The exponential sum (ES) is a linear combination of characters of an additive group $\mathbb C^n$. The exponential analytic set (EAS) is a set of common zeroes of a finite tuple of ESs. We consider ES and EAS as an analogs of Laurent polynomial and of algebraic variety in complex torus $(\mathbb{C}\setminus0)^n$. Respectively we construct the ring of conditions for $\mathbb C^n$ as an analog of the ring of conditions for $(\mathbb{C}\setminus0)^n$. The construction of this ring is based on the definition of associated to EAS algebraic subvariety of some multidimensional torus and on the applying tropical algebraic geometry to this subvariety. Just as in the case of a torus, the ring of conditions is generated by hypersurfaces. This preprint is an extended summary of the article proposed to "Izvestiya: Mathematics".
We show that the class of the affine line is a zero divisor in the Grothendieck ring of algebraic varieties over complex numbers. The argument is based on the Pfaffian-Grassmannian double mirror correspondence.
Lev A. Borisov has shown that the class of the affine line is a zero divisor in the Grothendieck ring of algebraic varieties over complex numbers. We improve the final formula by removing a factor.
Motivated by [Bor] and [Mar], we show the equality $\left([X] - [Y]\right) \cdot [\mathbb{A}^1] = 0$ in the Grothendieck ring of varieties, where $(X, Y)$ is a pair of Calabi-Yau 3-folds cut out from the pair of Grassmannians of type $G_2$.
We study affine group schemes over a discrete valuation ring $R$ using two techniques: Neron blowups and Tannakian categories. We employ the theory developed to define and study differential Galois groups of $\mathcal D$-modules on a scheme over a $R$. This throws light on how differential Galois groups of families degenerate.
In the 1st part of this work [DHdS18], we studied affine group schemes over a discrete valuation ring (DVR) by means of Neron blowups. We also showed how to apply these findings to throw light on the group schemes coming from Tannakian categories of D-modules. In the present work, we follow up this theme. We show that a certain class of affine group schemes of "infinite type", Neron blowups of formal subgroups, are quite typical. We also explain how these group schemes appear naturally in Tannakian categories of D-modules. To conclude, we isolate a Tannakian property of affine group schemes, n…
Among all affine, flat, finitely presented group schemes, we focus on those that are pure, this includes all groups which are extensions of a finite locally free group by a group with connected fibres. We prove that over an arbitrary base ring, pure group schemes have a classifying space satisfying the resolution property, an embedding into some GLn, a tensor generator for their category of finite type representations, and can be reconstructed from their category of projective finite type representations. In the case of an Artinian base ring, the same is true for all affine, flat, finitely pre…
For a field $k$, the ring of invariants of an action of the unipotent $k$-group $G_a$ on an affine $k$-variety is quasi-affine, but not generally affine. Cable algebras are introduced as a framework for studying these invariant rings. It is shown that the ring of invariants for the $G_a$-action on $A^5_k$ constructed by Daigle and Freudenburg is a monogenetic cable algebra. A generating cable is constructed for this ring, and a complete set of relations is given as a prime ideal in the infinite polynomial ring over $k$. In addition, it is shown that the ring of invariants for the well-known $G…
We show that the cohomology ring of Hilbert scheme of $n$-points in the affine plane is isomorphic to the coordinate ring of $\mathbb{G}_{m}$-fixed point scheme of the $n$-th symmetric product of $\mathbb{C}^{2}$ for a natural $\mathbb{G}_{m}$-action on it. This result can be seen as an analogue of a theorem of DeConcini, Procesi and Tanisaki on a description of the cohomology ring of Springer fiber of type A.
In this article, we give a proof for a geometric presentation theorem for any irreducible scheme $X$ smooth projective over a discrete valuation ring $R$. As a consequence, for any reductive $R$-group scheme $\mathbf{G}$, we prove that any generically trivial principal $\mathbf{G}$-bundle over $X$ glued to a principal $\mathbf{G}_U$-bundle over the affine line $\mathbb{A}^1_U$ for a semilocal affine scheme $U$.
