Mathematics

In this article a condition is given to detect the containment among thick subcategories of the bounded derived category of a commutative noetherian ring. More precisely, for a commutative noetherian ring R and complexes of R -modules with finitely generated homology M and N , we show N is in the thick subcategory generated by M if and only if the ghost index of N p with respect to M p is finite for each prime p of R . To do so, we establish a "converse coghost lemma" for the bounded derived category of a non-negatively graded DG algebra with noetherian homology.

In this paper we introduce the concept of clique disjoint edge sets in graphs. Then, for a graph G , we define the invariant η(G) as the maximum size of a clique disjoint edge set in G . We show that the regularity of the binomial edge ideal of G is bounded above by η(G) . This, in particular, settles a conjecture on the regularity of binomial edge ideals in full generality.

Let L ∗ be a family of finite subsets of N 0 having the following properties. (a). {0},{1}∈ L ∗ and all other sets of L ∗ lie in N ≥2 . (b). If L 1 , L 2 ∈ L ∗ , then the sumset L 1 + L 2 ∈ L ∗ . We show that there is a Dedekind domain D whose system of sets of lengths equals L ∗ .

In this paper, motivated by a work of Luk and Yau, and Huneke and Wiegand, we study various aspects of the cohomological rigidity property of tensor product of modules over commutative Noetherian rings. We determine conditions under which the vanishing of a single local cohomology module of a tensor product implies the vanishing of all the lower ones, and obtain new connections between the local cohomology modules of tensor products and the Tate homology. Our argument yields bounds for the depth of tensor products of modules, as well as criteria for freeness of modules over complete intersection rings. Along the way, we also give a splitting criteria for vector bundles on smooth complete intersections.

It is shown in a local strongly F -regular ring there exits natural number e 0 so that if M is any finitely generated maximal Cohen-Macaulay module then the pushforward of M under the e 0 th iterate of the Frobenius endomorphism contains a free summand. Consequently, the torsion subgroup of the divisor class group of a local strongly F -regular ring is finite.

Let a be a proper ideal of a commutative Noetherian ring R with identity. Using the notion of a-relative system of parameters, we introduce a relative variant of the notion of generalized Cohen-Macaulay modules. In particular, we examine relative analogues of quasi-Buchsbaum, Buchsbaum and surjective Buchsbaum modules. We reveal several interactions between these types of modules that extend some of the existing results in the classical theory to the relative one.

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.

Let A and B be integral domains. Suppose A is Noetherian and B is a finitely generated A-algebra that contains A. Denote by A' the integral closure of A in B. We show that A' is determined by finitely many unique discrete valuation rings. Our result generalizes Rees' classical valuation theorem for ideals. We also obtain a variant of Zariski's main theorem.

In this paper, we prove that a complete Noetherian local domain of mixed characteristic p>0 with perfect residue field has an integral extension that is an integrally closed, almost Cohen-Macaulay domain such that the Frobenius map is surjective modulo p . This result is seen as a mixed characteristic analogue of the fact that the perfect closure of a complete local domain in positive characteristic is almost Cohen-Macaulay. To this aim, we carry out a detailed study of decompletion of perfectoid rings and establish the Witt-perfect (decompleted) version of André's perfectoid Abhyankar's lemma and Riemann's extension theorem.

Let x – – = x 1 ,…, x k denote an ordered sequence of elements of a commutative ring R . Let M be an R -module. We recall the two notions that x – – is M -proregular given by Greenlees and May (see \cite{[5]}) and Lipman (see \cite{[1]}) and show that both notions are equivalent. As a main result we prove a cohomological characterization for x – – to be M -proregular in terms of Čech homology. This implies also that x – – is M -weakly proregular if it is M -proregular. A local-global principle for proregularity and weakly proregularity is proved. This is used for a result about prisms as introduced by Bhatt and Scholze (see \cite{[3]}).