scholarly journals Oriented bivariant theory, II: Algebraic cobordism of S-schemes

2019 ◽  
Vol 30 (06) ◽  
pp. 1950031
Author(s):  
Shoji Yokura
Keyword(s):  
Finite Type ◽  
Double Point ◽  
Base Field ◽  
Base Scheme ◽  
Bivariant Theory ◽  
Algebraic Cobordism

This is a sequel to our previous paper “Oriented bivariant theory, I”. In 2001, Levine and Morel constructed algebraic cobordism for (reduced) schemes [Formula: see text] of finite type over a base field [Formula: see text] in an abstract way and later Levine and Pandharipande reconstructed it more geometrically, using “double point degeneration”. In this paper in a similar manner to the construction of Levine–Morel, we construct an algebraic cobordism for a scheme [Formula: see text] over a fixed scheme [Formula: see text] in such a way that if the target scheme [Formula: see text] is the point [Formula: see text], then our algebraic cobordism is isomorphic to Levine–Morel’s algebraic cobordism. Our algebraic cobordism can be interpreted as “a family of algebraic cobordism” parametrized by the base scheme [Formula: see text].

scholarly journals Connective Algebraic K-theory

2014 ◽  
Vol 13 (1) ◽  
pp. 9-56 ◽  
Author(s):  
Shouxin Dai ◽  
Marc Levine
Keyword(s):  
Finite Type ◽  
Formal Group ◽  
Abelian Category ◽  
Homology Theory ◽  
Fundamental Class ◽  
Coherent Sheaves ◽  
Base Scheme ◽  
Stable Homotopy Category ◽  
Algebraic Cobordism ◽  
Coefficient Ring

AbstractWe examine the theory of connective algebraic K-theory, , defined by taking the −1 connective cover of algebraic K-theory with respect to Voevodsky's slice tower in the motivic stable homotopy category. We extend to a bi-graded oriented duality theory when the base scheme is the spectrum of a field k of characteristic zero. The homology theory may be viewed as connective algebraic G-theory. We identify for X a finite type k-scheme with the image of in , where is the abelian category of coherent sheaves on X with support in dimension at most n; this agrees with the (2n,n) part of the theory of connective algebraic K-theory defined by Cai. We also show that the classifying map from algebraic cobordism identifies with the universal oriented Borel-Moore homology theory having formal group law u + υ − βuυ with coefficient ring ℤ[β]. As an application, we show that every pure dimension d finite type k-scheme has a well-defined fundamental class [X]CK in ΩdCK(X), and this fundamental class is functorial with respect to pull-back for l.c.i. morphisms.

Variations on a theme of rationality of cycles

2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Nikita Karpenko
Keyword(s):  
Function Field ◽  
Algebraic Cycles ◽  
Base Field ◽  
Arbitrary Characteristic ◽  
Steenrod Operations ◽  
Cobordism Theory ◽  
Characteristic 2 ◽  
Algebraic Cobordism ◽  
Variations On A Theme

AbstractWe prove certain weak versions of some celebrated results due to Alexander Vishik comparing rationality of algebraic cycles over the function field of a quadric and over the base field. The original proofs use Vishik’s symmetric operations in the algebraic cobordism theory and work only in characteristic 0. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2. Our weak versions are still sufficient for existing applications. In particular, Vishik’s construction of fields of u-invariant 2r + 1, for r ≥ 3, is extended to arbitrary characteristic ≠ 2.

scholarly journals Around rationality of cycles

2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Raphaël Fino
Keyword(s):  
Function Field ◽  
Algebraic Cycles ◽  
Base Field ◽  
Steenrod Operations ◽  
Cobordism Theory ◽  
Characteristic 2 ◽  
Algebraic Cobordism

AbstractWe prove certain results comparing rationality of algebraic cycles over the function field of a quadric and over the base field. These results have already been obtained by Alexander Vishik in the case of characteristic 0, which allowed him to work with algebraic cobordism theory. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2.

scholarly journals ORIENTED BIVARIANT THEORIES, I

2009 ◽  
Vol 20 (10) ◽  
pp. 1305-1334 ◽  
Author(s):  
SHOJI YOKURA
Keyword(s):  
Singular Spaces ◽  
Geometric Type ◽  
Bivariant Theory ◽  
Algebraic Cobordism ◽  
Special Case

