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In this short paper, we combine the representability theorem introduced in [Porta and Yu, Representability theorem in derived analytic geometry, preprint, 2017, arXiv:1704.01683; Porta and Yu, Derived Hom spaces in rigid analytic geometry, preprint, 2018, arXiv:1801.07730] with the theory of derived formal models introduced in [António,
$p$
-adic derived formal geometry and derived Raynaud localization theorem, preprint, 2018, arXiv:1805.03302] to prove the existence representability of the derived Hilbert space
$\mathbf{R}\text{Hilb}(X)$
for a separated
$k$
-analytic space
$X$
. Such representability results rely on a localization theorem stating that if
$\mathfrak{X}$
is a quasi-compact and quasi-separated formal scheme, then the
$\infty$
-category
$\text{Coh}^{-}(\mathfrak{X}^{\text{rig}})$
of almost perfect complexes over the generic fiber can be realized as a Verdier quotient of the
$\infty$
-category
$\text{Coh}^{-}(\mathfrak{X})$
. Along the way, we prove several results concerning the
$\infty$
-categories of formal models for almost perfect modules on derived
$k$
-analytic spaces.
We define (iterated) coisotropic correspondences between derived Poisson stacks, and construct symmetric monoidal higher categories of derived Poisson stacks, where the
$i$
-morphisms are given by
$i$
-fold coisotropic correspondences. Assuming an expected equivalence of different models of higher Morita categories, we prove that all derived Poisson stacks are fully dualizable and so determine framed extended TQFTs by the Cobordism Hypothesis. Along the way, we also prove that the higher Morita category of
$E_{n}$
-algebras with respect to coproducts is equivalent to the higher category of iterated cospans.
Classification of AS-regular algebras is one of the main interests in noncommutative algebraic geometry. We say that a
$3$
-dimensional quadratic AS-regular algebra is of Type EC if its point scheme is an elliptic curve in
$\mathbb {P}^{2}$
. In this paper, we give a complete list of geometric pairs and a complete list of twisted superpotentials corresponding to such algebras. As an application, we show that there are only two exceptions up to isomorphism among all
$3$
-dimensional quadratic AS-regular algebras that cannot be written as a twist of a Calabi–Yau AS-regular algebra by a graded algebra automorphism.
Let k be an algebraically closed field of positive characteristic. For any integer
$m\ge 2$
, we show that the Hodge numbers of a smooth projective k-variety can take on any combination of values modulo m, subject only to Serre duality. In particular, there are no non-trivial polynomial relations between the Hodge numbers.
We contribute to the foundations of tropical geometry with a view toward formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework, the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves.
Our approach to tropical geometry permits tropical moduli problems—moduli of curves or otherwise—to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms.
We prove that a generic homogeneous polynomial of degree
$d$
is determined, up to a nonzero constant multiplicative factor, by the vector space spanned by its partial derivatives of order
$k$
for
$k\leqslant \frac{d}{2}-1$
.
In order to work with non-Nagata rings which are Nagata “up-to-completely-decomposed-universal-homeomorphism,” specifically finite rank Hensel valuation rings, we introduce the notions of pseudo-integral closure, pseudo-normalization, and pseudo-Hensel valuation ring. We use this notion to give a shorter and more direct proof that
$H_{\operatorname{cdh}}^{n}(X,F_{\operatorname{cdh}})=H_{l\operatorname{dh}}^{n}(X,F_{l\operatorname{dh}})$
for homotopy sheaves
$F$
of modules over the
$\mathbb{Z}_{(l)}$
-linear motivic Eilenberg–Maclane spectrum. This comparison is an alternative to the first half of the author’s volume Astérisque 391 whose main theorem is a cdh-descent result for Voevodsky motives. The motivating new insight is really accepting that Voevodsky’s motivic cohomology (with
$\mathbb{Z}[\frac{1}{p}]$
-coefficients) is invariant not just for nilpotent thickenings, but for all universal homeomorphisms.
