Abstract. Given surjective homomorphisms R → T ← S of local rings, and ideals in R and S that are isomorphic to some T -module V , the connected sum R# T S is defined to be ring obtained by factoring out the diagonal image of V in the fiber product R × T S. When T is Cohen-Macaulay of dimension d and V is a canonical module of T , it is proved that if R and S are Gorenstein of dimension d, then so is R# T S. This result is used to study how closely an artinian ring can be approximated by a Gorenstein ring mapping onto it. When T is regular, it is shown that R# T S almost never is a complete intersection ring. The proof uses a presentation of the cohomology algebra Ext * R# k S (k, k) as an amalgam of the algebras Ext * R (k, k) and Ext * S (k, k) over isomorphic polynomial subalgebras generated by one element of degree 2.
Let R be an isolated hypersurface singularity, and let M and N be finitely generated R-modules. As R is a hypersurface, the torsion modules of M against N are eventually periodic of period two (i.e., Tor R i (M, N ) ∼ = Tor R i+2 (M, N ) for i ≫ 0). Since R has only an isolated singularity, these torsion modules are of finite length for i ≫ 0. The theta invariant of the pair (M, N ) is defined by Hochster to be length(Tor R 2i (M, N )) − length(Tor R 2i+1 (M, N )) for i ≫ 0. H. Dao has conjectured that the theta invariant is zero for all pairs (M, N ) when R has even dimension and contains a field. This paper proves this conjecture under the additional assumption that R is graded with its irrelevant maximal ideal giving the isolated singularity. We also give a careful analysis of the theta pairing when the dimension of R is odd, and relate it to a classical pairing on the smooth variety Proj(R).
Abstract. The notion of a matrix factorization was introduced by Eisenbud in the commutative case in his study of bounded (periodic) free resolutions over complete intersections. Since then, matrix factorizations have appeared in a number of applications. In this work, we extend the notion of (homogeneous) matrix factorizations to regular normal elements of connected graded algebras over a field.Next, we relate the category of twisted matrix factorizations to an element over a ring and certain Zhang twists. We also show that in the AS-regular setting, every sufficiently high syzygy module is the cokernel of some twisted matrix factorization. Furthermore, we show that in this setting there is an equivalence of categories between the homotopy category of twisted matrix factorizations and the singularity category of the hypersurface, following work of Orlov.
Differential graded algebra techniques have played a crucial role in the development of homological algebra, especially in the study of homological properties of commutative rings carried out by Serre, Tate, Gulliksen, Avramov, and others. In this article, we extend the construction of the Koszul complex and acyclic closure to a more general setting. As an application of our constructions, we shine some light on the structure of the Ext algebra of quotients of skew polynomial rings by ideals generated by normal elements. As a consequence, we give a presentation of the Ext algebra when the elements generating the ideal form a regular sequence, generalizing a theorem of Bergh and Oppermann. It follows that in this case the Ext algebra is noetherian, providing a partial answer to a question of Kirkman, Kuzmanovich and Zhang.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.