Infinite Perfect Field Research Articles

Computing bases of complete intersection rings in Noether position

Let k be an effective infinite perfect field, k[ x 1,…, x n ] the polynomial ring in n variables and F∈ k[ x 1,…, x n ] M× M a square polynomial matrix verifying F 2= F. Suppose that the entries of F are polynomials given by a straight-line program of size L and their total degrees are bounded by an integer D. We show that there exists a well parallelizable algorithm which computes bases of the kernel and the image of F in time ( nL) O(1)( MD) O( n) . By means of this result we obtain a single exponential algorithm to compute a basis of a complete intersection ring in Noether position. More precisely, let f 1,…, f n− r ∈ k[ x 1,…, x n ] be a regular sequence of polynomials given by a slp of size ℓ, whose degrees are bounded by d. Let R≔ k[ x 1,…, x r ] and S≔ k[ x 1,…, x n ]/( f 1,…, f n− r ) such that S is integral over R; we show that there exists an algorithm running in time O( n)ℓ d O( n 2) which computes a basis of S over R. Also, as a consequence of our techniques, we show a single exponential well parallelizable algorithm which decides the freeness of a finite k[ x 1,…, x n ]-module given by a presentation matrix, and in the affirmative case it computes a basis.

Abstract Homomorphisms of Algebraic Groups

In this paper we are concerned with abstract homomorphisms from one algebraic group to another. That is, group homomorphisms where there is no assumption of rationality. Let K be an infinite perfect field and L an algebraically closed field of the same characteristic. Throughout G(K) will denote a Chevalley group of universal type over K.

On a question of nori: the local case

In this paper we consider an algebraic problem which was motivated by a topological problem posed by Nori, about the homotopy of sections of projective modules. We give an affirmative answer in the case of some local rings, namely when the ring is a powerseries ring k[[X1,… , Xn]] over a field k or when the ring is a regular k‐spot over an infinite perfect field k.

Bounds for traces in complete intersections and degrees in the Nullstellensatz

In this paper we obtain an effective Nullstellensatz using quantitative considerations of the classical duality theory in complete intersections. Letk be an infinite perfect field and let f1,...,f n−r∈k[X1,...,Xn] be a regular sequence with d:=maxj deg fj. Denote byA the polynomial ringk [X1,..., Xr] and byB the factor ring k[X1,...,Xn]/(f1,...,fnr); assume that the canonical morphism A→B is injective and integral and that the Jacobian determinantΔ with respect to the variables Xr+1,...,Xn is not a zero divisor inB. Let finally σ∈B*:=HomA(B, A) be the generator of B* associated to the regular sequence.

On the Existence of Modules Which Are neither Preprojectives nor Preinjectives

Let A be a finite-dimensional algebra over an infinite perfect field. Suppose that each indecomposable A-module is either a preprojective or a preinjective in the sense defined by Auslander and Smalø. It is proved here that under these hypotheses A is of finite representation type.

On a theorem of Ojanguren

Let k be a field of characteristic not two. It is proved that if a quadratic k[X1,...,Xn]-space is a hyperbolic space under an extension to the field ofrational functions k(X1,...,Xn, then the initial space is also hyperbolic. Earlier this result was obtained by Ojanguren under the additional assumption that k is an infinite perfect field (of characteristic not two).

Picard groups of singular affine curves over a perfect field

This was proved in [W1] for the case of an algebraically closed field k; in that case the conditions force R to be a Dedekind domain. For an infinite perfect ground field this is no longer the case. For example, the conditions are satisfied for the restricted polynomial ring k + X K [X l, where K/k is a finite algebraic extension and X is an indeterminate. A sharpened form of the main theorem, proved in the next section, says that a not Dedekind domain satisfying the conditions of (0.1) has exactly one singularity, and its singularity is analytically isomorphic to that of a restricted polynomial ring. We wilt show by example that (0.1.1) no longer implies (0.1.3) if "domain" is replaced by "reduced ring", or if "infinite" is deleted from the hypotheses on k. (I don't know whether perfection of the ground field k is essential.) We obtain a partial result for reduced rings: If D(R) is finitely generated, then R is seminormal and the intersection graph of the irreducible components of spec (R) is acyclic.

A Note on Complete Intersections

Let $R$ be a regular local ring and let $R[T]$ be a polynomial algebra in one variable over $R$. In this paper the author proves that every maximal ideal of $R[T]$ is complete intersection in each of the following cases: (1) $R$ is a local ring of an affine algebra over an infinite perfect field, (2) $R$ is a power series ring over a field.

A note on complete intersections

Let R R be a regular local ring and let R [ T ] R[T] be a polynomial algebra in one variable over R R . In this paper the author proves that every maximal ideal of R [ T ] R[T] is complete intersection in each of the following cases: (1) R R is a local ring of an affine algebra over an infinite perfect field, (2) R R is a power series ring over a field.

Representation of algebraic groups preserving quaternion skew-hermitian forms

Introduction. Let K be an infinite perfect field of characteristic different from 2, let e3 be a quaternion division algebra over K, and let tt-denote the canonical involution of the first kind on Z. Let V be a finite-dimensional right vector space over Z. A quaternion skew-hermitian form H over Z is a sesquilinear form on VX V, i.e., H is a map from VX V to Z such that (i) H(x, Yl+Y2) =H(x, yi) +H(x, Y2) and H(x, yax) =H(x, y)a for all x, y, Y1, y2 in V and ai in ; (ii) H(x, y) = -H(y, x) for all x, y in V. Let {xl, . . *, x.) be a basis for V over Q. We say that H is nondegenerate if the reduced norm in Mn(Z) of the matrix (H(xi, xj)) is not zero. Associated to such a nondegenerate form H are 3 invariants, the dimension of V over 0, dime(V), the discriminant of H, b(H), and the Clifford algebra of H, G. Let G be a simply connected semisimple algebraic group (in some GL(m, X)) which is defined over K and let p:G-?GL(V/Z) be an absolutely irreducible representation of G defined over K into the group of all nonsingular Z-linear endomorphisms of V. We shall assume that there is a nondegenerate quaternion skew-hermitian form H on V which is invariant with respect to p (G). The purpose of this paper is to describe the Clifford algebra of the invariant form H in terms of p, G, and the Steinberg group associated to G. In a previous paper, we have described dimsz(V) and b(H) in such a way and have indicated how representations such as p arise [2, Theorem I.2 ]. The invariant -y(G) plays an important role in our study and so we recall some of its properties in ?1. In 2, we define the invariant Y using the representation p. Jacobson first constructed the Clifford algebra of a quaternion skew-hermitian form [4]. In this paper, however, we shall follow a method due to Satake [6]. We give some examples in ?3 with special emphasis on the case where G is absolutely simple.