abelian variety used like module

Question

I am reading Hida's "p-adic automorphic forms on Shimura varieties", p99, toward the end of the page, $A$ is an abelian variety, and $O$ an endormorphism ring, $O \to End(A)$, $c^{-1}$ is an ideal, he uses a sequence $0 \to O \to c \to c/O \to 0$ and tensor with $A$ to get $0 \to Tor_1(c/O,A) \to A \to A \otimes_O c \to 0$, he refer to his book LFE 1.1, but the referenced section deals with modules, i understand the sequence if $A$ is module, but what is the meaning of the object $A \otimes_O c$ and $Tor_1(c/o,A)$ when $A$ is abelian variety? is there any reference?

Answer 1

For a base ring $B$, the category of schemes over $B$ can be identified as a covariant functor from the category $\textbf{B-Alg}$ of $B$-algebras into the category $\textbf{Sets}$ of sets. In other words, the category of schemes over $B$ is isomorphic to the full subcategory of covariant functors from $\textbf{B-Alg}$ nto $\textbf{Sets}$, Grothendieck's viewpoint; see Section 4 of notes from a graduate course of Hida here (or Section 4.1 of another Springer book of his published in 2013 here).

Hida, Haruzo. Elliptic curves and arithmetic invariants. Springer Monographs in Mathematics. Springer, New York, 2013. xviii+449 pp.

For example, this goes as follows, if $B$ is a field and $X$ is a variety over $B$, for any $B$-algebras $K$, one associates the set of $K$-rational points $X(K)$ of $X$; so, we get a covariant functor $K \mapsto X(K)$. If $A$ is an abelian variety whose endomorphism algebra containing $O$, $A(K)$ is an abelian group and actually an $O$-module (so, the functor has values in the category of $O$-modules). Thus if $M$ is an $O$-module, one can think of a new covariant functor $K \mapsto A(K) \otimes_O M$. If $M$ is finite type over $O$ as modules and $O$-flat, long ago, Serre showed that there exists an abelian variety $A'$ such that $A'(K)$ is isomorphic to $A(K) \otimes_O M$ for all $K$. This abelian variety $A'$; is the one written $A \otimes_O M$. Similarly $\text{Tor}_1^O(A, \mathfrak{c}/O)$ is a functor from $B$-algebras into finite abelian group sending $K$ to $\text{Tor}_1^O(A(K), \mathfrak{c}/O)$ which is again represented by a finite flat group subscheme of $A$. Thus $A \otimes_O \mathfrak{c}$ is the quotient of $A$ by a finite torsion subgroup. If $B$ is a field of characteristic $0$, as is well known (from the time of Weil) the quotient of an abelian variety over $B$ by a torsion subgroup is again an abelian variety (proving classically Serre's result in this special case).