arXiv subjects

Alexander Slavik

Publications and source records attributed to Alexander Slavik.

6 recordsLinked to original sources

arXiv2019

Flat morphisms of finite presentation are very flat

Principal affine open subsets in affine schemes are an important tool in the foundations of algebraic geometry. Given a commutative ring $R$, $\,R$-modules built from the rings of functions on principal affine open subschemes in $\operatorname{Spec}R$ using ordinal-indexed filtrations and direct summands are called very flat. The related class of very flat quasi-coherent sheaves over a scheme is intermediate between the classes of locally free and flat sheaves, and has serious technical advantages over both. In this paper we show that very flat modules and sheaves are ubiquitous in algebraic geometry: if $S$ is a finitely presented commutative $R$-algebra which is flat as an $R$-module, then $S$ is a very flat $R$-module. This proves a conjecture formulated in the February 2014 version of the long preprint arXiv:1209.2995. We also show that the (finite) very flatness property of a flat module satisfies descent with respect to commutative ring homomorphisms of finite presentation inducing surjective maps of the spectra.

math.AC↗

arXiv2019

Purity in categories of sheaves

We consider categorical and geometric purity for sheaves of modules over a scheme satisfying some mild conditions, both for the category of all sheaves and for the category of quasicoherent sheaves. We investigate the relations between these four purities and compute a number of examples, in particular describing both the geometric and categorical Ziegler spectra for the category of quasicoherent sheaves over the projective line over a field.

math.AG↗

arXiv2018

On strongly flat and weakly cotorsion modules

The aim of this paper is to describe the classes of strongly flat and weakly cotorsion modules with respect to a multiplicative subset or a finite collection of multiplicative subsets in a commutative ring. The strongly flat modules are characterized by a set of conditions, while the weakly cotorsion modules are produced by a generation procedure. Passing to the collection of all countable multiplicative subsets, we define quite flat and almost cotorsion modules, and show that, over a Noetherian ring with countable spectrum, all flat modules are quite flat and all almost cotorsion modules are cotorsion.

math.AC↗

arXiv2018

Quillen equivalent models for the derived category of flats and the resolution property

We investigate under which assumptions a subclass of flat quasi-coherent shea\-ves on a quasi-compact and semi-separated scheme allows to "mock" the homotopy category of projective modules. Our methods are based on module theoretic properties of the subclass of flat modules involved as well as their behaviour with respect to Zariski localizations. As a consequence we get that, for such schemes, the derived category of flats is equivalent to the derived category of very flats. If, in addition, the scheme satisfies the resolution property then both derived categories are equivalent to the derived category of infinite-dimensional vector bundles. The equivalences are inferred from a Quillen equivalence between the corresponding models.

math.AT↗

arXiv2016

Very flat, locally very flat, and contraadjusted modules

Very flat and contradjusted modules naturally arise in algebraic geometry in the study of contraherent cosheaves over schemes. Here, we investigate the structure and approximation properties of these modules over commutative noetherian rings. Using an analogy between projective and flat Mittag-Leffler modules on one hand, and very flat and locally very flat modules on the other, we prove that each of the following statements are equivalent to the finiteness of the Zariski spectrum Spec(R) of a noetherian domain R: (i) the class of all very flat modules is covering, (ii) the class of all locally very flat modules is precovering, and (iii) the class of all contraadjusted modules is enveloping. We also prove an analog of Pontryagin's criterion for locally very flat modules over Dedekind domains.

math.AC↗

arXiv2012

Approximations and locally free modules

For any set of modules S, we prove the existence of precovers (right approximations) for all classes of modules of bounded C-resolution dimension, where C is the class of all S-filtered modules. In contrast, we use infinite dimensional tilting theory to show that the class of all locally free modules induced by a non-sum-pure-split tilting module is not precovering. Consequently, the class of all locally Baer modules is not precovering for any countable hereditary artin algebra of infinite representation type.

math.RT↗