A counterexample to the Hermite ring conjecture
We show that there exists a stably free module over a polynomial ring which is not extended from the ground ring. This provides a counterexample to the Hermite ring conjecture.
math.AC↗
arXiv subjects
Daniel Schäppi
Publications and source records attributed to Daniel Schäppi.
Symplectic K-theory and a problem of Murthy
We compute low-dimensional K-groups of certain rings associated with the study of the Hermite ring conjecture. This includes a monoid ring whose low-dimensional K-groups were recently computed by Krishna and Sarwar in the case where the base ring is a regular ring containing the rationals. We are able to extend their result to an arbitrary regular base ring, thereby completing an answer to a question of Gubeladze. Our computation only relies on certain conveniently chosen analytic patching diagrams. These patching diagrams also allow us to investigate stably free modules appearing in a problem posed by Murthy. They make it possible to relate Murthy's problem to a stable question (about the relationship between symplectic and ordinary K-theory). More precisely, we show that Murthy's problem has a solution if the stable question has an affirmative answer, while a negative answer to the stable question implies that there are counterexamples to the Hermite ring conjecture. Since the module appearing in Murthy's problem has rank 2, the above mentioned argument relies on some theorems about patching with 2-by-2 matrices. These use so-called pseudoelementary 2-by-2 matrices, a notion we introduce which makes it possible to extend certain results valid for larger matrices to the case of 2-by-2 matrices (if we use pseudoelementary instead of elementary matrices). For example, we prove that the analogue of Vorst's theorem about matrices over polynomial extensions over a regular ring containing a field holds for 2-by-2 matrices.
math.AC↗
Flat functors in higher topos theory
For a small $n$-category $\mathscr{C}$ and an $n$-topos $\mathscr{X}$, we study necessary and sufficient conditions for a functor $f \colon \mathscr{C} \to \mathscr{X}$ to determine a geometric morphism from $\mathscr{X}$ to the $n$-topos $\mathcal{P}(\mathscr{C})_n$ of presheaves on $\mathscr{C}$ for any $n \geq 1$. These results generalize and unify results of Lurie for $n=\infty$ and classical characterizations of flat functors (Diaconescu's theorem) for $n=1$. Interestingly, for $n=\infty$, our analogue of Diaconescu's theorem requires hypercompleteness. As an application, we show that the $\infty$-topos associated to an $n$-site behaves as an $n$-localic $\infty$-topos with respect to hypercomplete $\infty$-topoi.
math.CT↗
Formal Laurent series rings and the Hermite ring conjecture
We study the question if projective modules over formal Laurent series rings are extended. We relate this question to the Bass-Quillen conjecture for commutative regular local rings and to the Hermite ring conjecture for all commutative local rings. Using our result about projective modules over formal Laurent series rings, we prove a reduction step for the Hermite ring conjecture. We show that the Hermite ring conjecture holds for all commutative local rings if and only if it holds for complete intersection rings which are also unique factorization domains.
math.AC↗
Flat replacements of homology theories
To a homology theory one can associate an additive site and a new homological functor with values in the category of additive sheaves on that site. If this category of sheaves can be shown to be equivalent to a category of comodules of a Hopf algebroid, then we obtain a new homology theory by composing with the underlying module functor. This new homology theory is always flat and we call it a flat replacement of the original theory. For example, Pstrągowski has shown that complex cobordism is a flat replacement of singular homology. In this article we study the basic properties of the sites associated to homology theories and we prove an existence theorem for flat replacements.
math.AT↗
Graded-Tannakian categories of motives
Given a rigid tensor-triangulated category and a vector space valued homological functor for which the Künneth isomorphism holds, we construct a universal graded-Tannakian category through which the given homological functor factors. We use this to (unconditionally) construct graded-Tannakian categories of pure motives associated to a fixed Weil cohomology theory, with a fiber functor realizing the given cohomology theory. For $\ell$-adic cohomology and a ground field which is algebraic over a finite field, this category is Tannakian. In this case, we obtain in particular motivic Galois groups which act naturally on $\ell$-adic cohomology without assuming any of the standard conjectures. We show that these graded-Tannakian categories are equivalent to Grothendieck's category of pure motives if the standard conjecture D holds.
math.AG↗
Uniqueness of fiber functors and universal Tannakian categories
The principal aim of this note is to give an elementary proof of the fact that any two fiber functors of a Tannakian category are locally isomorphic. This builds on an idea of Deligne concerning scalar extensions of Tannakian categories and implements a proof strategy which Deligne attributes to Grothendieck. Besides categorical generalities, the proof merely relies on basic properties of exterior powers and the classification of finitely generated modules over a principal ideal domain. Using related ideas (but less elementary means) we also present an alternative characterization of Tannakian categories among the more general weakly Tannakian categories. As an application of this result we can construct from any right exact symmetric monoidal abelian category with simple unit object a universal Tannakian category associated to it.
math.AG↗
When coproducts are biproducts
Among right-closed monoidal categories with finite coproducts, we characterise those with finite biproducts as being precisely those in which the initial object and the coproduct of the unit with itself admit right duals. This generalises Houston's result that any compact closed category with finite coproducts admits biproducts.
math.CT↗
Descent via Tannaka duality
Given a diagram of schemes, we can ask if a geometric object over one of them can be built from descent data (usually objects of the same type over the various other schemes in the diagram, together with compatibility isomorphisms). Using the language of moduli stacks, we can rephrase this as follows: saying that descent problems for a given diagram have essentially unique solutions amounts to saying that the diagram in question is a (bicategorical) colimit diagram in a certain 2-category of stacks. In this paper we use generalized Tannaka duality to explicitly compute certain colimits in the 2-category of Adams stacks. Using this we extend recent results of Bhatt from algebraic spaces to Adams stacks and a result of Hall-Rydh to non-noetherian rings. We conclude the paper with a global version of the Beauville-Laszlo theorem, which states that a large class of schemes and stacks can be built by gluing the open complement of an effective Cartier divisor with the infinitesimal neighbourhood of the divisor.
