Varieties of modules for Z/2Z x Z/2Z

Let k be an algebraically closed field of characteristic 2. We prove that the restricted nilpotent commuting variety C, that is the set of pairs of (n x n)-matrices (A, B) such that A(2) = B-2 = [A, B] = 0, is equidimensional. C can be identified with the 'variety of n-dimensional modules' for Z/2Z x Z/2Z, or equivalently, for k[X, Y]/(X-2, Y-2). On the other hand, we provide an example showing that the restricted nilpotent commuting variety is not equidimensional for fields of characteristic > 2. We also prove that if e(2) = 0 then the set of elements of the centralizer of e whose square is zero is equidimensional. Finally, we express each irreducible component of C as a direct sum of indecomposable components of varieties of Z/2Z x Z/2Z-modules. (c) 2007 Elsevier Inc. All rights reserved.

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2007
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https://infoscience.epfl.ch/record/179626?ln=fr

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Concepts associés (4)

Groupe nilpotent

En théorie des groupes, les groupes nilpotents forment une certaine classe de groupes contenue dans celle des groupes résolubles et contenant celle des groupes abéliens. Les groupes nilpotents apparaissent dans la théorie de Galois et dans la classification des groupes de Lie ou des groupes algébriques linéaires. Soit G un groupe noté multiplicativement, d'élément neutre e. Si A et B sont deux sous-groupes de G, on note [A,B] le sous-groupe engendré par les commutateurs de la forme [x,y] pour x dans A et y dans B.

Direct sum

The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the abelian group. The direct sum of two abelian groups and is another abelian group consisting of the ordered pairs where and . To add ordered pairs, we define the sum to be ; in other words addition is defined coordinate-wise.

Irreducible component

In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal (for set inclusion) for this property. For example, the set of solutions of the equation xy = 0 is not irreducible, and its irreducible components are the two lines of equations x = 0 and y =0.