A Sheaf of Modules is a Geometric Generalization of a Module over a Ring – A Case Derivative of Abelian Closure

A coherent sheaf is a generalization of, on the one hand, a module over a ring, and on the other hand, a vector bundle over a manifold. Indeed, the category of coherent sheaves is the “abelian closure” of the category of vector bundles on a variety.

Given a field which we always take to be the field of complex numbers C, an affine algebraic variety X is the vanishing locus

X = 􏰐(x₁,…, x_n) : f_i(x₁,…, x_n) = 0􏰑 ⊂ Aⁿ

of a set of polynomials f_i(x₁,…, x_n) in affine space Aⁿ with coordinates x₁,…, x_n. Associated to an affine variety is the ring A = C[X] of its regular functions, which is simply the ring C[x₁,…, x_n] modulo the ideal ⟨f_i⟩ of the defining polynomials. Closed subvarieties Z of X are defined by the vanishing of further polynomials and open subvarieties U = X \ Z are the complements of closed ones; this defines the Zariski topology on X. The Zariski topology is not to be confused with the complex topology, which comes from the classical (Euclidean) topology of Cⁿ defined using complex balls; every Zariski open set is also open in the complex topology, but the converse is very far from being true. For example, the complex topology of A¹ is simply that of C, whereas in the Zariski topology, the only closed sets are A¹ itself and finite point sets.

Projective varieties X ⊂ Pⁿ are defined similarly. Projective space Pⁿ is the set of lines in Aⁿ⁺¹ through the origin; an explicit coordinatization is by (n + 1)-tuples

(x₀,…, x_n) ∈ Cⁿ⁺¹ \ {0,…,0}

identified under the equivalence relation

(x₀,…, x_n) ∼ (λx₀,…, λx_n) for λ ∈ C^∗

Projective space can be decomposed into a union of (n + 1) affine pieces (Aⁿ)_i = 􏰐[x₀,…, x_n] : x_i ≠ 0􏰑 with n affine coordinates y_j = x_j/x_i. A projective variety X is the locus of common zeros of a set {f_i(x₁,…, x_n)} of homogeneous polynomials. The Zariski topology is again defined by choosing for closed sets the loci of vanishing of further homogeneous polynomials in the coordinates {x_i}. The variety X is covered by the standard open sets X_i = X ∩ (Aⁿ)_i ⊂ X, which are themselves affine varieties. A variety􏰭 X is understood as a topological space with a finite open covering X = ∪_i U_i, where every open piece U_i ⊂ Aⁿ is an affine variety with ring of global functions A_i = C[U_i]; further, the pieces U_i are glued together by regular functions defined on open subsets. The topology on X is still referred to as the Zariski topology. X also carries the complex topology, which again has many more open sets.

Given affine varieties X ⊂ Aⁿ, Y ⊂ A^m, a morphism f : X → Y is given by an m-tuple of polynomials {f₁(x₁, . . . , x_n), . . . , f_m(x₁, . . . , x_n)} satisfying the defining relations of Y. Morphisms on projective varieties are defined similarly, using homogeneous polynomials of the same degree. Morphisms on general varieties are defined as morphisms on their affine pieces, which glue together in a compatible way.

If X is a variety, points P ∈ X are either singular or nonsingular. This is a local notion, and hence, it suffices to define a nonsingular point on an affine piece U_i ⊂ Aⁿ. A point P ∈ U_i is nonsingular if, locally in the complex topology, a neighbourhood of P ∈ U_i is a complex submanifold of Cⁿ.

The motivating example of a coherent sheaf of modules on an algebraic variety X is the structure sheaf or sheaf of regular functions O_X. This is a gadget with the following properties:

1. On every open set U ⊂ X, we are given an abelian group (or even a commutative ring) denoted O_X(U), also written Γ(U, O_X), the ring of regular functions on U.
2. Restriction: if V ⊂ U is an open subset, a restriction map res_UV : O_X(U) → O_X(V) is defined, which simply associates to every regular function f defined over U, the restriction of this function to V. If W ⊂ V ⊂ U are open sets, then the restriction maps clearly satisfy res_UW = res_VW ◦ res_UV.
3. Sheaf Property: suppose that an open subset U ⊂ X is covered by a collection of open subsets {U_i}, and suppose that a set of regular functions f_i ∈ O_X(U_i) is given such that whenever U_i and U_j intersect, then the restrictions of f_i and f_j to U_i ∩ U_j agree. Then there is a unique function f ∈ O_X(U) whose restriction to U_i is f_i.

In other words, the sheaf of regular functions consists of the collection of regular functions on open sets, together with the obvious restriction maps for open subsets; moreover, this data satisfies the Sheaf Property, which says that local functions, agreeing on overlaps, glue in a unique way to a global function on U.

A sheaf F on the algebraic variety X is a gadget satisfying the same formal properties; namely, it is defined by a collection {F(U)} of abelian groups on open sets, called sections of F over U, together with a compatible system of restriction maps on sections res_UV : F(U) → F(V) for V ⊂ U, so that the Sheaf Property is satisfied: sections are locally defined just as regular functions are. But, what of sheaves of O_X-modules? The extra requirement is that the sections F(U) over an open set U form a module over the ring of regular functions O_X(U), and all restriction maps are compatible with the module structures. In other words, we multiply local sections by local functions, so that multiplication respects restriction. A sheaf of O_X-modules is defined by the data of an A-module for every open subset U ⊂ X with ring of functions A = O_X(U), so that these modules are glued together compatibly with the way the open sets glue. Hence, a sheaf of modules is indeed a geometric generalization of a module over a ring.

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