  We introduce the alternating Schur algebra $AS_F(n,d)$ as the commutant of
the action of the alternating group $A_d$ on the $d$-fold tensor power of an
$n$-dimensional $F$-vector space. When $F$ has characteristic different from
$2$, we give a basis of $AS_F(n,d)$ in terms of bipartite graphs, and a
graphical interpretation of the structure constants. We introduce the abstract
Koszul duality functor on modules for the even part of any $\mathbf Z/2\mathbf
Z$-graded algebra. The algebra $AS_F(n,d)$ is $\mathbf Z/2\mathbf Z$-graded,
having the classical Schur algebra $S_F(n,d)$ as its even part. This leads to
an approach to Koszul duality for $S_F(n,d)$-modules that is amenable to
combinatorial methods. We characterize the category of $AS_F(n,d)$-modules in
terms of $S_F(n,d)$-modules and their Koszul duals. We use the graphical basis
of $AS_F(n,d)$ to study the dependence of the behavior of derived Koszul
duality on $n$ and $d$.