In his work on deformation quantization of algebraic varieties Kontsevich introduced the notion of algebroid as a certain generalization of a sheaf of algebras. We construct algebroids which are given locally by NC-smooth thickenings in the sense of Kapranov, over two classes of smooth varieties: the bases of miniversal families of vector bundles on projective curves, and the bases of miniversal families of quiver representations.
We study Van den Bergh's non-commutative symmetric algebra 𝕊nc(M) (over division rings) via Minamoto's theory of Fano algebras. In particular, we show that 𝕊nc(M) is coherent, and its proj category ℙnc(M) is derived equivalent to the corresponding bimodule species. This generalizes the main theorem of [8], which in turn is a generalization of Beilinson's derived equivalence. As corollaries, we show that ℙnc(M) is hereditary and there is a structure theorem for sheaves on ℙnc(M) analogous to that for ℙ1.
in a symmetric monoidal
$(\infty ,2)$
-category
$\mathscr{E}$
where
$X,Y\in \mathscr{E}$
are dualizable objects and
$\unicode[STIX]{x1D711}$
admits a right adjoint we construct a natural morphism
$\mathsf{Tr}_{\mathscr{E}}(F_{X})\xrightarrow[{}]{~~~~~}\mathsf{Tr}_{\mathscr{E}}(F_{Y})$
between the traces of
$F_{X}$
and
$F_{Y}$
, respectively. We then apply this formalism to the case when
$\mathscr{E}$
is the
$(\infty ,2)$
-category of
$k$
-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah–Bott formula (also known as the Holomorphic Lefschetz fixed point formula).
We study the structure of the stable category
$\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$
of graded maximal Cohen–Macaulay module over
$S/(f)$
where
$S$
is a graded (
$\pm 1$
)-skew polynomial algebra in
$n$
variables of degree 1, and
$f=x_{1}^{2}+\cdots +x_{n}^{2}$
. If
$S$
is commutative, then the structure of
$\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$
is well known by Knörrer’s periodicity theorem. In this paper, we prove that if
$n\leqslant 5$
, then the structure of
$\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$
is determined by the number of irreducible components of the point scheme of
$S$
which are isomorphic to
$\mathbb{P}^{1}$
.
In this paper we use the theory of formal moduli problems developed by Lurie in order to study the space of formal deformations of a
$k$
-linear
$\infty$
-category for a field
$k$
. Our main result states that if
${\mathcal{C}}$
is a
$k$
-linear
$\infty$
-category which has a compact generator whose groups of self-extensions vanish for sufficiently high positive degrees, then every formal deformation of
${\mathcal{C}}$
has zero curvature and moreover admits a compact generator.
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 construct strictly commutative simplicial ring spectra representing a given cohomology theory and illustrate this with a strictly commutative motivic ring spectrum representing higher order products on Deligne cohomology. These results are obtained by first establishing Smith’s stable positive model structure for abstract spectra and then showing that this category of spectra possesses excellent model-theoretic properties: we show that all colored symmetric operads in symmetric spectra valued in a symmetric monoidal model category are admissible, i.e., algebras over such operads carry a model structure. This generalizes the known model structures on commutative ring spectra and
$\text{E}_{\infty }$
-ring spectra in simplicial sets or motivic spaces. We also show that any weak equivalence of operads in spectra gives rise to a Quillen equivalence of their categories of algebras. For example, this extends the familiar strictification of
$\text{E}_{\infty }$
-rings to commutative rings in a broad class of spectra, including motivic spectra. We finally show that operadic algebras in Quillen equivalent categories of spectra are again Quillen equivalent. This paper is also available at arXiv:1410.5699v2.
Gromov–Witten invariants have been constructed to be deformation invariant, but their behavior under other transformations is subtle. We show that logarithmic Gromov–Witten invariants are also invariant under appropriately defined logarithmic modifications.
Let
$X$
be a quasi-compact and quasi-separated scheme. There are two fundamental and pervasive facts about the unbounded derived category of
$X$
: (1)
$\mathsf{D}_{\text{qc}}(X)$
is compactly generated by perfect complexes and (2) if
$X$
is noetherian or has affine diagonal, then the functor
$\unicode[STIX]{x1D6F9}_{X}:\mathsf{D}(\mathsf{QCoh}(X))\rightarrow \mathsf{D}_{\text{qc}}(X)$
is an equivalence. Our main results are that for algebraic stacks in positive characteristic, the assertions (1) and (2) are typically false.
