Graded Rings of Theta Functions

Abstract

We shall collect several lemmas on graded rings, some of which will become necessary only in later chapters. If S is a commutative ring (with the identity) containing a sequence of additive subgroups S0, S1, … such that S is their direct sum and Sp Sq ⊂ Sp+q for every p, q, we say that S is a graded ring over S0; we observe that S0 is a subring of S and Sk an S0-module. If M is an S-module containing a sequence of additive subgroups M0, M1, … such that M is their direct sum and Sp Mq ⊂ Mp+q for every p, q, we say that M is a graded S-module with Mk as its homogeneous component of degree k. An element of Mk is called a homogeneous element of M of degree k; every element x of M can be written uniquely as x0 + x1 + … with xk in Mk, in which xk=0 for almost all k; we call xk the homogeneous part of x of degree k. A graded S-module N contained in M is called a graded submodule if it is a submodule and Nk = N ∩ Mk for k = 0, 1, …. We observe that S itself is a graded S-module. If K isa commutative ring and K [x] = K [x1, …, xn] is the ring of polynomials in n letters x1, …, xn with coefficients in K, K[x] can be considered as a graded ring over K with K[x]1= K x1 + … + K xn; a graded submodule of K[x] is called a homogeneous ideal of K [x]. A graded ring R contained in S is called a graded subring if it is a subring and a graded submodule. For any positive integer d, we shall denote by S(d) the subring of S generated by Skd; S(d) is the direct sum of S0, Sd, … and Spd Sqd ⊂ S(p + q)d for every p, q. Therefore S(d) can be considered as a graded ring over S0 with Skd as its homogeneous component of degree k.