The self-intersection formula and the ‘formule-clef’

1975 ◽  
Vol 78 (1) ◽  
pp. 117-123 ◽  
Author(s):  
A. T. Lascu ◽  
D. Mumford ◽  
D. B. Scott
Keyword(s):  
Ring Homomorphism ◽  
Equivalence Classes ◽  
The Self ◽  
Group Homomorphism ◽  
Closed Field ◽  
Chow Ring ◽  
Algebraically Closed Field ◽  
Rational Equivalence ◽  
Image Position ◽  
Intersection Formula

We shall consider exclusively algebraic non-singular quasi-projective irreducible varieties over an algebraically closed field. If V is such a variety will be the Chow ring of rational equivalence classes of cycles of Vand the group homomorphism defined by any proper morphism φ: V1 → V2. Alsodenotes the ring homomorphism defined by φ.

scholarly journals An algebraically closed field

1968 ◽  
Vol 9 (2) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Rayner
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Zero Characteristic ◽  
Image Position ◽  
Formal Power ◽  
Puiseux Expansions

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

On Matrix Commutators

1960 ◽  
Vol 12 ◽  
pp. 269-277 ◽  
Author(s):  
Marvin Marcus ◽  
Nisar A. Khan
Keyword(s):  
Linear Space ◽  
Elementary Divisor ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Elementary Divisors ◽  
Image Position

Let A, B, and X be n-square matrices over an algebraically closed field F of characteristic 0. Let [A, B] = AB — BA and set (A, B) = [A, [A, B]]. Recently several proofs (1; 3; 5) of the following result have appeared: if det (AB) ≠ 0 and (A,B) = 0 then A-1B-1AB - I is nilpotent. In (2) McCoy determined the general form of any X satisfying1.1in the case that A has a single elementary divisor corresponding to each eigenvalue, that is, A is non-derogatory. In Theorem 1 we determine the structure of any matrix X satisfying (1.1) and also give a formula for the dimension of the linear space of all such X in terms of the degrees of the elementary divisors of A.

Linear Relations Between Higher Matrix Commutators

1964 ◽  
Vol 16 ◽  
pp. 315-320 ◽  
Author(s):  
Nisar A. Khan
Keyword(s):  
Linear Relation ◽  
Closed Field ◽  
Research Papers ◽  
Algebraically Closed Field ◽  
Linear Relations ◽  
Image Position ◽  
Iterated Commutators

Let Mn denote the space of all n-square matrices over an algebraically closed field F. For A, B ∊ Mn, letdefine the iterated commutators of A and B. Recently several research papers (1, 2, 4, and 5) have appeared on these commutators. In (1), Kato and Taussky have proved that for n = 2 the iterated commutators of A and B satisfy the linear relation

Minimal Operators for Schrodinger-Type Differential Expressions with Discontinuous Principal Coefficients

1980 ◽  
Vol 32 (6) ◽  
pp. 1423-1437 ◽  
Author(s):  
M. Faierman ◽  
I. Knowles
Keyword(s):  
Basic Problem ◽  
Adjoint Operator ◽  
General Setting ◽  
Equivalence Classes ◽  
The Self ◽  
Singular Potentials ◽  
Image Position ◽  
Schrödinger Type Operators ◽  
Complex Valued

The objective of this paper is to extend the recent results [7, 8, 9] concerning the self-adjointness of Schrödinger-type operators with singular potentials to a more general setting. We shall be concerned here with formally symmetric elliptic differential expressions of the form1.1where x = (x1, …, xm) ∈ Rm (and m ≧ 1), i = (–1)1/2, ∂j = ∂/∂xj, and the coefficients ajk, bj and q are real-valued and measurable on Rm.The basic problem that we consider is that of deciding whether or not the formal operator defined by (1.1) determines a unique self-adjoint operator in the space L2(Rm) of (equivalence classes of) square integrable complex-valued functions on Rm.

scholarly journals Homology of SL2 over function fields I: Parabolic subcomplexes

2018 ◽  
Vol 2018 (739) ◽  
pp. 159-205
Author(s):  
Matthias Wendt
Keyword(s):  
Vector Bundles ◽  
Function Fields ◽  
Equivalence Classes ◽  
Isotropy Group ◽  
Closed Field ◽  
Explicit Formulas ◽  
Algebraically Closed Field ◽  
Group Homology ◽  
Equivariant Homology ◽  
Smooth Affine

