Monoid Scheme Research Articles

Absolute zeta functions arising from ceiling and floor Puiseux polynomials

For the [Formula: see text]-lift [Formula: see text] of a monoid scheme [Formula: see text] of finite type, Deitmar et al. calculated its absolute zeta function by interpolating [Formula: see text] for all prime powers [Formula: see text] using the Fourier expansion. This absolute zeta function coincides with the absolute zeta function of a certain polynomial. In this paper, we characterize the polynomial as a ceiling polynomial of the sequence [Formula: see text], which we introduce independently. Extending this idea, we introduce a certain pair of absolute zeta functions of a separated scheme [Formula: see text] of finite type over [Formula: see text] by means of a pair of Puiseux polynomials which estimate “[Formula: see text]” for sufficiently large [Formula: see text]. We call them the ceiling and floor Puiseux polynomials of [Formula: see text]. In particular, if [Formula: see text] is an elliptic curve, then our absolute zeta functions of [Formula: see text] do not depend on its isogeny class.

Hall Lie algebras of toric monoid schemes

Hall Lie algebras of toric monoid schemes

The points and localisations of the topos of $M$-sets

In previous papers, we were able to prove that, much like in classical algebraic geometry, it is possible to recover our monoid scheme X from the topos \mathsf{Qcoh}(X) . This was achieved using topos points and localisations of \mathsf{Qcoh}(X) . With this philosophy in mind, the aim of this paper is to study the topos of M -sets for non-commutative monoids, especially their points and localisations. We will classify the points and localisations of M -sets for finite monoids in terms of the idempotent elements of M and idempotent ideals of M , respectively. Some of the results obtained in this paper can already be found in previous works, in direct or indirect forms. See the last part of the introduction.

Vector bundles on tropical schemes

We define vector bundles for tropical schemes, and explore their properties. The paper largely consists of three parts (1) we study free modules over zero-sum free semirings, which provide the necessary algebraic background for the theory, (2) we relate vector bundles on tropical schemes to topological vector bundles and vector bundles on monoid schemes, and finally (3) we show that all line bundles on a tropical scheme can be lifted to line bundles on a usual scheme in the affine case.

Localization, monoid sets and K-theory

We develop the K-theory of sets with an action of a pointed monoid (or monoid scheme), analogous to the K-theory of modules over a ring (or scheme). In order to form localization sequences, we construct the quotient category of a nice regular category by a Serre subcategory.

Perverse 𝔽 p -sheaves on the affine Grassmannian

Abstract For a reductive group over an algebraically closed field of characteristic p > 0 {p>0} we construct the abelian category of perverse 𝔽 p {\mathbb{F}_{p}} -sheaves on the affine Grassmannian that are equivariant with respect to the action of the positive loop group. We show this is a symmetric monoidal category, and then we apply a Tannakian formalism to show this category is equivalent to the category of representations of a certain affine monoid scheme. We also show that our work provides a geometrization of the inverse of the mod p Satake isomorphism. Along the way we prove that affine Schubert varieties are globally F-regular and we apply Frobenius splitting techniques to the theory of perverse 𝔽 p {\mathbb{F}_{p}} -sheaves.

Algebraic $$K\!$$-theory and Grothendieck–Witt theory of monoid schemes

AbstractWe study the algebraic $$K\!$$ K -theory and Grothendieck–Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic $$K\!$$ K -theory space of an integral monoid scheme X in terms of its Picard group $${{\,\mathrm{Pic}\,}}(X)$$ Pic ( X ) and pointed monoid of regular functions $$\Gamma (X, {\mathcal {O}}_X)$$ Γ ( X , O X ) and a complete description of the Grothendieck–Witt space of X in terms of an additional involution on $${{\,\mathrm{Pic}\,}}(X)$$ Pic ( X ) . We also prove space-level projective bundle formulae in both settings.

Reconstruction theorem for monoid schemes

We aim to reconstruct a monoid scheme X from the category of quasi-coherent sheaves over it. This is much in the vein of Gabriel's original reconstruction theorem. Under some finiteness condition on a monoid schemes X , we show that the localisations of the topos Qc ( X ) of quasi-coherent sheaves on X are in a one-to-one correspondence with open subsets of X , while the elements of X correspond to the topos points of Qc ( X ) . This allows us to reconstruct X from Qc ( X ) .

