Appendices A Contramodules over Coalgebras over Fields Let C be a coassociative coalgebra with counit over a field k.Itiswellknown (see [82, Theorems 2.1.3 and 2.2.1] or [68, 5.1.1]) that C is the union of its finite- dimensional subcoalgebras and any C-comodule is a union of finite-dimensional comodules over finite-dimensional subcoalgebras of C. The dual assertion for C-contramodules is not true: for the most common of noncosemisimple infinite- dimensional coalgebras C there exist C-contramodules P such that the intersection of the images of Homk(C/U, P)inP over all finite-dimensional subcoalgebras U ⊂ C is nonzero. A weaker statement holds, however: if the contraaction map Homk(C/U, P) −−→ P is surjective for every finite-dimensional subcoalgebra U of C,thenP = 0. Besides, even though adic filtrations of contramodules are not in general separated, they are always complete. Using the related techniques we show that any contraflat C-con- tramodule is projective, generalizing the well-known result that any flat module over a finite-dimensional algebra is projective [7]. A.1 Counterexamples A.1.1 Let C be the coalgebra for which the dual algebra C∗ is isomorphic to the algebra of formal power series k[[x]]. Then a C-contramodule P can be equivalently defined as a k-vector space endowed with the following operation of summation of n sequences of vectors with formal coefficients x : for any elements p0, p1, ... in P, ∞ n an element of P denoted by n=0 x pn is defined. This operation should satisfy the following equations: ∞ ∞ ∞ n n n x (apn + bqn)=a x pn + b x qn n=0 n=0 n=0 for a, b ∈ k, pn, qn ∈ P (linearity); ∞ n x pn = p0 when p1 = p2 = ···=0 n=0 (counity); and ∞ ∞ ∞ i j n x x pij = x pij i=0 j=0 n=0 i+j=n for any pij ∈ P, i, j =0, 1,... (contraassociativity). In the latter equation, the interior summation sign in the right-hand side denotes the conventional finite sum 230 A. Contramodules over Coalgebras over Fields of elements of a vector space, while the three other summation signs refer to the contramodule infinite summation operation. The following examples of C-contramodules are revealing. Let E denote the free C-contramodule generated by the sequence of symbols e0, e1, . ; its elements ∞ can be represented as formal sums i=0 ai(x)ei,whereai(x) are formal power series in x such that ai(x) → 0 in the topology of k[[x]] as i →∞.LetF denote the free C-contramodule generated by the sequence of symbols f1, f2, ...; then C-contramodule homomorphisms from F to E correspond bijectively to sequences of elements of E that are images of the elements fi. We are interested in the map −→ i − ∞ g : F E sending fi to x ei e0; in other words, an element i=1 bi(x)fi of F ∞ i − ∞ is mapped to the element i=1 x bi(x)ei i=1 bi(x) e0. It is clear from this formula that the element e0 ∈ E does not belong to the image of g.LetP denote the cokernel of the morphism g and pi denote the images of the elements ei in P. n Then one has p0 = x pn in P; in other words, the element p0 belongs to the image of Homk(C/U, P) under the contraaction map Homk(C, P) −→ P for any finite-dimensional subcoalgebra U =(k[[x]]/xn)∗ of C. Now let E be the free C-contramodule generated by the symbols e1, e2, ..., −→ P denote the cokernel of the map g : F E sending fi to xiei,andpi denote the images of ei in P . Then the result of the contramodule infinite summation ∞ n n n=1 x pn is nonzero in P , even though every element x pn is equal to zero. Therefore, the contramodule summation operation cannot be understood as any kind of limit of finite partial sums. Actually, the C-contramodule P is just the n direct sum of the contramodules k[[x]]/x k[[ x]] over n = 1, 2, . in the category C ∞ n of -contramodules. Notice that the element n=1 x pn also belongs to the image of Homk(C/U, P )inP for any finite-dimensional subcoalgebra U ⊂ C. Remark. In the above notation, a C-contramodule structure on a k-vector space P is uniquely determined by the underlying structure of a module over the algebra of polynomials k[x]; the natural functor C-contra −→ k[x]-mod is fully faithful. ∞ n ∈ Indeed, for any p0, p1, ... in P the sequence qm = n=0 x pm+n P, m =0, 1, . is the unique solution of the system of equations qm = pm + xqm+1.The image of this functor is a full abelian subcategory closed under kernels, coker- nels, extensions, and infinite products; it consists