The paper consider regulous functions on the real affine space $\mathbb{R}^N$. We shall study some algebraic properties of the ring of those functions. It is presented a proof of the regulous version of Nullstellensatz based on the substitution property and the Artin-Lang property for the considered function ring. We prove that every maximal ideal in the ring of regulous functions on $\mathbb{R}^N$ when $N\geq 2$ is not finitely generated. Finally, we extend the latter result to an arbitrary, smooth, real affine algebraic variety of dimension $d\geq 2$.
A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed space, endowed with a finite open covering, produces a ringed finite space. We make a study of the homotopy of ringed finite spaces (that extends the homotopy of finite topological spaces) and a study of the quasi-coherent sheaves on a ringed finite space. This leads to set and develop basic notions on ringed finite spaces and morphisms, as being affine, sch…
We construct affine spaces, projective spaces and grassmannians in the ca\-te\-gory of ringed spaces. We show how finite posets and sheaves of rings on them appear in a natural way.
A general problem is to classify the real forms of a complex variety up to isomorphism. This paper introduces the polar group of a real form $X$ of a complex variety $Y$ as a tool to distinguish such real forms. This group is an invariant of $X$ which encodes information about the residual divisors in the coordinate ring of $Y$ over the coordinate ring of $X$. We calculate polar groups for various curves, including the real line and the algebraic 1-sphere.
We provide an explicit presentation of the equivariant cohomology ring of the compactified Jacobian $J_{q/p}$ of the rational curve $C_{q/p}$ with planar equation $x^{q}=y^{p}$ for $(p,q)=1$. We also prove analogous results for the closely related affine Springer fiber $Sp_{q/p}$ in the affine flag variety of $SL_{p}$. We show that the perverse filtration on the cohomology of $J_{q/p}$ is multiplicative, and the associated graded ring under the perverse filtration is a degeneration of the ring of functions on a moduli space of maps $\mathbb{P}^{1}\to C_{q/p}$. We also propose several conjectur…
In this article, we give a full description of the topology of the one dimensional affine analytic space $\mathbb{A}_R^1$ over a complete valuation ring $R$ (i.e. a valuation ring with "real valued valuation" which is complete under the induced metric), when its field of fractions $K$ is algebraically closed. In particular, we show that $\mathbb{A}_R^1$ is both connected and locally path connected. Furthermore, $\mathbb{A}_R^1$ is the completion of $K\times (1,\infty)$ under a canonical uniform structure. As an application, we describe the Berkovich spectrum $\mathfrak{M}(\mathbb{Z}_p[G])$ of …
Consider the generalized flag manifold $G/B$ and the corresponding affine flag manifold $\mathcal{Fl}_G$. In this paper we use curve neighborhoods for Schubert varieties in $\mathcal{Fl}_G$ to construct certain affine Gromov-Witten invariants of $\mathcal{Fl}_G$, and to obtain a family of "affine quantum Chevalley" operators $Λ_0, \ldots, Λ_n$ indexed by the simple roots in the affine root system of $G$. These operators act on the cohomology ring $\mathrm{H}^*(\mathcal{Fl}_G)$ with coefficients in $\mathbb{Z}[q_0, \ldots,q_n]$. By analyzing commutativity and invariance properties of these oper…
The goal of this paper is to establish a general rigidity statement for abstract representations of elementary subgroups of Chevalley groups of rank at least 2 over a class of commutative rings that includes the localizations of 1-generated rings and the coordinate rings of affine curves. This is achieved by developing the approach introduced in our previous work, and in particular by verifying condition (Z) over the class of rings at hand. Our main result implies, for example, that any finite-dimensional representation of SL_n(Z[X]) (for n at least 3) over an algebraically closed field of cha…