Topos points of quasi-coherent sheaves over monoid schemes

AbstractLet X be a monoid scheme. We will show that the stalk at any point of X defines a point of the topos of quasi-coherent sheaves over X. As it turns out, every topos point of is of this form if X satisfies some finiteness conditions. In particular, it suffices for M/M× to be finitely generated when X is affine, where M× is the group of invertible elements.This allows us to prove that two quasi-projective monoid schemes X and Y are isomorphic if and only if and are equivalent.The finiteness conditions are essential, as one can already conclude by the work of A. Connes and C. Consani [3]. We will study the topos points of free commutative monoids and show that already for ℕ∞, there are ‘hidden’ points. That is to say, there are topos points which are not coming from prime ideals. This observation reveals that there might be a more interesting ‘geometry of monoids’.

Picard groups for tropical toric schemes

From any monoid scheme $X$ (also known as an $\mathbb{F}_1$-scheme) one can pass to a semiring scheme (a generalization of a tropical scheme) $X_S$ by scalar extension to an idempotent semifield $S$. We prove that for a given irreducible monoid scheme $X$ (satisfying some mild conditions) and an idempotent semifield $S$, the Picard group $Pic(X)$ of $X$ is stable under scalar extension to $S$ (and in fact to any field $K$). In other words, we show that the groups $Pic(X)$ and $Pic(X_S)$ (and $Pic(X_K)$) are isomorphic. In particular, if $X_\mathbb{C}$ is a toric variety, then $Pic(X)$ is the same as the Picard group of the associated tropical scheme. The Picard groups can be computed by considering the correct sheaf cohomology groups. We also define the group $CaCl(X_S)$ of Cartier divisors modulo principal Cartier divisors for a cancellative semiring scheme $X_S$ and prove that $CaCl(X_S)$ is isomorphic to $Pic(X_S)$.

Quasicoherent sheaves on projective schemes over [formula omitted

Given a graded monoid A with 1, one can construct a projective monoid scheme MProj(A) analogous to Proj(R) of a graded ring R. This paper is concerned with the study of quasicoherent sheaves on MProj(A), and we prove several basic results regarding these. We show that:1.every quasicoherent sheaf F on MProj(A) can be constructed from a graded A-set in analogy with the construction of quasicoherent sheaves on Proj(R) from graded R-modules2.if F is coherent on MProj(A), then F(n) is globally generated for large enough n, and consequently, that F is a quotient of a finite direct sum of invertible sheaves3.if F is coherent on MProj(A), then Γ(MProj(A),F) is finitely generated over A0 (and hence a finite set if A0={0,1}). The last part of the paper is devoted to classifying coherent sheaves on P1 in terms of certain directed graphs and gluing data. The classification of these over F1 is shown to be much richer and combinatorially interesting than in the case of ordinary P1, and several new phenomena emerge.

Blue schemes, semiring schemes, and relative schemes after Toën and Vaquié

It is a classical insight that the Yoneda embedding defines an equivalence of schemes as locally ringed spaces with schemes as sheaves on the big Zariski site. Similarly, the Yoneda embedding identifies monoid schemes (or F1-schemes in the sense of Deitmar) with schemes relative to sets (in the sense of Toën and Vaquié).In this paper, we investigate the generalization to blue schemes and to semiring schemes. We establish Yoneda functors for both scheme theories. These functors fail, however, to be equivalences in both situations. The reason for this failure is a divergence in the Grothendieck pretopologies coming from schemes as topological spaces and schemes as sheaves.Restricted to blue schemes that are locally of finite type over a blue field, we construct an inverse to the Yoneda functor, which establishes an equivalence for this subclass of blue schemes. Moreover, we verify the compatibility of the Yoneda functors with the base extension from blue schemes to semiring schemes and with the base extension from semiring schemes to usual schemes.