Abstract The present paper studies the group homology of the groups {\operatorname{SL}_{2}(k[C])} and {\operatorname{PGL}_{2}(k[C])} , where {C=\overline{C}\setminus\{P_{1},\dots,P_{s}\}} is a smooth affine curve over an algebraically closed field k. It is well known that these groups act on a product of trees and the quotients can be described in terms of certain equivalence classes of rank two vector bundles on the curve {\overline{C}} . There is a natural subcomplex consisting of cells with suitably non-trivial isotropy group. The paper provides explicit formulas for the equivariant homology of this “parabolic subcomplex”. These formulas also describe group homology of {\operatorname{SL}_{2}(k[C])} above degree s, generalizing a result of Suslin in the case {s=1} .

scholarly journals On the Grothendieck ring of varieties

2015 ◽  
Vol 158 (3) ◽  
pp. 477-486
Author(s):  
AMIT KUBER
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Equivalence Classes ◽  
Closed Field ◽  
Associated Graded Ring ◽  
Grothendieck Ring ◽  
Algebraically Closed Field ◽  
Monoid Ring ◽  
Birational Equivalence ◽  
Multiplicative Monoid

AbstractLet K0(Vark) denote the Grothendieck ring of k-varieties over an algebraically closed field k. Larsen and Lunts asked if two k-varieties having the same class in K0(Vark) are piecewise isomorphic. Gromov asked if a birational self-map of a k-variety can be extended to a piecewise automorphism. We show that these two questions are equivalent over any algebraically closed field. If these two questions admit a positive answer, then we prove that its underlying abelian group is a free abelian group and that the associated graded ring of the Grothendieck ring is the monoid ring $\mathbb{Z}$[$\mathfrak{B}$] where $\mathfrak{B}$ denotes the multiplicative monoid of birational equivalence classes of irreducible k-varieties.

A Multivariable form of the Fundamental Theorem of Algebra

1983 ◽  
Vol 26 (3) ◽  
pp. 271-272
Author(s):  
Pablo M. Salzberg
Keyword(s):  
Fundamental Theorem ◽  
Homogeneous Polynomial ◽  
Inner Product ◽  
Closed Field ◽  
Sufficient Condition ◽  
Algebraically Closed Field ◽  
Fundamental Theorem Of Algebra ◽  
Image Position ◽  
Derivatives Of

AbstractLet H(x) be a homogeneous polynomial in n indeterminates over an algebraically closed field K. A necesssary and sufficient condition is given for H(x) to admit a factorization of the forma, b∈ Kn, and “∘” is the usual inner product. This condition involves the linear derivatives of H(x).

A NOTE ON DORMANT OPERS OF RANK IN CHARACTERISTIC

2018 ◽  
Vol 235 ◽  
pp. 115-126
Author(s):  
YUICHIRO HOSHI
Keyword(s):  
Smooth Curve ◽  
Equivalence Classes ◽  
Closed Field ◽  
Algebraically Closed Field

In this paper, we prove that the set of equivalence classes of dormant opers of rank $p-1$ over a projective smooth curve of genus ${\geqslant}2$ over an algebraically closed field of characteristic $p>0$ is of cardinality one.

scholarly journals Analytic Equivalence of Algebroid Curves

1959 ◽  
Vol 11 ◽  
pp. 1-17
Author(s):  
Andrew H. Wallace
Keyword(s):  
Power Series ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Image Position

Let k be an algebraically closed field and let x1, x2, . . . , xn be indeterminates. Denote by Rn the ring k[[x1, x2, … , xn]] of power series in the xi With coefficients in the field k. Let and be two ideals in this ring. Then and will be said to be analytically equivalent if there is an automorphism T of Rn such that T() = . and will be called analytically equivalent under T.The above situation can be described geometrically as follows. The ideals and can be regarded as defining algebroid varieties V and V' in (x1, x2, … , xn)-space, and these varieties will be said to be analytically equivalent under T.The automorphism T can be expressed by means of equations of the form :where the determinant is not zero and the fi are power series of order not less than two (that is to say, containing terms of degree two or more only).

A Semigroup Approach to Linear Algebraic Groups III. Buildings

1986 ◽  
Vol 38 (3) ◽  
pp. 751-768 ◽  
Author(s):  
Mohan S. Putcha
Keyword(s):  
Algebraic Groups ◽  
Usual Sense ◽  
Closed Field ◽  
Parabolic Subgroups ◽  
Algebraically Closed Field ◽  
Linear Algebraic Groups ◽  
Semigroup Approach ◽  
Image Position

Introduction. Let K be an algebraically closed field, G = SL(3, K) the group of 3 × 3 matrices over K of determinant 1. Let denote the monoid of all 3 × 3 matrices over K. If e is an idempotent in , thenare opposite parabolic subgroups of G in the usual sense [1], [28]. However the mapdoes not exhaust all pairs of opposite parabolic subgroups of G. Now consider the representation ϕ:G → SL(6, K) given by

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