Let A be a finite-dimensional k-algebra,where k is a fixed field. All modules of A are finitely generated left modules. Suppose X is an A-module. We denote by add(X) the full subcategory of A-modules consisting of all direct summands of direct sum of finitely many copies of X. $D$ is the usual k-duality $Hom\_k(-,k)$, $\nu \_A$ is the Nakayama functor $DHom\_A(-,\_{A}A)$


Recall that the module X is called a generator over A if $add(\_{A}A) \subseteq add(X)$, a cogenerator if $add(D(A\_A)) \subseteq add(X)$, and a generator-cogenerator if it is both a generator and a cogenerator over A.


Can anyone tell me how to get the following results:


1):Let V be a generator over A with $B := End\_A(V)$.Then $Hom\_A(V,I)$ is an injective B-module for every injective A-module I;


2): If V is a generator-cogenerator, then each projective-injective B-module is precisely of the form $Hom\_A(V,I)$ with I an injective A-module.



The projective and injective modules of $End_A(V)$?
to 1): V being a generator of mod-A implies that V is projective in mod-B. Now $Hom(V,D(A)) \cong Hom(A,D(V)) \cong D(V)$ is injective. Now use that every indecomposable injective I is a summand of D(A). A general injective module is a direct sum of indecomposables and thus the result follows.


2) This can be seen as a special case of lemma 3.1.(3) in Auslander, M.; Platzeck, M. I.; Todorov, G. Homological theory of idempotent ideals. Trans. Amer. Math. Soc. 332 (1992), no. 2, 667–692.


