From Canonical Forms to Locally Coherent Sheaves — University of Kyiv “Taras Shevchenko”
This brief work demonstrates some properties of modules and module morphisms from a homological perspective. This perspective is more general than the classification of modules via linear maps and invariant factors that gives rise to various canonical matrix forms. The invariant factors and elementary divisors of modules are derived from the decomposition of a linear transformation between vector spaces over a field. Given a linear transformation T: V → W between vector spaces V and W over a field, the invariant factors of T are a sequence of non-zero polynomials f₁(x), f₂(x), …, fₖ(x) such that f₁(x) divides f₂(x), f₂(x) divides f₃(x), and so on; the ideal generated by f₁(x) is the annihilator of T, (i.e. set of all polynomials g(x) such that g(T) = 0); and the product of the invariant factors is the minimal polynomial of T, denoted as m(x). Elementary divisors have the same properties, except they are powers of irreducible polynomials over the base field. These divisibility conditions are well-defined and satisfiable because the polynomials over a field must be a Principal Ideal Domain (PID) and, even more strongly, a Euclidean Domain.
Certain properties of algebraic varieties can be described via modules, but only when the theory of sheaves is introduced. Affine varieties are subsets of affine space defined as the common zeros of a set of polynomials. A projective variety is a subset of projective space that is defined as the common zeros of a set of homogeneous polynomials over a field F. Most authors take projective space to be an extension of the usual Euclidean space which adds a point ‘at infinity’ providing a compactification in which linear subspaces that are parallel are said to intersect at infinity. Projective and affine varieties are used in module theory in the construction and classification of sheaves. Indeed, if R is the coordinate ring of an algebraic variety (either projective or affine), then modules over R can be thought of as ‘vector bundles’ or ‘coherent sheaves’ over the variety. For example, the Hilbert polynomial of a graded module over a graded ring corresponds to the dimension of the space of global sections of the associated coherent sheave at each degree.
One can appreciate how sheaves are structures in their own right, independently of module representation on varieties. More generally, sheaves are topological tools that encode local information in a manner compatible with restrictions to smaller open sets and give rise to homological notions. A sheaf F on a topological space X assigns to each open set U in X a set or group (or even a ring) F(U), such that these satisfy certain “continuous” compatibility conditions when we restrict to smaller open sets. A coherent sheaf on a scheme or algebraic variety is itself an O-module (where O is the structure sheaf of the scheme or variety). These sorts of sheaves are the natural generalization in algebraic geometry of vector bundles and the concept of a locally free sheaf of finite rank on a compact complex manifold in complex geometry. While all vector bundles give rise to coherent sheaves (specifically, locally free sheaves), not all coherent sheaves arise from vector bundles. The concept of a coherent sheaf is broader and allows one to incorporate algebraic and geometric singularity. Informally, a vector bundle is a topological space that looks locally like a product of a topological vector space and an open set in the base space. In the language of sheaves, a vector bundle corresponds to a locally free sheaf of finite rank. This means that around every point, there is an open neighborhood where the sheaf looks like a direct sum of copies of the structure sheaf. Coherent sheaves merely generalize the concepts that pertain to vector bundles. While a vector bundle (or a locally free sheaf) looks locally like a direct sum of copies of the structure sheaf, a coherent sheaf only has to look like a quotient of a locally free sheaf. For example, the structure sheaf of a subvariety Y of X corresponds to a coherent sheaf on X, even though it may not be locally free if Y is not a divisor.
Fully categorical definitions of modules are often preferable to classical definitions in the context of homological algebra. A module P is said to be projective if, given any module homomorphism f : P → M and any epimorphism g : N → M, there exists a module homomorphism h : P → N making the diagram commute. In other words, any module homomorphism from a projective module can be lifted along onto maps. An equivalent definition states that a module P is projective if and only if every short exact sequence of the form 0 → A → B → P → 0 splits. That is, there exists a module homomorphism r : B → A such that the composition A → B → A is the identity on A . Similarly, a module Q is said to be injective if, given any module homomorphism f : N → Q and any endomorphism g : N → M, there exists a homomorphism h : M → Q making the diagram commute. In other words, any module homomorphism into an injective module can be extended along an injection. Equivalently (and unsurprisingly), a module Q is injective if and only if every short exact sequence of the form 0 → Q → A → B → 0 splits. That is, there exists a module homomorphism s : A → B such that the composition A → B → A is the identity on A. If M is a module over a ring R and N is a submodule of M, then the quotient module M/N is defined as the set of cosets of N in M, with addition and scalar multiplication defined in the natural way. The quotient module M/N is itself a module over R. Quotient modules often arise in the context of exact sequences. If I is an ideal of a ring R, then the factor ring R/I is a ring, and it acts on the quotient module M/IM in a natural way. This allows us to reduce questions about module M to questions about the smaller module M/IM.
