Mathematics

We show that compactly generated t-structures in the derived category of a commutative ring R are in a bijection with certain families of compactly generated t-structures over the local rings R m where m runs through the maximal ideals in the Zariski spectrum Spec(R) . The families are precisely those satisfying a gluing condition for the associated sequence of Thomason subsets of Spec(R) . As one application, we show that the compact generation of a homotopically smashing t-structure can be checked locally over localizations at maximal ideals. In combination with a result due to Balmer and Favi, we conclude that the ??-Telescope Conjecture for a quasi-coherent and quasi-separated scheme is a stalk-local property. Furthermore, we generalize the results of Trlifaj and ?ahinkaya and establish an explicit bijection between cosilting objects of cofinite type over R and compatible families of cosilting objects of cofinite type over all localizations R m at maximal primes.

We classify connected graphs G whose binomial edge ideal is Gorenstein. The proof uses methods in prime characteristic.

We characterize all Gorenstein rings generated by strongly stable sets of monomials of degree two. We compute their Hilbert series in several cases, which also provides an answer to a question by Migliore and Nagel.

In this paper we study graded Bourbaki ideals. It is a well-known fact that for torsionfree modules over Noetherian normal domains, Bourbaki sequences exist. We give criteria in terms of certain attached matrices for a homomorphism of modules to induce a Bourbaki sequence. Special attention is given to graded Bourbaki sequences. In the second part of the paper, we apply these results to the Koszul cycles of the residue class field and determine particular Bourbaki ideals explicitly. We also obtain in a special case the relationship between the structure of the Rees algebra of a Koszul cycle and the Rees algebra of its Bourbaki ideal.

Let R be a positively graded algebra over a field. We say that R is Hilbert-cyclotomic if the numerator of its reduced Hilbert series has all of its roots on the unit circle. Such rings arise naturally in commutative algebra, numerical semigroup theory and Ehrhart theory. If R is standard graded, we prove that, under the additional hypothesis that R is Koszul or has an irreducible h -polynomial, Hilbert-cyclotomic algebras coincide with complete intersections. In the Koszul case, this is a consequence of some classical results about the vanishing of deviations of a graded algebra.

Let $R=\oplus_{n\in \N_0}R_n$ be a standard graded ring, M be a finitely generated graded R -module and $R_+:=\oplus_{n\in \N}R_n$ denotes the irrelevant ideal of R . In this paper, considering the new concept of linkage of ideals over a module, we study the graded components $H^i_{\fa}(M)_n$ when $\fa$ is an h-linked ideal over M . More precisely, we show that $H^i_{\fa}(M)$ is tame in each of the following cases: \begin{itemize} \item [(i)] $i=f_{\fa}^{R_+}(M)$, the first integer $i$ for which $R_+\nsubseteq \sqrt{0:H^i_{\fa}(M)}$; \item [(ii)] $i=\cd(R_+,M)$, the last integer $i$ for which $H^{i}_{R_+}(M)\neq 0$, and $\fa=\fb+R_+$ where $\fb$ is an h-linked ideal with $R_+$ over $M$. \end{itemize} Also, among other things, we describe the components $H^i_{\fa}(M)_n$ where $\fa$ is radically h- M -licci with respect to R + of length 2.

Let A be a commutative noetherian ring. Let H(A) be the quotient of the Grothendieck group of finitely generated A-modules by the subgroup generated by pseudo-zero modules. Suppose that the real vector space H(A)_R = H(A) \otimes_Z R has finite dimension. Let C(A) (resp. C_r(A)) be the convex cone in H(A)_R spanned by maximal Cohen-Macaulay A-modules (resp. maximal Cohen-Macaulay A-modules of rank r). We explore the interior, closure and boundary, and convex polyhedral subcones of C(A). We provide various equivalent conditions for A to have only finitely many rank r maximal Cohen-Macaulay points in C_r(A) in terms of topological properties of C_r(A). Finally, we consider maximal Cohen-Macaulay modules of rank one as elements of the divisor class group Cl(A).

In 1980, White conjectured that the toric ideal of a matroid is generated by quadratic binomials corresponding to a symmetric exchange. In this paper, we compute Gröbner bases of toric ideals associated with matroids and show that, for every matroid on ground sets of size at most seven except for two matroids, Gröbner bases of toric ideals consist of quadratic binomials corresponding to a symmetric exchange.

We introduce the concept of a Gröbner nice pair of ideals in a polynomial ring and we present some applications.

Let S=K[ x 1 ,…, x n ] be a polynomial ring, where K is a field, and G be a simple graph on n vertices. Let J(G)⊂S be the vertex cover ideal of G . Herzog, Hibi and Ohsugi have conjectured that all powers of vertex cover ideals of chordal graph are componentwise linear. Here we establish the conjecture for the special case of trees. We also show that if G is a unicyclic vertex decomposable graph that does not contain C 3 or C 5 , then symbolic powers of J(G) are componentwise linear.