Let R be a real closed field. The Pierce-Birkhoff conjecture says that any piecewise polynomial function f on R^n can be obtained from the polynomial ring R[x_1,...,x_n] by iterating the operations of maximum and minimum. The purpose of this paper is twofold. First, we state a new conjecture, called the Connectedness conjecture, which asserts the existence of connected sets in the real spectrum of R[x_1,...,x_n] satisfying certain conditions. We prove that the Connectedness conjecture implies the Pierce-Birkhoff conjecture. Secondly, we construct a class of connected sets in the real spectrum which, though not in itself enough for the proof of the Pierce-Birkhoff conjecture, is the first and simplest example of the sort of connected sets we really need, and which constitutes a crucial step on the way to a proof of the Pierce-Birkhoff conjecture in dimension greater than 2, to appear in a subsequent paper.
We study the connectedness of the non-subklt locus over the spectrum of a formal power series ring. In dimension 3, we prove the existence and normality of the smallest lc centre, and apply it to the ACC for minimal log discrepancies greater than 1 on non-singular 3-folds.
We study the Chevalley-Warning problem in the Grothendieck ring $K_0(Var/k)$. We show that the $\A^1$-homotopy theory yields well defined invariants on $K_0(Var/k)/\bold{L}$, in particular the Brauer group is such an invariant. We use this to give a concrete counter-example to the Chevalley-Warning conjecture over a $C_1$-field (Brown and Schnetz, 2011). This also gives a negative answer to the question in Bilgin (2011).
We establish a "second vanishing theorem" for local cohomology modules over regular rings of unramified mixed characteristic, which relates the connectedness of the spectrum of a ring with the vanishing of local cohomology. Applying this, and new results on the mixed characteristic Lyubeznik numbers, we further study connectedness properties of the spectra of a certain class of rings.
We investigate the relationship between connectedness properties of spectra and the Lyubeznik numbers, numerical invariants defined via local cohomology. We prove that for complete equidimensional local rings, the Lyubeznik numbers characterize when connectedness dimension equals one. More generally, these invariants determine a bound on connectedness dimension. Additionally, our methods imply that the Lyubeznik number with indices (1,2) of the local ring at the vertex of the affine cone over a projective variety is independent of the choice of its embedding into projective space.
We show that the Kisin varieties associated to simple $ϕ$-modules of rank $2$ are connected in the case of an arbitrary cocharacter. This proves that the connected components of the generic fiber of the flat deformation ring of an irreducible $2$-dimensional Galois representation of a local field are precisely the components where the multiplicities of the Hodge-Tate weights are fixed.
This paper is a step in our program for proving the Piece-Birkhoff Conjecture for regular rings of any dimension (this would contain, in particular, the classical Pierce-Birkhoff conjecture which deals with polynomial rings over a real closed field). We first recall the Connectedness and the Definable Connectedness conjectures, both of which imply the Pierce - Birkhoff conjecture. Then we introduce the notion of a system of approximate roots of a valuation v on a ring A (that is, a collection Q of elements of A such that every v-ideal is generated by products of elements of Q). We use approxim…
This paper contains a partial result on the Pierce--Birkhoff conjecture on piece-wise polynomial functions defined by a finite collection {f 1,. .., f r} of polynomials. In the nineteen eighties, generalizing the problem from the polynomial ring to an artibtrary ring $Σ$, J. Madden proved that the Pierce--Birkhoff conjecture for $Σ$ is equivalent to a statement about an arbitrary pair of points $α$, $β$ $\in$ Sper $Σ$ and their separating ideal < $α$, $β$ >, we refer to this statement as the local Pierce-Birkhoff conjecture at $α$, $β$. In [8] we introduced a slightly stronger conjecture, also…
We consider the Kisin variety associated to a $n$-dimensional absolutely irreducible mod $p$ Galois representation $\barρ$ of a $p$-adic field $K$ and a cocharacter $μ$. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin's conjecture holds if $K$ is totally ramfied with $n=3$ or $μ$ is of a very particular form. As an application, we also get a connectedness result for the deformation ring associated to $\barρ$ of given Hodge-Tate weights. We also give counterexamples to show Kisin's conjecture does not hold in general.
