Simplicial Scheme Research Articles

Degree 2 cohomological invariants of linear algebraic groups

The paper deals with the cohomological invariants of smooth and connected linear algebraic groups over an arbitrary field. More precisely, we study degree $2$ invariants with coefficients $\mathbb{Q}/\mathbb{Z}(1)$, that is invariants taking values in the Brauer group. Our main tool is the \'etale cohomology of sheaves on simplicial schemes. We get a description of these invariants for \emph{every} smooth and connected linear groups, in particular for non reductive groups over an imperfect field.

Subtle characteristic classes for Spin-torsors

Extending (Smirnov and Vishik, Subtle Characteristic Classes, arXiv:1401.6661), we obtain a complete description of the motivic cohomology with {{,mathrm{mathbb {Z}},}}/2-coefficients of the Nisnevich classifying space of the spin group Spin_n associated to the standard split quadratic form. This provides us with very simple relations among subtle Stiefel–Whitney classes in the motivic cohomology of Čech simplicial schemes associated to quadratic forms from I^3, which are closely related to Spin_n-torsors over the point. These relations come from the action of the motivic Steenrod algebra on the second subtle Stiefel–Whitney class. Moreover, exploiting the relation between Spin_7 and G_2, we describe completely the motivic cohomology ring of the Nisnevich classifying space of G_2. The result in topology was obtained by Quillen (Math Ann 194:197–212, 1971).

On uniqueness ofp-adic period morphisms, II

We prove equality of the various rational$p$-adic period morphisms for smooth, not necessarily proper, schemes. We start with showing that the$K$-theoretical uniqueness criterion we had found earlier for proper smooth schemes extends to proper finite simplicial schemes in the good reduction case and to cohomology with compact support in the semistable reduction case. It yields the equality of the period morphisms for cohomology with compact support defined using the syntomic, almost étale, and motivic constructions. We continue with showing that the$h$-cohomology period morphism agrees with the syntomic and almost étale period morphisms whenever the latter morphisms are defined (and up to a change of Hyodo–Kato cohomology). We do it by lifting the syntomic and almost étale period morphisms to the$h$-site of varieties over a field, where their equality with the$h$-cohomology period morphism can be checked directly using the Beilinson Poincaré lemma and the case of dimension$0$. This also shows that the syntomic and almost étale period morphisms have a natural extension to the Voevodsky triangulated category of motives and enjoy many useful properties (since so does the$h$-cohomology period morphism).

First Nontrivial Group of Homologies for the Simplicial Schemes of Unimodular Frames Over the Dedekind Ring

We prove a theorem on the generation of the first nontrivial group of homologies of a simplicial scheme of unimodular frames over the Dedekind ring by standard cycles.

Subtle Characteristic Classes and Hermitian Forms

Following A. Smirnov and A. Vishik ["Subtle characteristic classes", Preprint, ], we compute the motivic cohomology ring of the Nisnevich classifying space of the unitary group associated to the standard split hermitian form of a quadratic extension. This provides us with subtle characteristic classes which take value in the motivic cohomology of the Čech simplicial scheme associated to a hermitian form. Comparing these new classes with subtle Stiefel-Whitney classes arising in the orthogonal case, we obtain relations among the latter ones holding in the motivic cohomology of the Čech simplicial scheme associated to a quadratic form divisible by a 1-fold Pfister form. Moreover, we present a description of the motive of the torsor corresponding to a hermitian form in terms of its subtle characteristic classes.

The Weights of the Tautological Classes of Character Varieties

I calculate the weights of the tautological classes of character varieties using the functorial mixed Hodge structure on simplicial schemes.

Preacyclicity Over the Rings with Infinite Fields of Residues

We consider a simplicial scheme, its geometric realization, and the required properties of the simplicial schemes for unimodal frames and prove both a sufficiently strong theorem on acyclicity for the simplicial schemes of unimodal frames and a theorem stating that the first nontrivial group of homologies is generated by standard cycles over the rings with infinite fields of residues.

Biharmonic optimization of piecewise planar surfaces

In this paper, we propose a method for optimizing geometric objects such as polygons and triangulated piecewise planar surfaces, for the purposes of greater detailing and providing visual smoothness of their images. The initial object is treated as a geometric realization of a simplicial scheme. The method involves refining the simplicial scheme and biharmonic interpolation of the embedding function of a subdivision of the simplicial scheme in Euclidean space.

