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.

In this paper, we prove the integrality conjecture for quotient stacks arising from weakly symmetric representations of reductive groups. Our main result is a decomposition of the cohomology of the stack into finite-dimensional components indexed by some equivalence classes of cocharacters of a maximal torus. This decomposition enables the definition of new enumerative invariants associated with the stack, which we begin to explore.

For each three-dimensional non-Lie Leibniz algebra over the complex numbers, we describe the algebra of polynomial invariants and determine its group of automorphisms. As a consequence, we establish that any two non-nilpotent three-dimensional non-Lie Leibniz algebras can be distinguished by the traces of degrees $\leqslant 2$ and by the dimensions of their automorphism groups.

We introduced positive cones in an earlier paper as a notion of ordering on central simple algebras with involution that corresponds to signatures of hermitian forms. In the current article, we describe signatures of hermitian forms directly out of positive cones, and also use this approach to rectify a problem that affected some results in the previously mentioned paper.

We construct pathological examples of MMP singularities in every positive characteristic using quotients by $\alpha _p$-actions. In particular, we obtain non-$S_3$ terminal singularities, as well as locally stable (respectively stable) families whose general fibers are smooth (respectively klt, Cohen–Macaulay, and F-injective) and whose special fibers are non-$S_2$. The dimensions of these examples are bounded below by a linear function of the characteristic.

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 n be a positive integer and f belong to the smallest ring of functions $\mathbb R^n\to \mathbb R$ that contains all real polynomial functions of n variables and is closed under exponentiation. Then there exists $d\in \mathbb N$ such that for all $m\in \{0,\dots , n\}$ and $c\in \mathbb R^{m}$, if $x\mapsto f(c,x)\colon \mathbb R^{n-m}\to \mathbb R$ is harmonic, then it is polynomial of degree at most d. In particular, f is polynomial if it is harmonic.

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 show that any $\mathbb {A}^2$-fibration over a discrete valuation ring which is also an $\mathbb {A}^2$-form is necessarily a polynomial ring. Further, we show that separable $\mathbb {A}^2$-forms over principal ideal domains are trivial.

Let $(R, \mathfrak{m})$ be a d-dimensional Noetherian local ring that is formally equidimensional, and let M be an arbitrary R-submodule of the free module $F = R^p$ with an analytic spread $s:=s(M)$. In this work, inspired by Herzog-Puthenpurakal-Verma in [10], we show the existence of a unique largest R-module Mk with $\ell_R(M_{k}/M) \lt \infty$ and $M\subseteq M_{s}\subseteq\cdots\subseteq M_{1}\subseteq M_{0}\subseteq q(M),$ such that $\deg(P_{M_{k}/M}(n)) \lt s-k,$ where q(M) is the relative integral closure of $M,$ defined by $q(M):=\overline{M}\cap M^{sat},$ where $M^{sat}=\cup_{n\geqslant 1}(M:_F\mathfrak{m}^n)$ is the saturation of M. We also provide a structure theorem for these modules. Furthermore, we establish the existence of coefficient modules between $I(M)M$ and M, where I(M) denotes the 0th Fitting ideal of $F/M$, and discuss their structural properties. Finally, we present some applications and discuss some properties.

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.

For an ideal I in a Noetherian ring R, the Fitting ideals $\mathrm{Fitt}_j(I)$ are studied. We discuss the question of when $\mathrm{Fitt}_j(I)=I$ or $\sqrt{\mathrm{Fitt}_j(I)}=\sqrt{I}$ for some j. A classical case is the Hilbert–Burch theorem when $j=1$ and I is a perfect ideal of grade 2 in a local ring.

A dimer model is a quiver with faces embedded in a surface. We define and investigate notions of consistency for dimer models on general surfaces with boundary which restrict to well-studied consistency conditions in the disk and torus case. We define weak consistency in terms of the associated dimer algebra and show that it is equivalent to the absence of bad configurations on the strand diagram. In the disk and torus case, weakly consistent models are nondegenerate, meaning that every arrow is contained in a perfect matching; this is not true for general surfaces. Strong consistency is defined to require weak consistency as well as nondegeneracy. We prove that the completed as well as the noncompleted dimer algebra of a strongly consistent dimer model are bimodule internally 3-Calabi-Yau with respect to their boundary idempotents. As a consequence, the Gorenstein-projective module category of the completed boundary algebra of suitable dimer models categorifies the cluster algebra given by their underlying quiver. We provide additional consequences of weak and strong consistency, including that one may reduce a strongly consistent dimer model by removing digons and that consistency behaves well under taking dimer submodels.

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.

We perform a general study of the structure of locally compact modules over compactly generated abelian groups. We obtain a dévissage result for such modules of the form ‘compact-by-sheer-by-discrete’, and then study more specifically the sheer part. The main typical example of a sheer module is a polycontractible module, that is, a finite direct product of modules, each of which is contracted by some group element. We show that every sheer module has a ‘large’ polycontractible submodule, in some suitable sense. We apply this to the study of compactly generated metabelian groups. For instance, we prove that they always have a maximal compact normal subgroup, and we extend the Bieri–Strebel characterization of compactly presentable metabelian groups from the discrete case to this more general setting.

Continuing our work on group-theoretic generalisations of the prime Ax–Katz Theorem, we give a lower bound on the p-adic divisibility of the cardinality of the set of simultaneous zeros $Z(f_1,f_2,\dots,f_r)$ of r maps $f_j\,{:}\,A\rightarrow B_j$ between arbitrary finite commutative groups A and $B_j$ in terms of the invariant factors of $A, B_1,B_2, \cdots,B_r$ and the functional degrees of the maps $f_1,f_2, \dots,f_r$.

We describe the modulo $2$ de Rham-Witt complex of a field of characteristic $2$, in terms of the powers of the augmentation ideal of the $\mathbb {Z}/2$-geometric fixed points of real topological restriction homology ${\mathrm {TRR}}$. This is analogous to the conjecture of Milnor, proved in [Kat82] for fields of characteristic $2$, which describes the modulo $2$ Milnor K-theory in terms of the powers of the augmentation ideal of the Witt group of symmetric forms. Our proof provides a somewhat explicit description of these objects, as well as a calculation of the homotopy groups of the geometric fixed points of ${\mathrm {TRR}}$ and of real topological cyclic homology, for all fields.

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”.