A pseudocompact ring A is a complete Hausdorff topological ring which admits a system of open neighborhoods of 0 consisting of two sided ideals I for which A/I is an Artin ring. A complete HausdorR topological A-module M is said to be pseudocompact if it has a system of open neighborhoods of 0 consisting of submodules N for which M/N has finite length. The category G?,, of pseudocompact (I-modules is an Abelian category with exact inverse limits and enough projectives. Such generalities, which are more or less well known, are gathered in the first section for the convenience of the reader. To get more interesting results, we must introduce some commutativity by assuming that, in addition, (1 is a pseudocompact algebra over a commutative pseudocompact ring Q (see definition in Section 2). We may then define a tensor product on %A and introduce its derived functor 554. The category %+,, of discrete cl-modules, which is dual to VA0 by Proposition 2.3, also plays an important role through the bifunctor Horn : V, x 5BA + go and its derived functor &zt. We then have the proper setting for doing homological algebra. This is done in Section 3 which generalizes the elementary results on homological dimension in complete Noetherian semilocal rings. As an immediate application we find that if s2({xi}} is the algebra of noncommutative formal power series in {xi} over Q, then gl dim J2{{xi}} = gl dim Sa + 1 (Theorem 3.9). We recall that a profinite group is a compact totally disconnected topological group, i.e., an inverse limit of finite groups. We define the complete group