Van Tuyl states that the Hilbert function of X is completely determined by its restriction to the product of the intervals $[0, s_i - 1]$. We extend this result to arbitrary zero-dimensional subschemes of $\mathbb {V}$.

We study the transfer of (co)silting objects in derived categories of module categories via the extension functors induced by a morphism of commutative rings. It is proved that the extension functors preserve (co)silting objects of (co)finite type. In many cases the bounded silting property descends along faithfully flat ring extensions. In particular, the notion of bounded silting complex is Zariski local.

Let d be a positive integer, and let $\mathfrak {a}$ be an ideal of a commutative Noetherian ring R. We answer Hartshorne’s question on cofiniteness of complexes posed in Hartshorne (1970, Invent. Math. 9, 145–164) in the cases $\mathrm {dim}R=d$ or $\mathrm {dim}R/\mathfrak {a}=d-1$ or $\mathrm {ara}(\mathfrak {a})=d-1$, show that if $d\leqslant 2$, then a complex $X\in \mathrm {D}_\sqsubset (R)$ is $\mathfrak {a}$-cofinite if and only if each homology module $\mathrm {H}_i(X)$ is $\mathfrak {a}$-cofinite; if R is regular local, $\mathfrak {a}$ is perfect and $d\leqslant 2$, then $X\in \mathrm {D}(R)$ is $\mathfrak {a}$-cofinite if and only if every $\mathrm {H}_i(X)$ is $\mathfrak {a}$-cofinite; if $d\geqslant 3$, then $X\in \mathrm {D}_\sqsubset (R)$ is $\mathfrak {a}$-cofinite and $\mathrm {Ext}^j_R(R/\mathfrak {a},\mathrm {H}_i(X))$ is finitely generated for $j\leqslant d-2$ and $i\in \mathbb {Z}$ if and only if every $\mathrm {H}_{i}(X)$ is $\mathfrak {a}$-cofinite.

In this article, we study the algebra of Veronese type. We show that the presentation ideal of this algebra has an initial ideal whose Alexander dual has linear quotients. As an application, we explicitly obtain the Castelnuovo–Mumford regularity of the Veronese type algebra. Furthermore, we give an effective upper bound on the multiplicity of this algebra.

We define and study a notion of G-dimension for DG-modules over a non-positively graded commutative noetherian DG-ring A. Some criteria for the finiteness of the G-dimension of a DG-module are given by applying a DG-version of projective resolution introduced by Minamoto [Israel J. Math. 245 (2021) 409-454]. Moreover, it is proved that the finiteness of G-dimension characterizes the local Gorenstein property of A. Applications go in three directions. The first is to establish the connection between G-dimensions and the little finitistic dimensions of A. The second is to characterize Cohen-Macaulay and Gorenstein DG-rings by the relations between the class of maximal local-Cohen-Macaulay DG-modules and a special G-class of DG-modules. The third is to extend the classical Buchweitz-Happel Theorem and its inverse from commutative noetherian local rings to the setting of commutative noetherian local DG-rings. Our method is somewhat different from classical commutative ring.

We prove the conjecture of Franceschini and Lorenzini [‘Fat points of $\mathbb P^n$ whose support is contained in a linear proper subspace’, J. Pure and Appl. Algebra 160 (2001), 169–182] about the regularity index of fat points of $\mathbb P^n$ whose support is contained in a linear proper subspace.

In this paper, we consider a conilpotent coalgebra $C$ over a field $k$. Let $\Upsilon :\ C{{-\mathsf{Comod}}}\longrightarrow C^*{{-\mathsf{Mod}}}$ be the natural functor of inclusion of the category of $C$-comodules into the category of $C^*$-modules, and let $\Theta :\ C{{-\mathsf{Contra}}}\longrightarrow C^*{{-\mathsf{Mod}}}$ be the natural forgetful functor. We prove that the functor $\Upsilon$ induces a fully faithful triangulated functor on bounded (below) derived categories if and only if the functor $\Theta$ induces a fully faithful triangulated functor on bounded (above) derived categories, and if and only if the $k$-vector space $\textrm {Ext}_C^n(k,k)$ is finite-dimensional for all $n\ge 0$. We call such coalgebras “weakly finitely Koszul”.

