Semiperfect coalgebras over rings Robert Wisbauer University of D¨ usseldorf, Germany e-mail: wisbauer@math.uni-duesseldorf.de Abstract Our investigation of coalgebras over commutative rings R is based on the close relationship between comodules over a coalgebra C and modules over the dual algebra C * . If C is projective as an R-module the category of right C-comodules can be identified with the category σ[C ∗ C] of left C * -modules which are subgenerated by C. In this context semiperfect coalgebras are described by results from module theory. Over QF rings semiperfect coalgebras are characterized by the exactness of the trace functor Tr(σ[C ∗ C], -). 1 Introduction Although there are many interesting examples of coalgebras over rings R, a large part of literature on the structure theory is restricted to coalgebras over fields. This is mainly due to the fact that for certain basic proofs the existence of an R-basis is needed. Here we work with coalgebras C over any commutative ring R. A good deal of the basic definitions and properties carry over from base fields to rings nearly verbatim. For such situations we do not repeat proofs. Of course, the properties of C as an R-module will be of importance. Without any restriction on the coalgebra C we observe that the category of right C-comodules Comod-C is subgenerated by C, i.e., every right C-comodule is a subcomodule of some comodule which is generated by C. It turns out that Comod-C is a Grothendieck category if and only if R C is flat. In the classical structure theory of C the dual algebra C ∗ plays an important part. To make sure that C ∗ is not trivial we will need the condition that R C is projective. Then there exists a dual basis for R C and this will allow to transfer proofs known for base fields. In particular we will show that in this case Comod-C can be identified with σ[ C ∗ C], the category of left C ∗ -modules which are subgenerated by C. A coalgebra C is called right semiperfect if every simple right comodule has a projective cover in Comod-C. If R C is projective this corresponds to the condition that in σ[ C ∗ C] every simple module has a projective cover in σ[ C ∗ C], a situation which was well studied in module theory. Over QF rings R we have relationships between right C-comodules and left C-comodules (via finitely presented modules). Based on this we characterize right semiperfect coalgebras C by the exactness of the trace functor Tr(σ[ C ∗ C], −): C ∗ -Mod → σ[ C ∗ C]. In fact most of the propositions known for coalgebras over fields carry over to coalgebras over QF rings R provided R C is projective. For introductory texts on coalgebras the reader is referred to Abe [1], Beidar [6], Kaplansky [11], Montgomery [14] and Sweedler [15]. The main references for module theory are [17] and [18]. The author is indebted to K.I. Beidar, S. Dˇ ascˇalescu and J. G´omez Torrecillas for inspiring discussions on the topic. 1