Pseudodeformations
From the text: ``Mazur's deformation theory of representations of (Galois) groups played a very important role in the striking progresses that have been made in algebraic number theory in the last fifteen years (to name a few: the proof of Shimura-Taniyama-Weil conjecture, of Serre's conjecture on the modularity of odd \(\mod p\) Galois representations, of Fontaine-Mazur's conjecture, and of Sato-Tate conjecture). The theory works as its best only when the representation to be deformed, \(\overline\rho: G\to \mathrm{GL}_d(\kappa)\) (where \(\kappa\) is a field), satisfies the following condition (1): \(\dim_\kappa \mathrm{End}_G(\overline\rho, \overline\rho)= 1\). By Schur's lemma, this happens for example if \(\overline\rho\) is absolutely irreducible. In this case, under some finiteness hypotheses on the group \(G\) that are satisfied in applications to Galois theory, Mazur's deformation functor \(A\to D_{\overline\rho}(A)\) which attaches to an Artinian local ring \(A\) with residue field \(\kappa\) the set of strict equivalence classes of deformations \(\rho: G\to\mathrm{GL}_d(A)\) of \(\overline\rho\) is pro-representable by a complete Noetherian local ring \(R\) (see [\textit{B. Mazur}, Math. Sci. Res. Inst. 16, 385--437 (1989; Zbl 0714.11076)]. The tangent space of this local ring is identified with the space \(\mathrm{Ext}^1_G(\overline\rho, \overline\rho)= H^1(G, \mathrm{ad\,}\overline\rho)\). Lifting questions are also solved by the construction of obstruction classes in \(\mathrm{Ext}^2_G(\overline\rho, \overline\rho)= H^2(G, \mathrm{ad\,}\overline\rho\)).'' ``We consider here the problem of finding analogues of those results in the absence of condition (1).'' ``We study the deformation functor of a reducible pseudo character. We show that there is a natural filtration (the complexity filtration) on such a functor, and that this filtration induces a filtration on the tangent space, whose graded pieces can be described in terms of the extension spaces between the irreducible components of the pseudo character. We also study the obstruction theory of that deformation problem.''
scientific article; zbMATH DE number 6031785