arXiv subjects

Anders Thorup

Publications and source records attributed to Anders Thorup.

4 recordsLinked to original sources

arXiv2013

Completeness of the ring of polynomials

Let $k$ be an uncountable field. We prove that the polynomial ring $R:=k[X_1,\dots,X_n]$ in $n\ge 2$ variables over $k$ is complete in its adic topology. In addition we prove that also the localization $R_{\goth m}$ at a maximal ideal $\goth m\subset R$ is adically complete. The first result settles an old conjecture of C. U. Jensen, the second a conjecture of L. Gruson. Our proofs are based on a result of Gruson stating (in two variables) that $R_{\goth m}$ is adically complete when $R=k[X_1,X_2]$ and $\goth m=(X_1,X_2)$.

math.AC↗

arXiv2012

Remarks on certain composita of fields

Let $L$ and $M$ be two algebraically closed fields contained in some common larger field. It is obvious that the intersection $C=L\cap M$ is also algebraically closed. Although the compositum $LM$ is obviously perfect, there is no reason why it should be algebraically closed except when one of the two fields is contained in the other. We prove that if the two fields are strictly larger that $C$, and linearly disjoint over $C$, then the compositum $LM$ is not algebraically closed; in fact we shall prove that the Galois group of the maximal abelian extension of $LM$ is the free pro-abelian group of rank $|LM|$, and that the free pro-nilpotent group of rank $|C|$ can be realized as a Galois group over $LM$. The above results may be considered as the main contribution of this article but we obtain some additional results on field composita that might be of independent interest.

math.AC↗

arXiv2011

On the invariants of the splitting algebra

For a given monic polynomial $p(t)$ of degree $n$ over a commutative ring $k$, the splitting algebra is the universal $k$-algebra in which $p(t)$ has $n$ roots, or, more precisely, over which $p(t)$ factors, $p(t)=(t-ξ_1)...(t-ξ_n)$. The symmetric group $S_r$ for $1\le r\le n$ acts on the splitting algebra by permuting the first $r$ roots $ξ_1,...,ξ_r$. We give a natural, simple condition on the polynomial $p(t)$ that holds if and only if there are only trivial invariants under the actions. In particular, if the condition on $p(t)$ holds then the elements of $k$ are the only invariants under the action of $S_n$. We show that for any $n\ge 2$ there is a polynomial $p(t)$ of degree $n$ for which the splitting algebra contains a nontrivial element invariant under $S_n$. The examples violate an assertion by A. D. Barnard from 1974.

math.AC↗

arXiv1995

Mixed Buchsbaum--Rim Multiplicities

We prove the results about mixed Buchsbaum--Rim multiplicities announced in (9.10)(ii) on p.224 of our recent paper [J.Alg.(1994)], including a general mixed-multiplicity formula. In addition, we identify these multiplicities as the coefficients of the ``leading form'' of the appropriate Buchsbaum-Rim polynomial in three variables, and we prove a positivity theorem. In fact, we define the multiplicities as the degrees of certain zero-dimensional ``mixed twisted'' Segre classes, and we develop an encompassing general theory of these new rational equivalence classes in all dimensions. In parallel, we develop a theory of pure ``twisted'' Segre classes, and we recover the main results in [J.Alg.(1994)] about the pure Buchsbaum--Rim multiplicities, the polar multiplicities, and so forth. Moreover, we identify the additivity theorem [J.Alg.(1994), (6.7b)(i), p.205] as giving a sort of residual-intersection formula, and we show its (somewhat unexpected) connection to the mixed-multiplicity formula. Also, we work in a more general setup than before, and we develop a new approach, based on the completed normal cone.

alg-geom↗