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 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.

Consider a pair of elements f and g in a commutative ring Q. Given a matrix factorization of f and another of g, the tensor product of matrix factorizations, which was first introduced by Knörrer and later generalized by Yoshino, produces a matrix factorization of the sum $f+g$. We will study the tensor product of d-fold matrix factorizations, with a particular emphasis on understanding when the construction has a non-trivial direct sum decomposition. As an application of our results, we construct indecomposable maximal Cohen–Macaulay and Ulrich modules over hypersurface domains of a certain form.

In this paper, we introduce new classes of gluing of complex analytic space germs, called weakly large, large, and strongly large. We describe their Poincaré series and, as applications, we give numerical criteria to determine when these classes of gluing of germs of complex analytic spaces are smooth, singular, complete intersections and Gorenstein, in terms of their Betti numbers. In particular, we show that the gluing of the same germ of complex analytic space along any subspace is always a singular germ.

In the derived category of a commutative noetherian ring, we explicitly construct a silting object associated with each sp-filtration of the Zariski spectrum satisfying the “slice” condition. Our new construction is based on local cohomology and it allows us to study when the silting object is tilting. For a ring admitting a dualizing complex, this occurs precisely when the sp-filtration arises from a codimension function on the spectrum. In the absence of a dualizing complex, the situation is more delicate and the tilting property is closely related to the condition that the ring is a homomorphic image of a Cohen–Macaulay ring. We also provide dual versions of our results in the cosilting case.

We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions.

We characterize the finite codimension sub-${\mathbf {k}}$-algebras of ${\mathbf {k}}[\![t]\!]$ as the solutions of a computable finite family of higher differential operators. For this end, we establish a duality between such a sub-algebras and the finite codimension ${\mathbf {k}}$-vector spaces of ${\mathbf {k}}[u]$, this ring acts on ${\mathbf {k}}[\![t]\!]$ by differentiation.

Valuation rings and perfectoid rings are examples of (usually non-Noetherian) rings that behave in some sense like regular rings. We give and study an extension of the concept of regular local rings to non-Noetherian rings so that it includes valuation and perfectoid rings and it is related to Grothendieck’s definition of formal smoothness as in the Noetherian case. For that, we have to take into account the topologies. We prove a descent theorem for regularity along flat homomorphisms (in fact for homomorphisms of finite flat dimension), extending some known results from the Noetherian to the non-Noetherian case, as well as generalizing some recent results in the non-Noetherian case, such as the descent of regularity from perfectoid rings by B. Bhatt, S. Iyengar and L. Ma.

We prove that double Schubert polynomials have the saturated Newton polytope property. This settles a conjecture by Monical, Tokcan and Yong. Our ideas are motivated by the theory of multidegrees. We introduce a notion of standardization of ideals that enables us to study nonstandard multigradings. This allows us to show that the support of the multidegree polynomial of each Cohen–Macaulay prime ideal in a nonstandard multigrading, and in particular, that of each Schubert determinantal ideal is a discrete polymatroid.

In this paper, we are concerned with certain invariants of modules, called reducing invariants, which have been recently introduced and studied by Araya–Celikbas and Araya–Takahashi. We raise the question whether the residue field of each commutative Noetherian local ring has finite reducing projective dimension and obtain an affirmative answer for the question for a large class of local rings. Furthermore, we construct new examples of modules of infinite projective dimension that have finite reducing projective dimension and study several fundamental properties of reducing dimensions, especially properties under local homomorphisms of local rings.

Let $({\cal{A}},{\cal{E}})$ be an exact category. We establish basic results that allow one to identify sub(bi)functors of ${\operatorname{Ext}}_{\cal{E}}(-,-)$ using additivity of numerical functions and restriction to subcategories. We also study a small number of these new functors over commutative local rings in detail and find a range of applications from detecting regularity to understanding Ulrich modules.

The existence of Ulrich modules for (complete) local domains has been a difficult and elusive open question. For over thirty years, it was unknown whether complete local domains always have Ulrich modules. In this paper, we answer the question of existence for both Ulrich modules and weakly lim Ulrich sequences – a weaker notion recently introduced by Ma – in the negative. We construct many local domains in all dimensions $d \geq 2$ that do not have any Ulrich modules. Moreover, we show that when $d = 2$, these local domains do not have weakly lim Ulrich sequences.