The genus of the configuration spaces for Artin groups of affine type

Let (\mathbf W,S) be a Coxeter system, S finite, and let \mathbf G_{\mathbf W} be the associated Artin group. One has _ configuration spaces_ \mathbf Y,\ \mathbf Y_{\mathbf W}, where \mathbf G_{\mathbf W}=\pi_1(\mathbf Y_{\mathbf W}), and a natural \mathbf W -covering f_{\mathbf W}:\ \mathbf Y\to\mathbf Y_{\mathbf W}. The_Schwarz genus_ g(f_{\mathbf W}) is a natural topological invariant to consider. In [DS00] it was computed for all finite-type Artin groups, with the exception of case A_n (for which see [Vas92],[DPS04]). In this paper we generalize this result by computing the Schwarz genus for a class of Artin groups, which includes the affine-type Artin groups. Let K=K(\mathbf W,S) be the simplicial scheme of all subsets J\subset S such that the parabolic group \mathbf W_J is finite. We introduce the class of groups for which \mathrm{dim}(K) equals the homological dimension of K, and we show that g(f_{\mathbf W}) is always the maximum possible for such class of groups. For affine Artin groups, such maximum reduces to the rank of the group. In general, it is given by \mathrm{dim}(\mathbf X_{\mathbf W})+1, where \mathbf X_{\mathbf W}\subset \mathbf Y_{\mathbf W} is a well-known CW -complex which has the same homotopy type as \mathbf Y_{\mathbf W} .

On the motive of the group of units of a division algebra

AbstractWe consider the algebraic group GL1 (A), where A is a division algebra of prime degree over a field F, and the associated motive in the Voevodsky category of motivic complexes (F). We relate the motive of GL1 (A) to the motive of the Čech simplicial scheme χ, associated to the Severi-Brauer variety of A, and compute the second differential in the resulting spectral sequence for motivic cohomology.

Chow cohomology groups of algebraic surfaces

For an algeraic surface with isolated singularities we consider its higher Chow group and Chow cohomology (the latter defined by the author). We study the canonical map from the Chow cohomology to the higher Chow group by relating it to the canonical map from the higher Chow group to Chow cohomology of the exceptional divisor of a desingularization of the surface. Introduction. For a quasi-projective variety S over a field, S. Bloch defined its higher Chow groups CH(S, n) as the homology of a certain complex Z(S, ·) called the cycle complex, [Bl 1], [Bl 2], [Bl 3]. One may view this as a Borel-Moore homology theory; for example it is covariantly functorial for proper maps, and contravariantly functorial for open immersions (more generally for flat maps). For a quasi-projective variety S over a field of characteristic zero, using resolution of singularities and Bloch’s cycle complexes, we defined the Chow cohomology groups CHC(S, n) (see [Ha 2] for details). To briefly recall it, we take a cubical hyperresolution X• → S, which gives a strict truncated simplicial scheme; it consists of smooth varieties Xa for 0 ≤ a ≤ N with some N , and the face maps di : Xn → Xn−1, i = 0, . . . , n, satisfying the usual identities. (It differs from a simplicial scheme in that there are only face maps and no degeneracies, and there are only finite many terms.) We then form the double complex Z(X0, ·) d∗ −−−→Z(X1, ·) d∗ −−−→· · · d ∗ −−−→Z(XN , ·) where the a-th column is the cycle complex of Xa, and the horizontal differentials d ∗ : Z(Xa, ·) → Z(Xa+1, ·) are the alternating sums of the pull-backs di by the face maps. (Strictly speaking one must take appropriate quasi-isomorphic subcomplexes for d∗ be defined, see §1.) The total complex of this double complex is denoted Z(X•, ·)∗, and called the cohomological cycle complex of S. Then CHC(S, n) is by definition the (−n)-th cohomology of Z(X•, ·)∗. It is proven in [Ha 2] that CHC(S, n) is well-defined up to canonical isomorphism, independent of the choice of a hyperresolution, and that the association S 7→ CHC(S, n) is contravariantly functorial for all maps. The condition the characteristic being zero is unnecessary if dimS ≤ 2, since its desingularization (and thus its cubical hyperresolution) exist. In that case we exhibit the cubical hyperresolution and the cohomological cycle complex of S explicitly in §§1 and 2. (The relationship of our Chow cohomology to the motivic cohomology of [FV] will be discussed in a separate paper.) 2010Mathematics Subject Classification. Primary 14C25; Secondary 14C15, 14C35.

Presenting higher stacks as simplicial schemes

We show that an n-geometric stack may be regarded as a special kind of simplicial scheme, namely a Duskin n-hypergroupoid in affine schemes, where surjectivity is defined in terms of covering maps, yielding Artin n-stacks, Deligne–Mumford n-stacks and n-schemes as the notion of covering varies. This formulation adapts to all HAG contexts, so in particular works for derived n-stacks (replacing rings with simplicial rings). We exploit this to describe quasi-coherent sheaves and complexes on these stacks, and to draw comparisons with Kontsevich’s dg-schemes. As an application, we show how the cotangent complex controls infinitesimal deformations of higher and derived stacks.

