Basic Definitions from Hartshorne's Algebraic Geometry
Robin Hartshorne's "Algebraic Geometry" is a fundamental text that has shaped modern approaches to algebraic geometry. It builds on the foundations laid by Grothendieck, Serre, and others, providing a comprehensive treatment of schemes and their applications. This page introduces some of the essential definitions that form the foundation of this deep mathematical discipline.
Affine Varieties
Definition: Let k be an algebraically closed field. An affine algebraic set is the set V k of common zeros of a collection of polynomials f,...,f_m in k[x,...,x]. An affine variety is an irreducible affine algebraic set.
Affine varieties serve as the local building blocks for more general algebraic varieties. They naturally carry the Zariski topology, where closed sets are exactly the affine algebraic subsets. The correspondence between affine algebraic sets over an algebraically closed field and radical ideals (the Nullstellensatz) establishes the duality between geometry and algebra fundamental to algebraic geometry.
The coordinate ring A(V) = k[x,...,x]/I(V) of an affine variety V consists of regular functions on V. This ring is an integral domain because V is irreducible. The dimension of V is the Krull dimension of A(V).
Example: The parabola y = x in k is an affine variety defined as the zero locus of the polynomial f(x,y) = y - x. Its coordinate ring A(V) k[x] under the mapping x x and y x, showing that V is a one-dimensional variety.
Projective Varieties
Definition: The n-dimensional projective space over an algebraically closed field k is the set of all 1-dimensional subspaces of k. A projective algebraic set is the set of common zeros in of a collection of homogeneous polynomials in k[x,...,x]. A projective variety is an irreducible projective algebraic set.
Projective varieties resolve several limitations of affine varieties, particularly the lack of "compactness" and the inability to treat "points at infinity." They are crucial for a global study of algebraic geometry. Like affine varieties, projective varieties carry the Zariski topology, which in this case is Noetherian and compact.
Every projective variety X can be covered by a finite number of affine varietiesthe standard affine opens D+(f_i) = {[x] : f_i(x) 0}. This demonstrates the close relationship between affine and projective geometry.
Example: The projective line consists of points [x:x] where (x,x) (0,0) and [x:x] = [x:x] for all nonzero k. It can be covered by two affine lines: D+(x) Spec(k[x/x]) and D+(x) Spec(k[x/x]).
Schemes
Definition: A locally ringed space (X, O_X) consists of a topological space X together with a sheaf of rings O_X such that for each point x X, the stalk O_{X,x} is a local ring. For any ring A, Spec A denotes the set of prime ideals of A with the Zariski topology and the structure sheaf O_{Spec A}. An affine scheme is a locally ringed space isomorphic to Spec A for some ring A. A scheme is a locally ringed space (X, O_X) such that X can be covered by open subsets U_i where (U_i, O_X|U_i) Spec A_i for some rings A_i.
Schemes, introduced by Grothendieck, provide a vast generalization of varieties. They allow working over arbitrary rings (not just algebraically closed fields), incorporate nilpotent elements (providing infinitesimal information), and unify the treatment of affine and projective cases. The category of schemes has excellent categorical properties, including existence of fibered products, which are crucial for developing relative notions.
Theorem: For any ring A, there is a naturally induced equivalence of categories between affine schemes Spec A and the opposite category of commutative rings. This is why morphisms of schemes are often described by ring homomorphisms in the opposite direction.
Example: Spec contains all prime ideals (p) for prime numbers p, the zero ideal (0), and the ideal (0). The generic point (0) is dense in the whole space, while each (p) is a closed point. The fiber over (p) in the morphism Spec [x] Spec is isomorphic to Spec _p[x].
Morphisms
Definition: A morphism of schemes f: X Y consists of a continuous map f: X Y of topological spaces together with a morphism of sheaves f#: O_Y f_*O_X on Y. Equivalently, for each open V Y, we have a ring homomorphism f#(V): O_Y(V) O_X(f^{-1}(V)) compatible with restriction maps.
Morphisms preserve the local ring structure: for each x X, if y = f(x), then f# induces a local homomorphism O_{Y,y} O_{X,x}. In the affine case, a morphism Spec B Spec A corresponds precisely to a ring homomorphism A B, demonstrating the contravariance between rings and schemes.
Particularly important classes of morphisms include:
- Closed immersions: morphisms f: X Y that are homeomorphisms onto a closed subset of Y, with a surjective sheaf morphism O_Y f_*O_X.
