The paper presents a further step within a comprehensive program of “Dualizing Universal Algebra” [4, 7, 10, 11, 12]. We analyze the construction of subalgebras generated by a given subset (of generators) of the carrier of an algebra and reformulate it in a systematic categorical way. Dualizing the new categorical description, we obtain then the construction of quotient coalgebras “cogenerated”...

It was proved by Beligiannis and Krause that over certain Artin algebras, there are Gorenstein flat modules which are not direct limits of finitely generated Gorenstein projective modules. That is, these algebras have no Gorenstein analogue of the Govorov-Lazard Theorem. We show that, in fact, there is a large class of rings without such an analogue. Namely, let R be a commutative local noether...

To a topological group G, we assign a naive G-spectrum DG, called the dualizing spectrum of G. When the classifying space BG is finitely dominated, we show that DG detects Poincaré duality in the sense that BG is a Poincaré duality space if and only ifDG is a homotopy finite spectrum. Secondly, we show that the dualizing spectrum behavesmultiplicatively on certain topological group extensions. ...

In this paper we present a new approach to Grothendieck duality on schemes. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. We obtain most of the important features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results apply to finite t...

Definition 1.1. A curved L∞ algebra over A consists of a locally free finitely generated graded A]-module V, together with a cohomological degree 1 and square zero derivation d : Ŝym(V∨[−1])→ Ŝym(V∨[−1]) where V∨ is the A]-linear dual and the completed symmetric algebra is also over A]. There are two additional requirements on the derivation d: (1) The derivation d makes Ŝym(V∨[−1]) into a dga ...

For a one-parameter degeneration of reduced compact complex analytic spaces dimension $n$, we prove the invariance frontier Hodge numbers $h^{p,q}$ (that is, with $pq(n-p)(n-q)=0$) for intersection cohomology fibers and also their desingularizations, assuming that central fiber is reduced, projective, has only rational singularities. This can be shown to equivalent structure sheaf (which known ...

We prove that the shifted vanishing cycles and nearby cycles commute with Verdier dualizing up to a natural isomorphism, even when the coefficients are not in a field.