In a paper in 1962, Golod proved that the Betti sequence of the residue field of a local ring attains an upper bound given by Serre if and only if the homology algebra of the Koszul complex of the ring has trivial multiplications and trivial Massey operations. This is the origin of the notion of Golod ring. Using the Koszul complex components he also constructed a minimal free resolution of the residue field. In this article, we extend this construction up to degree five for any local ring. We describe how the multiplicative structure and the triple Massey products of the homology of the Koszul algebra are involved in this construction. As a consequence, we provide explicit formulas for the first six terms of a sequence that measures how far the ring is from being Golod.
For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the $n$-th valued hyperfields of $K_1$ and $K_2$ are isomorphic over $p$ for each $n\ge1$, then $K_1$ and $K_2$ are isomorphic. More generally, for $n_1,n_2\ge 1$, if $n_2$ is large enough, then any homomorphism, which is over $p$, from the $n_1$-th valued hyperfield of $K_1$ to the $n_2$-th valued hyperfield of $K_2$ can be lifted to a homomorphism from $K_1$ to $K_2$. We compute such $n_2$ effectively, which depends only on the ramification indices of $K_1$ and $K_…
It is shown that every dp-minimal integral domain $R$ is a local ring and for every non-maximal prime ideal $\mathfrak p $ of $R$, the localization $R_{\mathfrak p }$ is a valuation ring and $\mathfrak{p}R_{\mathfrak{p}}=\mathfrak{p}$. Furthermore, a dp-minimal integral domain is a valuation ring if and only if its residue field is infinite or its residue field is finite and its maximal ideal is principal.
Let R be a commutative noetherian local ring with residue field k and assume that it is not Gorenstein. In the minimal injective resolution of R, the injective envelope E of the residue field appears as a summand in every degree starting from the depth of R. The number of copies of E in degree i equals the k-vector space dimension of the cohomology module Ext^i(k,R). These dimensions, known as Bass numbers, form an infinite sequence of invariants of R about which little is known. We prove that it is non-decreasing and grows exponentially if R is Golod, a non-trivial fiber product, or Teter, or…
We derive basic properties of minimal extensions of local rings and their restrictions to subrings. Some applications are included to subrings of truncated polynomial rings.
Forman's Discrete Morse theory is studied from an algebraic viewpoint. Analogous to independent work of Emil Skoeldberg we show that this theory can be extended to chain complexes of free modules over a ring. We provide three applications of this theory: We construct new resolutions of the residue class field $k$ over $A$, where $A$ is the quotient of a (i) commutative polynomial ring or (ii) non-commutaitve polynomial ring by a (twosided) ideal and (iii) we construct a new resolution of $A$ as an $A \otimes A^{op}$-module in the situation (ii). In either case we prove minimality of the resolu…
Minimal pairs of definition were introduced by Alexandru, Popescu and Zaharescu to study residue transcendental extensions. In this paper we obtain analogous results in the value transcendental case. We introduce the notion of minimal fields of definition for valuation transcendental extensions and show that they share some common ramification theoretic properties. The connection between minimal fields of definition and implicit constant fields is also investigated. Further, we explore the relationship between valuation transcendental extensions and pseudo-Cauchy sequences.
The covering number of an associative ring $R$ is the minimal number of proper subrings whose union is $R$. We establish a strategy to classify unital rings of a given finite covering number, and obtain a classification of unital rings whose covering number is four. Along the way we compute the covering number of every finite local ring whose residue field has prime order.
Expanding on the work of Fouli and Vassilev \cite{FV}, we determine a formula for the *-$\rm{core}$ of an ideal in two different settings: (1) in a Cohen--Macaulay local ring of characteristic $p>0$, perfect residue field and test ideal of depth at least two, where the ideal has a minimal *-reduction that is a parameter ideal and (2) in a normal local domain of characteristic $p>0$, perfect residue field and $\m$-primary test ideal, where the ideal is a sufficiently high Frobenius power of an ideal. We also exhibit some examples where our formula fails if our hypotheses are not met.
Let $(A, \mathfrak{m})$ be a $d$-dimensional Cohen-Macaulay local ring with infinite residue field and let $J$ be a minimal reduction of $\mathfrak{m}$. We show that $λ(\mathfrak{m}^3/J\mathfrak{m}^2)$ is independent of $J$.