On cohomology and vector bundles over monoid schemes

The aim of this paper is to study the cohomology theory of monoid schemes in general and apply it to vector and line bundles. We will prove that over separated monoid schemes, any vector bundle is a coproduct of line bundles and that the Pic functor respects finite products.Next we will introduce the notion of s-cancellative monoids and show that if X is locally of that type, OX⁎ can be embed injectively in a constant sheaf. We develop the theory of s-divisors, which generalise the Cartier divisors, and we prove that for s-cancellative monoid schemes, s-divisors describe the group Pic(X).We then generalise smooth monoid schemes with s-smooth monoid schemes, and prove that for them Hi(X,OX⁎)=0 for all i≥2. Furthermore we show that it is a local property and respects finite products.Finally we investigate the relationship between line bundles over a monoid scheme X and over its geometric realisation Xk, where k is a commutative ring. We prove that if k is an integral domain (resp. PID) and X is a cancellative and torsion-free (resp. seminormal and torsion-free) monoid scheme, then the induced map Pic(X)→Pic(Xk) is a monomorphism (resp. isomorphism).

Picard groups and class groups of monoid schemes

We study the Picard group of a monoid scheme and the class group of a normal monoid scheme. To do so, we develop some ideal theory for (pointed abelian) noetherian monoids, including primary decomposition and discrete valuations. The normalization of a monoid turns out to be a monoid scheme, but not always a monoid.

Toric varieties, monoid schemes and cdh descent

Abstract We give conditions for the Mayer–Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties and schemes, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to topological cyclic homology in characteristic p. To achieve our goals, we develop many notions for monoid schemes based on classical algebraic geometry, such as separated and proper maps and resolution of singularities.

CONGRUENCE SCHEMES

A new category of algebro-geometric objects is defined which contains the category of monoid schemes, or schemes over 𝔽1, and the category of Grothendieck schemes as subcategories.

Categorification of Hopf algebras of rooted trees

AbstractWe exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec ℕ) whose semiring of functions is (a P-version of) the Connes-Kreimer bialgebra H of rooted trees (a Hopf algebra after base change to ℤ and collapsing H 0). The monoidal structure is itself given by a polynomial functor, represented by three easily described set maps; we show that these maps are the same as those occurring in the polynomial representation of the free monad on P.

On the Hall algebra of coherent sheaves on [formula omitted] over [formula omitted

We define and study the category C o h n ( P 1 ) of normal coherent sheaves on the monoid scheme P 1 (equivalently, the M 0 -scheme P 1 / F 1 in the sense of Connes–Consani–Marcolli, Connes (2009) [2]). This category resembles in most ways a finitary abelian category, but is not additive. As an application, we define and study the Hall algebra of C o h n ( P 1 ) . We show that it is isomorphic as a Hopf algebra to the enveloping algebra of the product of a non-standard Borel in the loop algebra L g l 2 and an abelian Lie algebra on infinitely many generators. This should be viewed as a ( q = 1 ) version of Kapranov’s result relating (a certain subalgebra of) the Ringel–Hall algebra of P 1 over F q to a non-standard quantum Borel inside the quantum loop algebra U ν ( s l ̂ 2 ) , where ν 2 = q .

Reynolds operator on functors

Let G = Spec A be an affine R -monoid scheme. We prove that the category of dual functors (over the category of commutative R -algebras) of G -modules is equivalent to the category of dual functors of A ∗ -modules. We prove that G is invariant exact if and only if A ∗ = R × B ∗ as R -algebras and the first projection A ∗ → R is the unit of A . If M is a dual functor of G -modules and w G ≔ ( 1 , 0 ) ∈ R × B ∗ = A ∗ , we prove that M G = w G ⋅ M and M = w G ⋅ M ⊕ ( 1 − w G ) ⋅ M ; hence, the Reynolds operator can be defined on M .

Divided powers in Chow rings and integral Fourier transforms

We prove that for any monoid scheme M over a field with proper multiplication maps M × M → M , we have a natural PD-structure on the ideal CH > 0 ( M ) ⊂ CH ∗ ( M ) with regard to the Pontryagin ring structure. Further we investigate to what extent it is possible to define a Fourier transform on the motive with integral coefficients of the Jacobian of a curve. For a hyperelliptic curve of genus g with sufficiently many k-rational Weierstrass points, we construct such an integral Fourier transform with all the usual properties up to 2 N -torsion, where N = 1 + ⌊ log 2 ( 3 g ) ⌋ . As a consequence we obtain, over k = k ¯ , a PD-structure (for the intersection product) on 2 N ⋅ a , where a ⊂ CH ( J ) is the augmentation ideal. We show that a factor 2 in the properties of an integral Fourier transform cannot be eliminated even for elliptic curves over an algebraically closed field.