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 S_p S_{q ⊂ 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 S_k 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 S_p M_q ⊂ M_p+q for every p, q, we say that M is a graded S-module with M_k as its homogeneous component of degree k. An element of M_k is called a homogeneous element of M of degree k; every element x of M can be written uniquely as x₀ + x₁ + … with x_k in M_k, in which x_k=0 for almost all k; we call x_k 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 N_k = N ∩ M_k for k = 0, 1, …. We observe that S itself is a graded S-module. If K isa commutative ring and K [x] = K [x₁, …, x_n] is the ring of polynomials in n letters x₁, …, x_n with coefficients in K, K[x] can be considered as a graded ring over K with K[x]₁= K x₁ + … + K x_n; 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 S_kd; S^(d) is the direct sum of S₀, S_d, … and S_pd S_qd ⊂ S_{(p + q)d} for every p, q. Therefore S^(d) can be considered as a graded ring over S₀ with S_kd as its homogeneous component of degree k.