Let k be a field of characteristic zero and let $\Omega_{A/k}$ be the universally finite differential module of a k-algebra A, which is the local ring of a closed point of an algebraic or algebroid curve over k. A notorious open problem, known as Berger’s Conjecture, predicts that A must be regular if $\Omega_{A/k}$ is torsion-free. In this paper, assuming the hypotheses of the conjecture and observing that the module ${\rm Hom}_A(\Omega_{A/k}, \Omega_{A/k})$ is then isomorphic to an ideal of A, say $\mathfrak{h}$, we show that A is regular whenever the ring $A/a\mathfrak{h}$ is Gorenstein for some parameter a (and conversely). In addition, we provide various characterizations for the regularity of A in the context of the conjecture.

Hilbert–Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild assumptions. A natural question is for what singular rings these invariants are closest to one. For Hilbert–Kunz multiplicity this question was first considered by the last two authors and attracted significant attention. In this paper, we study this question, i.e., an upper bound, for F-signature and revisit lower bounds on Hilbert–Kunzmultiplicity.

Let R be a Cohen–Macaulay local K-algebra or a standard graded K-algebra over a field K with a canonical module $\omega _R$. The trace of $\omega _R$ is the ideal $\operatorname {tr}(\omega _R)$ of R which is the sum of those ideals $\varphi (\omega _R)$ with ${\varphi \in \operatorname {Hom}_R(\omega _R,R)}$. The smallest number s for which there exist $\varphi _1, \ldots , \varphi _s \in \operatorname {Hom}_R(\omega _R,R)$ with $\operatorname {tr}(\omega _R)=\varphi _1(\omega _R) + \cdots + \varphi _s(\omega _R)$ is called the Teter number of R. We say that R is of Teter type if $s = 1$. It is shown that R is not of Teter type if R is generically Gorenstein. In the present paper, we focus especially on zero-dimensional graded and monomial K-algebras and present various classes of such algebras which are of Teter type.

Let $(A,\mathfrak m)$ be an excellent two-dimensional normal local domain. In this paper, we study the elliptic and the strongly elliptic ideals of A with the aim to characterize elliptic and strongly elliptic singularities, according to the definitions given by Wagreich and Yau. In analogy with the rational singularities, in the main result, we characterize a strongly elliptic singularity in terms of the normal Hilbert coefficients of the integrally closed $\mathfrak m$-primary ideals of A. Unlike $p_g$-ideals, elliptic ideals and strongly elliptic ideals are not necessarily normal and necessary, and sufficient conditions for being normal are given. In the last section, we discuss the existence (and the effective construction) of strongly elliptic ideals in any two-dimensional normal local ring.

Let $f\colon Y \to X$ be a proper flat morphism of locally noetherian schemes. Then the locus in $X$ over which $f$ is smooth is stable under generization. We prove that, under suitable assumptions on the formal fibers of $X$, the same property holds for other local properties of morphisms, even if $f$ is only closed and flat. Our proof of this statement reduces to a purely local question known as Grothendieck's localization problem. To answer Grothendieck's problem, we provide a general framework that gives a uniform treatment of previously known cases of this problem, and also solves this problem in new cases, namely for weak normality, seminormality, $F$-rationality, and the ‘Cohen–Macaulay and $F$-injective’ property. For the weak normality statement, we prove that weak normality always lifts from Cartier divisors. We also solve Grothendieck's localization problem for terminal, canonical, and rational singularities in equal characteristic zero.

A local ring R is regular if and only if every finitely generated R-module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category $\mathsf {D}^{\mathsf f}(R)$, which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer–Greenlees–Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in $\mathsf {D}^{\mathsf f}(R)$ is proxy small. In this paper, we study a return to the world of R-modules, and search for finitely generated R-modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley–Reisner rings.

We relate the analytic spread of a module expressed as the direct sum of two submodules with the analytic spread of its components. We also study a class of submodules whose integral closure can be expressed in terms of the integral closure of its row ideals, and therefore can be obtained by means of a simple computer algebra procedure. In particular, we analyze a class of modules, not necessarily of maximal rank, whose integral closure is determined by the family of Newton polyhedra of their row ideals.

Unfortunately, there is a mistake in [PS, Lemma 3.10] which invalidates [PS, Theorem 3.12]. We show that the theorem still holds if the ring is assumed to be Gorenstein.