Let $(A,\mathfrak{m} )$ be a hypersurface local ring of dimension $d \geq 1$ and let I be an $\mathfrak{m} $-primary ideal. We show that there is a integer rI$\geq\;-1$ (depending only on I) such that if M is any non-free maximal Cohen–Macaulay (= MCM) A-module the function $n \rightarrow \ell(\operatorname{Tor}^A_1(M, A/I^{n+1}))$ (which is of polynomial type) has degree rI. Analogous results hold for Hilbert polynomials associated to Ext-functors. Surprisingly, a key ingredient is the classification of thick subcategories of the stable category of MCM A-modules (obtained by Takahashi, see [11, 6.6]).

We develop analogues of Green’s $N_p$ conditions for subvarieties of weighted projective space, and we prove that such $N_p$ conditions are satisfied for high degree embeddings of curves in weighted projective space. A key technical result links positivity with low degree (virtual) syzygies in wide generality, including cases where normal generation fails.

We consider the Bernstein–Sato polynomial of a locally quasi-homogeneous polynomial $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$. We construct, in the analytic category, a complex of $\mathscr{D}_{X}[s]$-modules that can be used to compute the $\mathscr{D}_{X}[s]$-dual of $\mathscr{D}_{X}[s] f^{s-1}$ as the middle term of a short exact sequence where the outer terms are well understood. This extends a result by Narváez Macarro where a freeness assumption was required. We derive many results about the zeros of the Bernstein–Sato polynomial. First, we prove each nonvanishing degree of the zeroth local cohomology of the Milnor algebra $H_{\mathfrak{m}}^{0} (R / (\partial f))$ contributes a root to the Bernstein–Sato polynomial, generalizing a result of M. Saito (where the argument cannot weaken homogeneity to quasi-homogeneity). Second, we prove the zeros of the Bernstein–Sato polynomial admit a partial symmetry about $-1$, extending a result of Narváez Macarro that again required freeness. We give applications to very small roots, the twisted logarithmic comparison theorem, and more precise statements when f is additionally assumed to be homogeneous. Finally, when f defines a hyperplane arrangement in $\mathbb{C}^{3}$ we give a complete formula for the zeros of the Bernstein–Sato polynomial of f. We show all zeros except the candidate root $-2 + (2 / \deg(f))$ are (easily) combinatorially given; we give many equivalent characterizations of when the only noncombinatorial candidate root $-2 + (2/ \deg(f))$ is in fact a zero of the Bernstein–Sato polynomial. One equivalent condition is the nonvanishing of $H_{\mathfrak{m}}^{0}( R / (\partial f))_{\deg(f) - 1}$.

We define a local homomorphism $(Q,k)\to (R,\ell )$ to be Koszul if its derived fiber $R\otimes ^{\mathsf {L}}_Q k$ is formal, and if $\operatorname {Tor}^{Q}(R,k)$ is Koszul in the classical sense. This recovers the classical definition when Q is a field, and more generally includes all flat deformations of Koszul algebras. The non-flat case is significantly more interesting, and there is no need for examples to be quadratic: all complete intersection and all Golod quotients are Koszul homomorphisms. We show that the class of Koszul homomorphisms enjoys excellent homological properties, and we give many more examples, especially various monomial and Gorenstein examples. We then study Koszul homomorphisms from the perspective of $\mathrm {A}_{\infty }$-structures on resolutions. We use this machinery to construct universal free resolutions of R-modules by generalizing a classical construction of Priddy. The resulting (infinite) free resolution of an R-module M is often minimal and can be described by a finite amount of data whenever M and R have finite projective dimension over Q. Our construction simultaneously recovers the resolutions of Shamash and Eisenbud over a complete intersection ring, and the bar resolutions of Iyengar and Burke over a Golod ring, and produces analogous resolutions for various other classes of local rings.

We prove new statistical results about the distribution of the cokernel of a random integral matrix with a concentrated residue. Given a prime p and a positive integer n, consider a random $n \times n$ matrix $X_n$ over the ring $\mathbb{Z}_p$ of p-adic integers whose entries are independent. Previously, Wood showed that as long as each entry of $X_n$ is not too concentrated on a single residue modulo p, regardless of its distribution, the distribution of the cokernel $\mathrm{cok}(X_n)$ of $X_n$, up to isomorphism, weakly converges to the Cohen–Lenstra distribution, as $n \rightarrow \infty$. Here on the contrary, we consider the case when $X_n$ has a concentrated residue $A_n$ so that $X_n = A_n + pB_n$. When $B_n$ is a Haar-random $n \times n$ matrix over $\mathbb{Z}_p$, we explicitly compute the distribution of $\mathrm{cok}(P(X_n))$ for every fixed n and a non-constant monic polynomial $P(t) \in \mathbb{Z}_p[t]$. We deduce our result from an interesting equidistribution result for matrices over $\mathbb{Z}_p[t]/(P(t))$, which we prove by establishing a version of the Weierstrass preparation theorem for the noncommutative ring $\mathrm{M}_n(\mathbb{Z}_p)$ of $n \times n$ matrices over $\mathbb{Z}_p$. We also show through cases the subtlety of the “universality” behavior when $B_n$ is not Haar-random.