of all k[x]-modules P such that i −1 D Extk[x](k[x, x ],P)=0fori = 0, 1. It follows that if is a coalgebra for which the dual algebra D∗ is isomorphic to a quotient algebra of the algebra of formal power series k[[x1,...,xm]] in a finite number of (commuting) variables by a closed ideal, then the natural functor D-contra −→ k[x1,...,xm]-mod is fully faithful. A.1.2 Now let us give an example of finite-dimensional (namely, two-dimen- sional) contramodule P over a coalgebra C such that the intersection of the images of Homk(C/U, P)inP is nonzero. Notice that for any coalgebra C there are natu- ral left C∗-module structures on any left C-comodule and any left C-contramodule; that is there are natural faithful functors C-comod −−→ C∗-mod and C-contra −−→ C∗-mod A.1. Counterexamples 231 (where C∗ is considered as an abstract algebra without any topology). The functor C-comod −→ C∗-mod is fully faithful, while the functor C-contra −→ C∗-mod is fully faithful on finite-dimensional contramodules. Let V be a vector space and C be the coalgebra such that the dual algebra ∗ ∗ C has the form ki2 ⊕ i2V i1 ⊕ ki1,wherei1 and i2 are idempotent elements with ∗ i1i2 = i2i1 =0andi1 + i2 =1.ThenleftC -modules are essentially pairs of k-vector spaces M1, M2 endowed with a map ∗ V ⊗k M1 −−→ M2, left C-comodules are pairs of vector spaces M1, M2 endowed with a map M1 −−→ V ⊗k M2, and left C-contramodules are pairs of vector spaces P1, P2 endowed with a map Homk(V,P1) −−→ P2. In particular, the functor C-contra −→ C∗-mod is not surjective on morphisms of infinite-dimensional objects, while the functor C-comod −→ C∗-mod is not surjec- tive on the isomorphism classes of finite-dimensional objects. (Neither is in general the functor C-contra −→ C∗-mod, as one can see in the example of an analogous coalgebra with three idempotent linear functions instead of two and three vector spaces instead of one; when k is a finite field and C is the countable direct sum of copies of the coalgebra k, there even exists a one-dimensional C∗-module which comes from no C-comodule or C-contramodule.) Let P be the C-contramodule with P1 = k = P2 corresponding to a linear function V ∗ −→ k coming from no element of V . Then the intersection of the images of Homk(C/U, P)inP over all finite-dimensional subcoalgebras U ⊂ C is equal to P2. More generally, for any coalgebra C any finite-dimensional left C-comodule M has a natural left C-contramodule structure given by the composition ∗ ∗ Homk(C, M) C ⊗k M −−→ C ⊗k C ⊗k M −−→ M of the map induced by the C-coaction in M and the map induced by the pairing ∗ C ⊗k C −→ k. The category of finite-dimensional left C-comodules is isomorphic to a full subcategory of the category of finite-dimensional left C-contramodules; a finite-dimensional C-contramodule comes from a C-comodule if and only if it comes from a contramodule over a finite-dimensional subcoalgebra of C.Wewillseebelow that every irreducible C-contramodule is a finite-dimensional contramodule over a finite-dimensional subcoalgebra of C; it follows that the above functor provides a bijective correspondence between irreducible left C-comodules and irreducible left C-contramodules. 232 A. Contramodules over Coalgebras over Fields Comparing the cobar complex for comodules with the bar complex for con- tramodules, one discovers that for any finite-dimensional left C-comodules L and M there is a natural isomorphism C,i i ∗∗ Ext (L, M) ExtC(L, M) . In other words, the Ext spaces between finite-dimensional C-comodules in the cate- gory of arbitrary C-contramodules are the completions of the Ext spaces in the cat- egory of finite-dimensional C-comodules with respect to the profinite-dimensional topology. A.2 Nakayama’s Lemma The exposition below is based on the structure theory of coalgebras, see [82, Chap- ters VIII–IX] or [68, Chapter 5]. A coalgebra is called cosimple if it has no nontrivial proper subcoalgebras. A coalgebra C is called cosemisimple if it is a union of finite-dimensional coalgebras dual to semisimple k-algebras, or equivalently, if the abelian category of (left or right) C-comodules is semisimple. Any cosemisimple coalgebra can be decomposed into an (infinite) direct sum of cosimple coalgebras in a unique way. For any coalgebra C,letCss denote its maximal cosemisimple subcoalgebra; it contains all other cosemisimple subcoalgebras of C.