Serre duality for non-commutative P^1-bundles

Non-commutative Mori contractions and $\PP^1$-bundles

We give a method for constructing maps from a non-commutative scheme to a commutative projective curve. With the aid of Artin-Zhang's abstract Hilbert schemes, this is used to construct analogues of the extremal contraction of a $K$-negative curve with self-intersection zero on a smooth projective surface. This result will hopefully be useful in studying Artin's conjecture on the birational classification of non-commutative surfaces. As a non-trivial example of the theory developed, we look at non-commutative ruled surfaces and, more generally, at non-commutative $\PP^1$-bundles. We show in pa…

Coherent rings, fp-injective modules, dualizing complexes, and covariant Serre-Grothendieck duality

For a left coherent ring A with every left ideal having a countable set of generators, we show that the coderived category of left A-modules is compactly generated by the bounded derived category of finitely presented left A-modules (reproducing a particular case of a recent result of Stovicek with our methods). Furthermore, we present the definition of a dualizing complex of fp-injective modules over a pair of noncommutative coherent rings A and B, and construct an equivalence between the coderived category of A-modules and the contraderived category of B-modules. Finally, we define the notio…

On Serre dimension of monoid algebras and Segre extensions

Let $R$ be a commutative noetherian ring of dimension $d$ and $M$ be a commutative$,$ cancellative$,$ torsion-free monoid of rank $r$. Then $S$-$dim(R[M]) \leq max\{1, dim(R[M])-1 \} = max\{1, d+r-1 \}$. Further$,$ we define a class of monoids $\{\mathfrak{M}_n\}_{n \geq 1}$ such that if $M \in \mathfrak{M}_n$ is seminormal$,$ then $S$-$dim(R[M]) \leq dim(R[M]) - n= d+r-n,$ where $1 \leq n \leq r$. As an application, we prove that for the Segre extension $S_{mn}(R)$ over $R,$ $S$-$dim(S_{mn}(R)) \leq dim(S_{mn}(R)) - \Big[\frac{m+n-1}{min\{m,n\}}\Big] = d+m+n-1 - \Big[\frac{m+n-1}{min\{m,n\}}\…

Homological duality for covering groups of reductive $p$-adic groups

In this largely expository paper we extend properties of the homological duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is the Hecke algebra of a reductive $p$-adic group, to the case where it is the Hecke algebra of a finite central extension of a reductive $p$-adic group. The most important properties being that $RHom_{\mathcal H}(-,{\mathcal H})$ is concentrated in a single degree for irreducible representations and that it gives rise to Schneider--Stuhler duality for Ext groups (a Serre functor like property). Along the way we also study Grothendieck--Serre dualit…

Non-commutative algebraic geometry of semi-graded rings

In this paper we introduce the semi-graded rings, which extend graded rings and skew PBW extensions. For this new type of non-commutative rings we will discuss some basic problems of non-commutative algebraic geometry. In particular, we will prove some elementary properties of the generalized Hilbert series, Hilbert polynomial and Gelfand-Kirillov dimension. We will extended the notion of non-commutative projective scheme to the case of semi-graded rings and we generalize the Serre-Artin-Zhang-Verevkin theorem. Some examples are included at the end of the paper.

有界投射维数模的有限型与Serre条件

The finite type of modules of bounded projective dimension and Serre's conditions

本文证明了交换noetherian环上投射维数以$k$为界的模类为有限型当且仅当满足Serre条件$(S_k)$，并推广证明了Govorov-Lazard定理的$k$维版本。

Duality and Rational Modules in Hopf Algebras over Commutative Rings

Let $A$ be an algebra over a commutative ring $R$. If $R$ is noetherian and $A^\circ$ is pure in $R^A$, then the categories of rational left $A$-modules and right $A^\circ$-comodules are isomorphic. In the Hopf algebra case, we can also strengthen the Blattner-Montgomery duality theorem. Finally, we give sufficient conditions to get the purity of $A^\circ$ in $R^A$.

A dualizing object approach to non-commutative Stone duality

The aim of the present paper is to extend the dualizing object approach to Stone duality to the non-commutative setting of skew Boolean algebras. This continues the study of non-commutative generalizations of different forms of Stone duality initiated in recent papers by Bauer and Cvetko-Vah, Lawson, Lawson and Lenz, Resende, and also the current author. In particular, we construct a series of dual adjunctions between the categories of Boolean spaces and skew Boolean algebras, unital versions of which are induced by dualizing objects ${0,1,..., n+1}$, $n\geq 0$. We describe Eilenberg-Moore cat…

A non-commutative Priestley duality

We prove that the category of left-handed strongly distributive skew lattices with zero and proper homomorphisms is dually equivalent to a category of sheaves over local Priestley spaces. Our result thus provides a non-commutative version of classical Priestley duality for distributive lattices and generalizes the recent development of Stone duality for skew Boolean algebras. From the point of view of skew lattices, Leech showed early on that any strongly distributive skew lattice can be embedded in the skew lattice of partial functions on some set with the operations being given by restrictio…

Commutative and non-commutative bialgebras of quasi-posets and applications to Ehrhart polynomials

To any poset or quasi-poset is attached a lattice polytope, whose Ehrhart polynomial we study from a Hopf-algebraic point of view. We use for this two interacting bialgebras on quasi-posets. The Ehrhart polynomial defines a Hopf algebra morphism taking its values in Q[X], we deduce from the interacting bialgebras an algebraic proof of the duality principle, a generalization and a new proof of a result on B-series due to Whright and Zhao, using a monoid of characters on quasi-posets, and a generalization of Faulhaber's formula. We also give non-commutative versions of these results: polynomials…