Étale homotopy types and bisimplicial hypercovers

Suppose $(C,x)$ is a pointed locally connected small Grothendieck site, and let $(X,z)$ denote any connected locally fibrant simplicial sheaf $X$ equipped with a geometric point $z$. Following Artin-Mazur, an etale homotopy type of $X$ may then be defined via the geometrically pointed hypercovers of $X$ to yield a pro-object of the homotopy category, but this is not the only possible definition. In Etale homotopy of simplicial schemes, Friedlander defined another etale homotopy type of a simplicial scheme $X$ by taking diagonals of geometrically pointed bisimplicial hypercovers. In this paper, these two types are shown to be pro-isomorphic by means of a direct comparison of the associated cocycle categories. Friedlander's construction of etale homotopy types as actual pro-simplicial sets relies on a rigidity property of the etale topology that may not always be available for arbitrary sites; the cocycle methods employed here do not have this limitation. By consequence, the associated homotopy types constructed from hypercovers and bisimplicial hypercovers are shown to be pro-isomorphic on any locally connected small Grothendieck site, and the comparison at the level of cocycles shows, in particular, that both abelian and non-abelian sheaf cohomology may be computed via bisimplicial hypercovers on arbitrary small Grothendieck sites.

Motives over simplicial schemes

AbstractThis paper was written as a part of [8] and is intended primarily to provide the definitions and results concerning motives over simplicial schemes, which are used in the proof of the Bloch-Kato conjecture.

Homology of linear groups via cycles in BG× X

For X a smooth separated scheme of finite type over a field k, and for G a linear algebraic group over k, we construct homology-type functors H i(X,G) from cycles in the simplicial scheme BG× X. When X=Spec( k) and k is algebraically closed, these groups are the ordinary homology groups H i ( G( k)) of the discrete group of k-rational points of G. We construct a spectral sequence beginning with our groups and converging to the étale cohomology of the simplicial scheme BG, thus relating this theory to conclude with some calculations in the case where X= Spec( R) .

Etale realization on the [formula omitted]-homotopy theory of schemes

We compare Friedlander's definition of the étale topological type for simplicial schemes to another definition involving realizations of pro-simplicial sets. This can be expressed as a notion of hypercover descent for étale homotopy. We use this result to construct a homotopy invariant functor from the category of simplicial presheaves on the étale site of schemes over S to the category of pro-spaces. After completing away from the characteristics of the residue fields of S, we get a functor from the Morel–Voevodsky A 1 -homotopy category of schemes to the homotopy category of pro-spaces.

Finiteness of coherent cohomology for proper 𝑓𝑝𝑝𝑓 stacks

We prove the result in the title for Noetherian fppf stacks, avoiding the use of Chow’s lemma. Instead we employ simplicial schemes and shrinking-methods.

Pavages des simplexes, schémas de graphes recollés et compactification des PGL r n+1 /PGL r

Some equivariant compactifications of the quotients PGLrn+1/PGLr are constructed. Each one is decomposed into locally closed strata which are smooth, are indexed by the entire convex pavings of the simplex of dimension n and admit a modular interpretation deduced from that of the Grassmann varieties. Together, they form a simplicial scheme which “compactifies” the classifying simplicial scheme of PGLr consisting of all the quotients PGLrn+1/PGLr, n≥0.

Projective graph theory and configurations of lines

The spaces of disjoint configurations of k-dimensional subspaces in ℝP 2k+1 (for example, lines in ℝP 3) are studied. These spaces are modeled by various simplicial schemes, and the homology groups of the latter are computed in certain cases. We use the fact that every configuration can be assigned a so-called projective graph, which is a class of graphs with respect to a certain equivalence relation. Bibliography: 5 titles.

Simultaneous structural analysis and design based on augmented Lagrangian duality

Solution procedures in structural optimization are commonly based on a nested approach where approximations of the analysis and design problems are solved alternately in an iterative scheme. In this paper, we study a simultaneous approach based on an integrated formulation of the analysis and design problems. An advantage of the simultaneous approach, when compared to the nested one, is that the dependence between the analysis and design variables is imposed explicitly. In the nested approach, this dependence is implicitly determined through the solution of the analysis problem. Earlier simultaneous approaches mostly utilize various penalty function reformulations. In this paper, we make use of two augmented Lagrangian schemes, which avoid the numerical ill-conditioning inherent in penalty reformulations. These schemes give rise to Lagrangian subproblems with somewhat different properties, and two efficient techniques are adapted for their solution. The first is a projected Newton method, and the second is a simplicial decomposition scheme. Computational results for bar-truss structures show that the proposed schemes are viable approaches for solving the integrated formulation, and that they are promising for future developments.