From a categorical perspective, every module can be presented as a quotient of a free module by a submodule. If M is a module over a ring R, then we can choose a set of generators for M, which gives us a surjective homomorphism from a free module F to M. The kernel of this homomorphism is a submodule of F, and M is isomorphic to F/Ker(f). Moreover, the quotients are always well-defined: The first isomorphism theorem states that if f: M -> N is a module homomorphism, then Ker(f) is a submodule of M and Im(f) is isomorphic to M/Ker(f).
Free modules are implicit in the homological characterization of modules given in the previous paragraph. To understand this notion abstractly, one considers the generalization of Euclidean products to tensor products of modules M and N over an arbitrary ring R. This product, denoted as M ⊗R N (or simply M ⊗ N when the ring is clear from the context), is defined in the following manner. Begin with the free module generated by the elements of M × N. This free module is the set of all formal linear combinations of pairs (m, n), where m ∈ M and n ∈ N, with coefficients from R. We desire to impose bilinearity on the tensor product so that it is a satisfactory generalization of the Cartesian product. Impose the relations:
a) (m + m’, n) = (m, n) + (m’, n) for all m, m’ ∈ M and n ∈ N.
b) (m, n + n’) = (m, n) + (m, n’) for all m ∈ M and n, n’ ∈ N.
c) (rm, n) = r(m, n) for all r ∈ R, m ∈ M, and n ∈ N.
The tensor product M ⊗R N is the quotient module obtained by taking the quotient of the free module by the submodule generated by all elements of the form (m + m’, n) — (m, n) — (m’, n), (m, n + n’) — (m, n) — (m, n’), and (rm, n) — r(m, n), where m, m’ ∈ M, n, n’ ∈ N, and r ∈ R. In homological terms, the tensor product satisfies the universal property that for any module L and any bilinear map f: M × N → L, there exists a unique module homomorphism g: M ⊗R N → L such that g(m ⊗ n) = f(m, n) for all m ∈ M and n ∈ N. This property is a complete characterization up to isomorphism.
Let us finally consider some applications of module theory to the definition of locally coherent sheaves. The derived category of a ring is a category that encodes the homological properties of modules over that ring. It consists of complexes of modules with morphisms given by chain maps up to homotopy, and its structure reflects the Ext groups between modules. The “Ext” and “Tor” functors measure the extent to which exact sequences fail to remain exact under the Hom and tensor product operations, respectively. These functors are defined using resolutions: For example, to compute Ext^i(M, N) for modules M and N, we take a “projective resolution” of M, apply Hom(-, N) to it, and take the ith homology of the resulting complex. A chain complex that ‘approximates’ a given module. By projective and injective resolutions, we mean sequences of projective or injective modules and morphisms between them. The length of the shortest projective (or injective) resolution of a module is called its projective (or injective) dimension. A locally coherent sheaf is a sheaf of modules on a scheme that is coherent when restricted to open sets. More formally, assign to each open subset U of a topological space (a scheme, perhaps) a module of some type. A module sheaf assigns a module to each open subset of the scheme in a compatible manner. A sheaf F of modules on a scheme is said to be coherent if it satisfies two conditions: It must be locally finitely generated, by which we mean that for every open subset U in the scheme, there exists a covering {Ui} of U such that for each Ui, the restriction of F to Ui, denoted as F|Ui, is a finitely generated module. It must furthermore exhibit locally zeroth cohomology in the sense that for every open subset U in the scheme, the cohomology groups H^i(U, F) vanish for i > 0, where H^i(U, F) denotes the i-th cohomology group of F over U. A sheaf of modules F on a scheme is called locally coherent if for every point p in the scheme, there exists an open neighborhood U of p such that the restriction of F to U, denoted as F|U, is a coherent sheaf.
