A note on Gorenstein spaces

Advances in Mathematics, 1988

Topology, 1986

International Mathematics Research Notices, 2022

We introduce the cohomological blowup of a graded Artinian Gorenstein algebra along a surjective map, which we term BUG (blowup Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blowup of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are complete intersections. We have included many examples throughout this manuscript.

Arxiv preprint math/0702590, 2007

International Journal of Algebra, 2014

In this paper we continue the study of non connected graded Gorenstein algebras initiated in [13], the main result is the proof of a version of the Local Cohomology formula.

Russian Mathematical Surveys, 1980

Communications in Algebra, 2013

Let K be an algebraically closed field of characteristic 0, and let A be an Artinian Gorenstein local commutative and Noetherian K-algebra, with maximal ideal m. In the present paper we prove a structure theorem describing such kind of K-algebras satisfying m 4 = 0. We use this result in order to prove that such a K-algebra A has rational Poincaré series and it is always smoothable in any embedding dimension, if dim K m 2 /m 3 ≤ 4. We also prove that the generic Artinian Gorenstein local K-algebra with socle degree three has rational Poincaré series, in spite of the fact that such algebras are not necessarily smoothable.

Journal of Mathematical Sciences, 2007

In the survey, we deal with the following situation. Let A be a graded algebra or a differential graded algebra. Adjoining a set x of free (in any sense) indeterminates, we make a new differential graded algebra A x by setting the differential values d : x → A on x. In the general case, such a construction is called the Shafarevich complex. Beginning with classical examples like the bar-complex, Koszul complex, and Tate resolution, we discuss noncommutative (and sometimes even nonassociative) versions of these notions. The comparison with the Koszul complex leads to noncommutative regular sequences and complete intersections; Tate's process of killing cycles gives noncommutative DG resolutions and minimal models. The applications include the Golod-Shafarevich theorem, growth measures for graded algebras, characterizations of algebras of low homological dimension, and a homological description of Gröbner bases. The same constructions for categories of algebras with identities (like Lie or Jordan algebras) allow one to give a homological description of extensions and deformations of PI-algebras.

Mathematische Zeitschrift, 1979

Definition 1.1. We say that two spaces X and Y of finite type (see [3]) have the same genus if for every prime p, X~p) is homotopy equivalent to Y(p). The set of all homotopy types which have the same genus as X is denoted by G(X). Definition 1.2 (see [6]). Let X be a CW complex. We say that X is of type FP if the singular chain complex C, (J~) of the universal cover 3) of X is Z [rci(X)]chain homotopy equivalent to a complex of a finite length 0-, e,-~ e,_ 1- , ...-, ~ ~ eo ~ 0 with P/finitely generated projective Z [re a (X)]-modules. If X is of type FP the finiteness obstruction of C.T.C. Wall is defined and it is equal to co(x)= ~ (-1)' [e,]e tCo(Z [~a(x)]). i-O Suppose that X is a nilpotent CW complex of type FP with a finite fundamental group. We show: Proposition A. If YEG(X) then Y is also of type FP. Our main result is: Theorem B. Suppose that X is a nilpotent CW complex of type FP with a finite fundamental group. If Ye G(X) then co(X)-j, co(Y)eim(6: KI(Z/[rh(X)I)-, Ko(Z [rci(X)]) where j: ~i(Y)-+~i(X) is an isomorphism, Ircl(X)l is the order of ~zi(X) and 6 is the connecting homomorphism of the Mayer-Vietoris exact sequence associated to the cartesian square

Manuscripta Mathematica, 2013

We compute the Nakayama automorphism of a PBW-deformation of a Koszul Artin-Schelter Gorenstein algebra of finite global dimension, and give a criterion for an augmented PBW-deformation of a Koszul Calabi-Yau algebra to be Calabi-Yau. The relations between the Calabi-Yau property of augmented PBW-deformations and that of non-augmented cases are discussed. The Nakayama automorphisms of PBW-deformations of Koszul Artin-Schelter Gorenstein algebras of global dimensions 2 and 3 are given explicitly. We show that if a PBW-deformation of a graded Calabi-Yau algebra is still Calabi-Yau, then it is defined by a potential under some mild conditions. Some classical results are also recovered. Our main method used in this paper is elementary and based on linear algebra.