Unless specified otherwise, by “manifolds” and “varieties” we shall always mean, respectively, complex manifolds and complex algebraic varieties. Consequently, “dimension” (of a manifold or variety) always refers to the complex dimension. We also fix, once and for all, a commutative Noetherian ring k of finite global dimension as our ring of “coefficients”; the reader is welcome to take k = Z,Q...

Given a homomorphism of commutative noetherian rings R → S and an S–module N , it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals sup {m ∈ Z | TorRm(E,N) 6= 0}, where E is the injective hull of the residue field of R. This r...

Let R be a commutative Noetherian ring and let G be the category of modules of G-dimension zero over R. We denote the associated stable category by G. We show that the functor category modG is a Frobenius category and we argue how this property could characterize G as a subcategory of modR.

Considering modules of finite complete intersection dimension over commutative Noetherian local rings, we prove (co)homology vanishing results in which we assume the vanishing of nonconsecutive (co)homology groups. In fact, the (co)homology groups assumed to vanish may be arbitrarily far apart

For a Noetherian local ring, the prime ideals in the singular locus completely determine the category of finitely generated modules up to direct summands, extensions and syzygies. From this some simple homological criteria are derived for testing whether an arbitrary module has finite projective dimension.

The left and right homological integrals are introduced for a large class of infinite dimensional Hopf algebras. Using the homological integrals we prove a version of Maschke’s theorem for infinite dimensional Hopf algebras. The generalization of Maschke’s theorem and homological integrals are the keys to studying noetherian regular Hopf algebras of Gelfand-Kirillov dimension one.

Introduction Let S be a Noetherian separated scheme of finite dimension, and let SH(S) denote the motivic stable homotopy category of Morel and Voevodsky. In order to get a motivic version of the Postnikov tower, Voevodsky [23] constructs a filtered family of triangulated subcategories of SH(S):

We investigate width and Krull–Gabriel dimension over commutative Noetherian rings which are “tame” according to the Klingler–Levy analysis in [4], [5] and [6], in particular over Dedekind-like rings and their homomorphic images. We show that both are undefined in most cases.

Let $R$ be a Noetherian local ring. We prove that is regular of dimension at most four if, and only every prime ideal, defining Gorenstein quotient ring, syzygetic. deduce characterization these rings in terms the Andr\'e-Quillen homology.