This paper represents a step in our program towards the proof of the Pierce--Birkhoff conjecture. In the nineteen eighties J. Madden proved that the Pierce-Birkhoff conjecture for a ring A$is equivalent to a statement about an arbitrary pair of points $α,β\in\sper\ A$ and their separating ideal $<α,β>$; we refer to this statement as the Local Pierce-Birkhoff conjecture at $α,β$. In this paper, for each pair $(α,β)$ with $ht(<α,β>)=\dim A$, we define a natural number, called complexity of $(α,β)$. Complexity 0 corresponds to the case when one of the points $α,β$ is monomial; this case was alrea…
A classical theorem by Hartshorne states that the dual graph of any arithmetically Cohen--Macaulay projective scheme is connected. We give a quantitative version of Hartshorne's result, in terms of Castelnuovo--Mumford regularity. If $X \subset \mathbb{P}^n$ is an arithmetically Gorenstein projective scheme of regularity $r+1$, and if every irreducible component of $X$ has regularity $\le r'$, we show that the dual graph of $X$ is $\lfloor{\frac{r+r'-1}{r'}}\rfloor$-connected. The bound is sharp. We also provide a strong converse to Hartshorne's result: Every connected graph is the dual graph …
In this paper, we study splitting (or toric) non-commutative crepant resolutions (= NCCRs) of some toric rings. In particular, we consider Hibi rings, which are toric rings arising from partially ordered sets, and show that Gorenstein Hibi rings with class group $\mathbb{Z}^2$ have a splitting NCCR. In the appendix, we also discuss Gorenstein toric rings with class group $\mathbb{Z}$, in which case the existence of splitting NCCRs is already known. We especially observe the mutations of modules giving splitting NCCRs for the three dimensional case, and show the connectedness of the exchange gr…
We will prove that the Pierce-Birkhoff Conjecture holds for non-singular two-dimensional affine real algebraic varieties over real closed fields, i.e., if W is such a variety, then every piecewise polynomial function on W can be written as suprema of infima of polynomial functions on W. More precisely, we will give a proof of the so-called Connectedness Conjecture for the coordinate rings of such varieties, which implies the Pierce-Birkhoff Conjecture.
We prove a Tannaka duality theorem for $(\infty,1)$-categories. This is a duality between certain derived group stacks, or more generally certain derived gerbes, and symmetric monoidal $(\infty,1)$-categories endowed with particular structure. This duality theorem is defined over commutative ring spectra and subsumes the classical statement. We show how the classical theory, and its extension over arbitrary rings, arises as a special case of our more general theory. The application to perfect complexes is explored.
We show that Shipley's "detection functor" for symmetric spectra generalizes to motivic symmetric spectra. As an application, we construct motivic strict ring spectra representing morphic cohomology, semi-topological $K$-theory, and semi-topological cobordism for complex varieties. As a further application to semi-topological cobordism, we show that it is related to semi-topological $K$-theory via a Conner-Floyd type isomorphism and that after inverting a lift of the Friedlander-Mazur $s$-element in morphic cohomology, semi-topological cobordism becomes isomorphic to periodic complex cobordism…
We show that the cellular objects in the module category over a motivic E infinity ring spectrum E can be described as the module category over a graded topological spectrum if E is strongly periodizable in our language. A similar statement is proven for triangulated categories of motives. Since MGL is strongly periodizable we obtain topological incarnations of motivic Landweber spectra. Under some categorical assumptions the unit object of the model category for triangulated motives is as well strongly periodizable giving motivic cochains whose module category models integral triangulated cat…
We develop homotopical algebraic geometry (see math.AG/0207028) in the special context where the base symmetric monoidal model category is the category S of spectra, i.e. what might be called, after Waldhausen, ``brave new algebraic geometry''. We discuss various model topologies on the model category of commutative algebras in S, the associated theories of geometric S-stacks (a geometric S-stack being an analog of Artin notion of algebraic stack in Algebraic Geometry), and finally show how to define global moduli spaces of associative ring spectra structures and a moduli space related to topo…
We initiate the theory of graded commutative 2-rings, a categorification of graded commutative rings. The goal is to provide a systematic generalization of Paul Balmer's comparison maps between the spectrum of tensor-triangulated categories and the Zariski spectra of their central rings. By applying our constructions, we compute the spectrum of the derived category of perfect complexes over any graded commutative ring, and we associate to every scheme with an ample family of line bundles an embedding into the spectrum of an associated graded commutative 2-ring.
The injective spectrum is a topological space associated to a ring $R$, which agrees with the Zariski spectrum when $R$ is commutative noetherian. We consider injective spectra of right noetherian rings (and locally noetherian Grothendieck categories) and establish some basic topological results and a functoriality result, as well as links between the topology and the Krull dimension of the ring (in the sense of Gabriel and Rentschler). Finally, we use these results to compute a number of examples.
We construct geometric lifts of the Bost-Connes algebra to Grothendieck rings and to the associated assembler categories and spectra, as well as to certain categories of Nori motives. These categorifications are related to the integral Bost-Connes algebra via suitable Euler characteristic type maps and zeta functions, and in the motivic case via fiber functors. We also discuss aspects of F1-geometry, in the framework of torifications, that fit into this general setting.
This paper sets up the foundations for derived algebraic geometry, Goerss--Hopkins obstruction theory, and the construction of commutative ring spectra in the abstract setting of operadic algebras in symmetric spectra in an (essentially) arbitrary model category. We show that one can do derived algebraic geometry a la Toën--Vezzosi in an abstract category of spectra. We also answer in the affirmative a question of Goerss and Hopkins by showing that the obstruction theory for operadic algebras in spectra can be done in the generality of spectra in an (essentially) arbitrary model category. We c…
Let $G$ be a finite group of order $n$, and $ξ$ an $n$-th primitive root of unity. Consider the affine scheme $C:=\mbox{Spc}({\mathbb Z}[ξ]\otimes_{\mathbb Z} R(G))$ where $R(G)$ is the representation ring of $G$. We study the fibers of the formal tangent sheaf of $C$ by computing their dimension and also finding (and measuring) the singularities of $C$. We present explicit computations for noncommutative groups of small order, and develop practical methods to compute these invariants for an arbitrary finite group.