math.AG↗
Constructing colimits by gluing vector bundles
As already observed by Gabriel, coherent sheaves on schemes obtained by gluing affine open subsets can be described by a simple gluing construction. An example due to Ferrand shows that this fails in general for pushouts along closed immersions, though the gluing construction still works for flat coherent sheaves. We show that by further restricting this gluing construction to vector bundles, we can construct pushouts along arbitrary morphisms (and more general colimits) of certain algebraic stacks called Adams stacks. The proof of this fact uses generalized Tannaka duality and a variant of Deligne's argument for the existence of fiber functors which works in arbitrary characteristic. We use this version of Deligne's existence theorem for fiber functors as a novel way of recognizing stacks which have atlases. It differs considerably from Artin's algebraicity results and their generalizations: rather than studying conditions on the functor of points which ensure the existence of an atlas, our theorem identifies conditions on the category of quasi-coherent sheaves of the stack which imply that an atlas exists.
math.AG↗
Which abelian tensor categories are geometric?
For a large class of geometric objects, the passage to categories of quasi-coherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a characterization of tensor categories in the image of this embedding. However, this notion requires additional structure to be present, namely a fiber functor. For the case of classical Tannakian categories in characteristic zero, Deligne has found intrinsic properties - expressible entirely within the language of tensor categories - which are necessary and sufficient for the existence of a fiber functor. In this paper we generalize Deligne's result to weakly Tannakian categories in characteristic zero. The class of geometric objects whose tensor categories of quasi-coherent sheaves can be recognized in this manner includes both the gerbes arising in classical Tannaka duality and more classical geometric objects such as projective varieties over a field of characteristic zero. Our proof uses a different perspective on fiber functors, which we formalize through the notion of geometric tensor categories. A second application of this perspective allows us to describe categories of quasi-coherent sheaves on fiber products.
math.AG↗
Ind-abelian categories and quasi-coherent sheaves
We study the question when a category of ind-objects is abelian. Our answer allows a further generalization of the notion of weakly Tannakian categories introduced by the author. As an application we show that, under suitable conditions, the category of coherent sheaves on the product of two schemes with the resolution property is given by the Deligne tensor product of the categories of coherent sheaves of the two factors. To do this we prove that the class of quasi-compact and semi-separated schemes with the resolution property is closed under fiber products.
math.AG↗
A characterization of categories of coherent sheaves of certain algebraic stacks
Under certain conditions, a scheme can be reconstructed from its category of quasi-coherent sheaves. The Tannakian reconstruction theorem provides another example where a geometric object can be reconstructed from an associated category, in this case the category of its finite dimensional representations. Lurie's result that the pseudofunctor which sends a geometric stack to its category of quasi-coherent sheaves is fully faithful provides a conceptual explanation for why this works. In this paper we prove a generalized Tannakian recognition theorem, in order to characterize a part of the image of the extension of the above pseudofunctor to algebraic stacks in the sense of Naumann. This allows us to further investigate a conjecture by Richard Pink about categories of filtered modules, which were defined by Fontaine and Laffaille to construct p-adic Galois representations. In order to do this we give a new characterization of Adams Hopf algebroids, which also allows us to answer a question posed by Mark Hovey.
math.AG↗
The formal theory of Tannaka duality
A Tannakian category is an abelian tensor category equipped with a fiber functor and additional structures which ensure that it is equivalent to the category of representations of some affine groupoid scheme acting on the spectrum of a field extension. If we are working over an arbitrary commutative ring rather than a field, the categories of representations cease to be abelian. We provide a list of sufficient conditions which ensure that an additive tensor category is equivalent to the category of representations of an affine groupoid scheme acting on an affine scheme, or, more generally, to the category of representations of a Hopf algebroid in a symmetric monoidal category. In order to do this we develop a "formal theory of Tannaka duality" inspired by Ross Street's "formal theory of monads." We apply our results to certain categories of filtered modules which are used to study p-adic Galois representations.
math.CT↗
Tannaka duality for comonoids in cosmoi
A classical result of Tannaka duality is the fact that a coalgebra over a field can be reconstructed from its category of finite dimensional representations by using the forgetful functor which sends a representation to its underlying vector space. There is also a corresponding recognition result, which characterizes those categories equipped with a functor to finite dimensional vector spaces which are equivalent to the category of finite dimensional representations of a coalgebra. In this paper we study a generalization of these questions to an arbitrary cosmos, that is, a complete and cocomplete symmetric monoidal closed category. Instead of representations on finite dimensional vector spaces we look at representations on objects of the cosmos which have a dual. We give a necessary and sufficient condition that ensures that a comonoid can be reconstructed from its representations, and we characterize categories of representations of certain comonoids. We apply this result to certain categories of filtered modules which are used to study p-adic Galois representations.
math.CT↗
Singular (Lipschitz) homology and homology of integral currents
We compare the homology groups $H_n ^{IC}(X)$ of the chain complex of integral currents with compact support of a metric space $X$ with the singular Lipschitz homology $H^L_n (X)$ and with ordinary singular homology. If $X$ satisfies certain cone inequalities all these homology theories coincide. On the other hand, for the Hawaiian Earring the homology of integral currents differs from the singular Lipschitz homology and it differs also from the classical singular homology $H_n(X)$.
math.DG↗