The notion of Vasconcelos invariant, known in the literature as v-number, of a homogeneous ideal in a polynomial ring over a field was introduced in 2020 to study the asymptotic behavior of the minimum distance of projective Reed–Muller type codes. We initiate the study of this invariant for graded modules. Let R be a Noetherian $\mathbb {N}$-graded ring and M be a finitely generated graded R-module. The v-number $v(M)$ can be defined as the least possible degree of a homogeneous element x of M for which $(0:_Rx)$ is a prime ideal of R. For a homogeneous ideal I of R, we mainly prove that $v(I^nM)$ and $v(I^nM/I^{n+1}M)$ are eventually linear functions of n. In addition, if $(0:_M I)=0$, then $v(M/I^{n}M)$ is also eventually linear with the same leading coefficient as that of $v(I^nM/I^{n+1}M)$. These leading coefficients are described explicitly. The result on the linearity of $v(M/I^{n}M)$ considerably strengthens a recent result of Conca which was shown when R is a domain and $M=R$, and Ficarra–Sgroi where the polynomial case is treated.

We analyze infinitesimal deformations of morphisms of locally free sheaves on a smooth projective variety X over an algebraically closed field of characteristic zero. In particular, we describe a differential graded Lie algebra controlling the deformation problem. As an application, we study infinitesimal deformations of the pairs given by a locally free sheaf and a subspace of its sections with a view toward Brill-Noether theory.

Let $(R,\mathfrak {m})$ be a Noetherian local ring and I an ideal of R. We study how local cohomology modules with support in $\mathfrak {m}$ change for small perturbations J of I, that is, for ideals J such that $I\equiv J\bmod \mathfrak {m}^N$ for large N, under the hypothesis that $R/I$ and $R/J$ share the same Hilbert function. As one of our main results, we show that if $R/I$ is generalized Cohen–Macaulay, then the local cohomology modules of $R/J$ are isomorphic to the corresponding local cohomology modules of $R/I$, except possibly the top one. In particular, this answers a question raised by Quy and V. D. Trung. Our approach also allows us to prove that if $R/I$ is Buchsbaum, then so is $R/J$. Finally, under some additional assumptions, we show that if $R/I$ satisfies Serre’s property $(S_n)$, then so does $R/J$.

In this paper, we show existence of bountiful examples of Gorenstein local rings A and B such that there is a triangle equivalence between the stable categories CM(A), CM(B).

We prove the existence of a power structure over the Grothendieck ring of geometric dg categories. We show that a conjecture by Galkin and Shinder (proved recently by Bergh, Gorchinskiy, Larsen and Lunts) relating the motivic and categorical zeta functions of varieties can be reformulated as a compatibility between the motivic and categorical power structures. Using our power structure, we show that the categorical zeta function of a geometric dg category can be expressed as a power with exponent the category itself. We give applications of our results for the generating series associated with Hilbert schemes of points, categorical Adams operations and series with exponent a linear algebraic group.

For a reduced hyperplane arrangement, we prove the analytic Twisted Logarithmic Comparison Theorem, subject to mild combinatorial arithmetic conditions on the weights defining the twist. This gives a quasi-isomorphism between the twisted logarithmic de Rham complex and the twisted meromorphic de Rham complex. The latter computes the cohomology of the arrangement’s complement with coefficients from the corresponding rank one local system. We also prove the algebraic variant (when the arrangement is central), and the analytic and algebraic (untwisted) Logarithmic Comparison Theorems. The last item positively resolves an old conjecture of Terao. We also prove that: Every nontrivial rank one local system on the complement can be computed via these Twisted Logarithmic Comparison Theorems; these computations are explicit finite-dimensional linear algebra. Finally, we give some $\mathscr {D}_{X}$-module applications: For example, we give a sharp restriction on the codimension one components of the multivariate Bernstein–Sato ideal attached to an arbitrary factorization of an arrangement. The bound corresponds to (and, in the univariate case, gives an independent proof of) M. Saito’s result that the roots of the Bernstein–Sato polynomial of a non-smooth, central, reduced arrangement live in $(-2 + 1/d, 0).$