- Finite morphisms: morphisms that are affine and have finite coordinate rings.
- Proper morphisms: the scheme-theoretic analogue of compact maps in topology.
Example: The projection onto the first factor is a proper morphism. This projection exemplifies how families of schemes (in this case, an for each point of the base ) can be treated uniformly in the scheme framework.
Sheaves
Definition: A presheaf F on a topological space X is a contravariant functor from the category of open subsets of X (with inclusions as morphisms) to the category of sets (or abelian groups, rings, etc.). A sheaf is a presheaf satisfying: (1) (locality) if s and t are sections over U and s|V_i = t|V_i for an open cover {V_i} of U, then s = t; (2) (gluing) if {s_i} are sections over sets V_i forming a cover of U with s_i|V_iV_j = s_j|V_iV_j for all i,j, then there exists a unique section s over U with s|V_i = s_i for each i.
Sheaves provide a systematic way to study local-to-global properties. The structure sheaf O_X of a scheme is a fundamental sheaf, assigning to each open set U the ring of regular functions on U. Particularly important are quasi-coherent sheaves, which locally look like modules over the structure sheaf.
Theorem: On an affine scheme Spec A, there is an equivalence of categories between the category of A-modules and the category of quasi-coherent sheaves on Spec A. This shows how algebra and geometry are deeply intertwined in the theory of schemes.
Example: The sheaf of differentials _X on a smooth variety X encodes infinitesimal information and plays a crucial role in duality theory. On , the sheaf of differentials ^{1}_{} corresponds to the twist O(-2) of the structure sheaf.
Cohomology
Definition: Given a sheaf F on a topological space X, the sheaf cohomology groups H^i(X, F) are the right derived functors of the global section functor (X, -): F (X, F) = F(X).
Cohomology quantifies the obstruction to patching local data into global data. Hartshorne focuses primarily on coherent sheaf cohomology, which has excellent properties on Noetherian schemes. The higher cohomology groups H^i(X, F) for i > 0 measure how far the global section functor is from being exact.
Key properties and results in sheaf cohomology include:
- H^i(X, F) = 0 for all i > 0 and all quasi-coherent sheaves F if and only if X is an affine scheme.
- The Serre duality theorem for smooth proper varieties X of dimension n: H^i(X, F) H^{n-i}(X, F _X)*, where _X is the canonical sheaf.
- Riemann-Roch theorems relating cohomology dimensions to geometric invariants.
Example: For the projective line and the structure sheaf O(k) twisted by an integer k, we have H^0(, O(k)) = {polynomials of degree k} if k 0 and = 0 if k < 0. Likewise, H^1(, O(k)) = 0 if k -2, but H^1(, O(k)) k if k -3, providing a concrete calculation of sheaf cohomology.
Divisors
Definition: A Weil divisor on a normal Noetherian scheme X is a formal finite linear combination D = n_i Y_i where Y_i are prime divisors (integral closed subschemes of codimension 1) and n_i . A Cartier divisor is a global section of the sheaf K^*/O_X^* where K^* is the sheaf of total quotient rings. For locally factorial Noetherian schemes, the groups of Weil and Cartier divisors are naturally isomorphic.
Divisors encode linear subvarieties of codimension 1 and provide a framework for studying maps to projective spaces. The divisor class group Cl(X) = Div(X)/Prin(X) measures the extent to which divisors fail to be principal (where D is principal if D = div(f) for some f in the function field). For a smooth variety, the divisor class group is isomorphic to the Picard group, the group of isomorphism classes of line bundles.
Theorem: On a complete variety X over an algebraically closed field, a divisor D defines a complete linear system |D| whose dimension is dim H^0(X, O_X(D)) - 1. This linear system gives rise to a rational map to a projective space (|D|): X (H^0(X, O_X(D))*).
Example: On a smooth projective curve C, divisors are formal sums of points. The degree of a divisor D = n_i P_i is deg(D) = n_i. A divisor is principal if and only if its degree is 0, illustrating the deep connection between algebraic and topological properties.
Conclusion
These definitions represent only the foundation of Hartshorne's comprehensive treatment of algebraic geometry. The depth and elegance of the subject unfolds as one explores the interconnected theories of schemes, cohomology, and divisors, leading eventually to sophisticated applications across mathematics. The power of algebraic geometry lies in its ability to translate geometric questions into algebraic ones, solve them using powerful algebraic techniques, and interpret the results back in geometric terms.
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