This paper is dedicated to a problem raised by Jacquet Tits in 1956: the Weyl group of a Chevalley group should find an interpretation as a group over what is nowadays called
$\mathbb{F}_{1}$
, the field with one element. Based on Part I of The geometry of blueprints, we introduce the class of Tits morphisms between blue schemes. The resulting Tits category
$\text{Sch}_{{\mathcal{T}}}$
comes together with a base extension to (semiring) schemes and the so-called Weyl extension to sets. We prove for
${\mathcal{G}}$
in a wide class of Chevalley groups—which includes the special and general linear groups, symplectic and special orthogonal groups, and all types of adjoint groups—that a linear representation of
${\mathcal{G}}$
defines a model
$G$
in
$\text{Sch}_{{\mathcal{T}}}$
whose Weyl extension is the Weyl group
$W$
of
${\mathcal{G}}$
. We call such models Tits–Weyl models. The potential of Tits–Weyl models lies in (a) their intrinsic definition that is given by a linear representation; (b) the (yet to be formulated) unified approach towards thick and thin geometries; and (c) the extension of a Chevalley group to a functor on blueprints, which makes it, in particular, possible to consider Chevalley groups over semirings. This opens applications to idempotent analysis and tropical geometry.
Noetherian dimer algebras form a prominent class of examples of noncommutative crepant resolutions (NCCRs). However, dimer algebras that are noetherian are quite rare, and we consider the question: how close are nonnoetherian homotopy dimer algebras to being NCCRs? To address this question, we introduce a generalization of NCCRs to nonnoetherian tiled matrix rings. We show that if a noetherian dimer algebra is obtained from a nonnoetherian homotopy dimer algebra A by contracting each arrow whose head has indegree 1, then A is a noncommutative desingularization of its nonnoetherian centre. Furthermore, if any two arrows whose tails have indegree 1 are coprime, then A is a nonnoetherian NCCR.
Let
$\Bbbk$
be a field of characteristic zero. For any positive integer
$n$
and any scalar
$a\in \Bbbk$
, we construct a family of Artin–Schelter regular algebras
$R(n,a)$
, which are quantizations of Poisson structures on
$\Bbbk [x_{0},\ldots ,x_{n}]$
. This generalizes an example given by Pym when
$n=3$
. For a particular choice of the parameter
$a$
we obtain new examples of Calabi–Yau algebras when
$n\geqslant 4$
. We also study the ring theoretic properties of the algebras
$R(n,a)$
. We show that the point modules of
$R(n,a)$
are parameterized by a bouquet of rational normal curves in
$\mathbb{P}^{n}$
, and that the prime spectrum of
$R(n,a)$
is homeomorphic to the Poisson spectrum of its semiclassical limit. Moreover, we explicitly describe
$\operatorname{Spec}R(n,a)$
as a union of commutative strata.
We develop a theory of unbounded derived categories of quasi-coherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks that either have finite stabilizers or are local quotient stacks. We also extend Toën and Antieau–Gepner’s results on derived Azumaya algebras and compact generation of sheaves on linear categories from derived schemes to derived Deligne–Mumford stacks. These are all consequences of our main theorem: compact generation of a presheaf of triangulated categories on an algebraic stack is local for the quasi-finite flat topology.
Let
$A\rightarrow B$
be a morphism of Artin local rings with the same embedding dimension. We prove that any
$A$
-flat
$B$
-module is
$B$
-flat. This freeness criterion was conjectured by de Smit in 1997 and improves Diamond’s criterion [The Taylor–Wiles construction and multiplicity one, Invent. Math. 128 (1997), 379–391, Theorem 2.1]. We also prove that if there is a nonzero
$A$
-flat
$B$
-module, then
$A\rightarrow B$
is flat and is a relative complete intersection. Then we explain how this result allows one to simplify Wiles’s proof of Fermat’s last theorem: we do not need the so-called ‘Taylor–Wiles systems’ any more.