We construct a multiplicative spectral sequence converging to the symplectic cohomology ring of any affine variety $X$, with first page built out of topological invariants associated to strata of any fixed normal crossings compactification $(M,\mathbf{D})$ of $X$. We exhibit a broad class of pairs $(M,\mathbf{D})$ (characterized by the absence of relative holomorphic spheres or vanishing of certain relative GW invariants) for which the spectral sequence degenerates, and a broad subclass of pairs (similarly characterized) for which the ring structure on symplectic cohomology can also be describ…
Green and Tao famously proved in a 2008 paper that there are arithmetic progressions of prime numbers of arbitrary lengths. Soon after, analogous statements were proved by Tao for the ring of Gaussian integers and by Lê for the polynomial rings over finite fields. In 2020 this was extented to orders of arbitrary number fields by Kai-Mimura-Munemasa-Seki-Yoshino. We settle the case of the coordinate rings of affine curves over finite fields. The main contribution of this paper is subtle choice of a polynomial subring of the given ring which plays the role of $\mathbb Z$ in the number field case…
We provide a generalization of pseudo-Frobenius numbers of numerical semigroups to the context of the simplicial affine semigroups. In this way, we characterize the Cohen-Macaulay type of the simplicial affine semigroup ring $\mathbb{K}[S]$. We define the type of $S$, $\operatorname{type}$, in terms of some Apéry sets of $S$ and show that it coincides with the Cohen-Macaulay type of the semigroup ring, when $\mathbb{K}[S]$ is Cohen-Macaulay. If $\mathbb{K}[S]$ is a $d$-dimensional Cohen-Macaulay ring of embedding dimension at most $d+2$, then $\operatorname{type}\leq 2$. Otherwise, $\operatorn…
As an extension of previous ungraded work, we define a graded $p$-polar ring to be an analog of a graded commutative ring where multiplication is only allowed on $p$-tuples (instead of pairs) of elements of equal degree. We show that the free affine $p$-adic group scheme functor, as well as the free formal group functor, defined on $k$-algebras for a perfect field $k$ of characteristic $p$, factors through $p$-polar $k$-algebras. It follows that the same is true for any affine $p$-adic or formal group functor, in particular for the functor of $p$-typical Witt vectors. As an application, we sho…
We develop algebraic geometry for general Segal's Gamma-rings and show that this new theory unifies two approaches we had considered earlier on (for a geometry under Spec Z). The starting observation is that the category obtained by gluing together the category of commutative rings and that of pointed commutative monoids, that we used in our previous work to define F1-schemes, is naturally a full subcategory of the category of Segal's Gamma-rings. In this paper we develop the affine case of this general algebraic geometry: one distinctive feature is that the spectrum Spec(A) of a Gamma-ring is…
We show that the functor of $p$-typical co-Witt vectors on commutative algebras over a perfect field $k$ of characteristic $p$ is defined on, and in fact only depends on, a weaker structure than that of a $k$-algebra. We call this structure a $p$-polar $k$-algebra. By extension, the functors of points for any $p$-adic affine commutative group scheme and for any formal group are defined on, and only depend on, $p$-polar structures. In terms of abelian Hopf algebras, we show that a cofree cocommutative Hopf algebra can be defined on any $p$-polar $k$-algebra $P$, and it agrees with the cofree co…
Given an algebraic stack $X$, one may compare the derived category of quasi-coherent sheaves on $X$ with the category of dg-modules over the dg-ring of functions on $X$. We study the analogous question in stable homotopy theory, for derived stacks that arise via realizations of diagrams of Landweber-exact homology theories. We identify a condition (quasi-affineness of the map to the moduli stack of formal groups) under which the two categories are equivalent, and study applications to topological modular forms. In particular, we provide new examples of Galois extensions of ring spectra and van…
Due to Rentschler, Miyanishi and Kojima, the invariant ring for a ${\bf G}_a$-action on the affine plane over an arbitrary field is generated by one coordinate. In this note, we give a new short proof for this result using the automorphism theorem of Jung and van der Kulk.