In 1981 W. Fulton and R. MacPherson introduced the notion of bivariant theory (BT), which is a sophisticated unification of covariant theories and contravariant theories. This is for the study of singular spaces. In 2001 M. Levine and F. Morel introduced the notion of algebraic cobordism, which is a universal oriented Borel–Moore functor with products (OBMF) of geometric type, in an attempt to understand better V. Voevodsky's (higher) algebraic cobordism. In this paper we introduce a notion of oriented bivariant theory (OBT), a special case of which is nothing but the oriented Borel–Moore functor with products. The present paper is a first one of the series to try to understand Levine–Morel's algebraic cobordism from a bivariant theoretical viewpoint, and its first step is to introduce OBT as a unification of BT and OBMF.

scholarly journals Recent Progress on Submersions: A Survey and New Properties

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Gabriel Picavet
Keyword(s):  
Finite Type ◽  
Recent Progress ◽  
Valuation Domain ◽  
Base Scheme ◽  
Dominant Property ◽  
Dual Notion

This paper is a survey about recent progress on submersive morphisms of schemes combined with new results that we prove. They concern the class of quasicompact universally subtrusive morphisms that we introduced about 30 years ago. They are revisited in a recent paper by Rydh, with substantial complements and key results. We use them to show Artin-Tate-like results about the 14th problem of Hilbert, for a base scheme either Noetherian or the spectrum of a valuation domain. We look at faithfully flat morphisms and get “almost” Artin-Tate-like results by considering the Goldman (finite type) points of a scheme. Bjorn Poonen recently proved that universally closed morphisms are quasicompact. By introducing incomparable morphisms of schemes, we are able to characterize universally closed surjective morphisms that are either integral or finite. Next we consider pure morphisms of schemes introduced by Mesablishvili. In the quasicompact case, they are universally schematically dominant morphisms. This leads us to a characterization of universally subtrusive morphisms by purity. Some results on the schematically dominant property are given. The paper ends with properties of monomorphisms and topological immersions, a dual notion of submersions.

scholarly journals Structure de poids à la Bondarko sur les motifs de Beilinson

2011 ◽  
Vol 147 (5) ◽  
pp. 1447-1462 ◽  
Author(s):  
David Hébert
Keyword(s):  
Finite Type ◽  
Base Scheme ◽  
Weight Structure

AbstractBondarko defines and studies the notion of weight structure and he shows that there exists a weight structure over the category of Voevodsky motives with rational coefficients (over a field of characteristic 0). In this paper we extend this weight structure to the category of Beilinson motives (for any scheme of finite type over a base scheme which is excellent of dimension at most two) introduced and studied by Cisinsky and Déglise. We also check the weight exactness of the Grothendieck operations.

Cross Double Point Discharge as Enhanced Excitation Source for Highly Sensitive Determination of Arsenic, Mercury and Lead by Optical Emission Spectrometry

2021 ◽  
Author(s):  
Lin He ◽  
Peixia Li ◽  
Kai Li ◽  
Tao Lin ◽  
Jin Luo ◽  
...  
Keyword(s):  
Double Point ◽  
Hydride Generation ◽  
Optical Emission ◽  
Sample Introduction ◽  
Excitation Source ◽  
Emission Spectrometry ◽  
Highly Sensitive ◽  
Point Discharge ◽  
Sensitive Determination

A new cross double point discharge (CrossPD) microplasma was designed as an excitation source to construct a miniaturized optical emission spectrometer with hydride generation (HG) for sample introduction. The CrossPD...

Regular Hom-algebras admitting a multiplicative basis

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Antonio J. Calderón Martín
Keyword(s):  
Arbitrary Dimension ◽  
Base Field ◽  
Direct Sum ◽  
Multiplicative Basis ◽  
Arbitrary Base ◽  
The Family

AbstractLet {({\mathfrak{H}},\mu,\alpha)} be a regular Hom-algebra of arbitrary dimension and over an arbitrary base field {{\mathbb{F}}}. A basis {{\mathcal{B}}=\{e_{i}\}_{i\in I}} of {{\mathfrak{H}}} is called multiplicative if for any {i,j\in I}, we have that {\mu(e_{i},e_{j})\in{\mathbb{F}}e_{k}} and {\alpha(e_{i})\in{\mathbb{F}}e_{p}} for some {k,p\in I}. We show that if {{\mathfrak{H}}} admits a multiplicative basis, then it decomposes as the direct sum {{\mathfrak{H}}=\bigoplus_{r}{{\mathfrak{I}}}_{r}} of well-described ideals admitting each one a multiplicative basis. Also, the minimality of {{\mathfrak{H}}} is characterized in terms of the multiplicative basis and it is shown that, in case {{\mathcal{B}}}, in addition, it is a basis of division, then the above direct sum is composed by means of the family of its minimal ideals, each one admitting a multiplicative basis of division.

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