The join-meet ideal was introduced by Takayuki Hibi in 1987. It is binomial ideals that are defined by finite lattices. We study the join-meet ideal of non-distributive finite lattices that do not always satisfy modular. In particular, we work on the case of Crystal lattice which is one of them. It was introduced by Yohei Oshida in 2022. The Crystal module in the title is the residue class ring which is in relation to the join-meet ideal of Crystal lattice. In this paper, we give important inequalities about the minimal free resolution of the Crystal module. The important point about this resu…
We find the minimal dimension for a truncated polynomial algebra over an arbitrary field for which there exists a "non-thin" subalgebra. Moreover, we discuss examples of subalgebras, and count them in low dimensions.
Let $(R,m)$ be a Cohen-Macaulay local ring of positive dimension $d$ and infinite residue field. Let $I$ be an m-primary ideal and $J$ a minimal reduction of $I$. In this paper, we show that $\widetilde{r_J(I)}\leq r_J(I)$. This answer to a question that made by M.E. Rossi and I. Swanson in [\ref{Rs}, Question 4.6].
Let $ R $ be a $ d $-dimensional Cohen-Macaulay (CM) local ring of minimal multiplicity. Set $ S := R/({\bf f}) $, where $ {\bf f} := f_1,\ldots,f_c $ is an $ R $-regular sequence. Suppose $ M $ and $ N $ are maximal CM $ S $-modules. It is shown that if $ \mathrm{Ext}_S^i(M,N) = 0 $ for some $ (d+c+1) $ consecutive values of $ i \geqslant 2 $, then $ \mathrm{Ext}_S^i(M,N) = 0 $ for all $ i \geqslant 1 $. Moreover, if this holds true, then either $ \mathrm{projdim}_R(M) $ or $ \mathrm{injdim}_R(N) $ is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, …
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…
Two ideals $I$ and $J$ are called transverse if $I \cap J = IJ$. We show that the obstructions defined by Avramov for classes of (sequentially) transverse ideals in regular local rings are always $0$. In particular, we compute an explicit free resolution and Koszul homology for all such ideals. Moreover, we construct an explicit trivial Massey operation on the associated Koszul complex and hence (by Golod's construction) a minimal free resolution of the residue field over the quotient defined by the product of transverse ideals. We conclude with questions about the existence of associative mul…
Let $\mathcal{K}=(K,v,\ldots)$ be a dp-minimal expansion of a non-trivially valued field of characteristic $0$ and $\mathcal{F}$ an infinite field interpretable in $\mathcal{K}$. Assume that $\mathcal{K}$ is one of the following: (i) $V$-minimal, (ii) power bounded $T$-convex, or (iii) $P$-minimal (assuming additionally in (iii) generic differentiability of definable functions). Then $\mathcal{F}$ is definably isomorphic to a finite extension $K$ or, in cases (i) and (ii), its residue field. In particular, every infinite field interpretable in $\mathbb{Q}_p$ is definably isomorphic to a finite…
Let G be a unitary group of a signed-Hermitian form h given over a non-Archimedian local field k of residue characteristic not two. Let V be the vector space on which h is defined. We consider minimal skew-strata, more precisely pairs (b,a) consisting of a Lie algebra element b and a hereditary order $a$ stable under the adjoint involution of h, such that b generates a field whose multiplicative group is a subset of the normalizer of $a$, and some more conditions. We prove that if two minimal skew-strata (b_i,a), i=1,2 interwine by an element of G, then they are conjugate under G, and we give …
We study the multigraded Poincaré-Betti series of $A=S/\aaf$, where $S$ is the ring of polynomials in $n$ indeterminates divided by the monomial ideal $\aaf$. There is a conjecture about the multigraded Poincaré-Betti series by Charalambous and Reeves which they proved in the case, where the Taylor resolution is minimal. We introduce a conjecture about the minimal $A$-free resolution of the residue class field and show that this conjecture implies the conjecture of Charalambous and Reeves. We prove our conjecture for several classes of algebras. The conjecture implies that $A$ is Golod if and …
We prove a generalized Dade's Lemma for quotients of local rings by ideals generated by regular sequences. That is, given a pair of finitely generated modules over such a ring with algebraically closed residue field, we prove a sufficient (and necessary) condition for the vanishing of all higher Ext or Tor of the modules. This condition involves the vanishing of all higher Ext or Tor of the modules over all quotients by a minimal generator of the ideal generated by the regular sequence.
