Z Q R C H P vp setlengthheadheight0bp setlengthheadsep0bp setlengthtopmargin36bp setlengthtextheight704bp Univ at Albany: Math: W. F. Hammond: Courses: Math 825 Spring Semester, 2006 1 Note: If you found this document through a web search engine, you may not be aware of its href="http:math.albany.edumathpershammondcoursemat825s2006"other presentation formats

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Fri.,May.5: A 1949 paper by Andre Weil gave evidence for the existence of topological cohomology in algebraic geometry linked to the notion of zeta function for a nonsingular projective algebraic variety defined over a finite field Let X be a scheme of finite type over Z For each element x X the residue field at x is the fraction field of an algebra of finite type over Z Thus, the residue field at a closed element x is a field that is an algebra of finite type over Z, i.e., a finite field One defines the zeta function of X by _X(s) _{x closed in X} 11 N(x)^s where N(x) (x) is the number of elements of the residue field of X at x (Ignore questions of convergence for now.) When X Spec Z, _X(s) is Riemanns zeta function When X is a scheme of finite type over F_q, each residue field at a closed element is a finite extension field of F_q, and, therefore, N(x) q^d(x) where d(x) is the extension degree With t q^s one writes _X(s) Z_X(t) _{x closed} 11 t^d(x) With the condition Z_X(0) 1 the Z form of the zeta function is determined by its logarithmic derivative ddt log Z_X(t) _{x closed} d(x) t^d(x)11 t^d(x) 1t _{r 1 x closedd(x) r} r t^r1 t^r 1t _{r 1} r c_r t^r1 t^r 1t _{r 1} r c_{r s 1} t^rs 1t _{1 r divides} r c_r t ₁ N t¹ where c_r denotes the number of closed elements in X with d(x) r and N denotes the number of iref="fieldpoint"points of X with values in the degree extension of F_q For a beginning example, when X Aⁿ, one has N qⁿ, and, therefore, Z_Aⁿ(t) 11qⁿt Of course, Aⁿ is not a projective variety for n 1 When F is a field, the set of Fvalued points of Pⁿ is the disjoint union of A⁰(F), A¹(F), , Aⁿ(F) Therefore, Dlog Z_Pⁿ(t) (over F_q) is the sum of Dlog Z_A^j(t) for 0 j n Hence, Z_Pⁿ(t) 1(1 t) (1 q t) (1 qⁿ t) For P¹ P¹, one has N (1 q)², and, therefore Z_{(P¹ P¹)}(t) 1(1 t)(1 q t)²(1 q² t) For curves of genus 1 defined over finite fields, the shape of its Z function was established before the time of Weils conjectures For example, in the case of the curve E given by the Weierstrass equation y² x³ 2 x over the field F₅, simply by counting points to see that E(F₅) 10, it is a consequence of the theoretical framework that Z(t) is the rational function Z_E(t) 1 4 t 5t²(1 t) (1 5t) For each of these last examples Pⁿ, P¹ P¹, and E one may observe that Z_X(t), relative to the field F_q is a rational function in one variable and that: the denominator is the product of polynomials whose degrees are the classical topological Betti numbers of the base extension X_C of X for even dimensions the numerator is the product of polynomials whose degrees are the classical topological Betti numbers of the base extension X_C of X for odd dimensions the polynomial factor corresponding to classical cohomology in dimension j has the form of the characteristic polynomial of a linear endomorphism of the form det(1 t) with complex reciprocal roots all of absolute value q^j2 Wed.,May.3: Beyond the theory of curves of genus 1 a good bit of what is involved in the study of curves and of complete nonsingular varieties in general is studying the group Div(X)Div(X) For curves one has Div(X) Div₀(X) Div(X) where the quotient for the second step is the discrete group Z when Div₀(X) is defined as the group of divisors of degree 0 It turns out that the quotient for the first step is a complete irreducible group variety of dimension g, and, thus, one cannot study curves in depth without studying varieties of higher dimension For varieties of dimension greater than 1, defining the degree of a divisor as the sum of its coefficients will not lead in the right direction One would like a definition of Div₀(X) such that the first step is a complete irreducible variety and the second step a finitelygenerated abelian group, but there is no hope with these two conditions that the second step will always be cyclic since for the case X Pk1 Pk1 one will find that Div(X)Div(X) Z Z For the purpose of gaining insight about Div(X)Div(X) in the theory of curves while at the same time beginning to understand what might be required for defining Div₀(X) when dim(X) 1, consider what is available with transcendental methods when k C Complex exponentiation provides the short exact sequence of abelian sheaves for the classical (locally Euclidean) topology on X: 0 Z O_hol e O_hol 0 where e(f) e^{2 i f} In the long cohomology sequence the 0 stage splits off since H⁰(X, O_hol) C GAGA tells us that coherent module cohomology matches, and although O is certainly not an Omodule, its H¹ in both algebraic and transcendental theories viewed through Czech theory classifies isomorphism classes of invertible coherent modules One has the exact sequence: 0 H¹(X, Z) H¹(X, O_hol) H¹(X, O_hol) H²(X, Z) If dim(X) 1, then H²(X, Z) Z, and one finds that the last map in this sequence, a connecting homomorphism, sends the isomorphism class of an invertible O_holmodule to its degree Therefore, remembering that Div(X)Div(X) H¹(X, O), one has H¹(X, O_hol)H¹(X, Z) Div₀(X)Div(X) and, in fact, the left side is the quotient of a gdimensional vector space over C by a lattice Thus, Div₀(X)Div(X) is a gdimensional complex torus; it is, moreover, a complete group variety over C For dim(X) 1 the kernel of the connecting homomorphism will provide a correct notion of degree 0 For working over an arbitrary algebraically closed field, one sees that something is needed to replace classical cohomology Because constant sheaves are flasque in the Zariski topology, their Zariskibased cohomology cannot be used Mon.,May.1: Continuing with the discussion of the previous hour: If p, q, r are any three points of X(k), then the triple sum p q r, like any point of X(k) is characterized by the linear equivalence class of the associated one point divisor One has the relation of linear equivalence p q r p q r 2 o Therefore, p q r o p q r 3 o p q r div(h) 3 o for some h L(3o) p q r div(s) for some s H⁰(X, O(3o)) p q r div(a x u³ b y u³ c u³), some (a: b: c) Pk2 p q r f¹(D), D div(a X b Y c Z) Div(Pk2) where f : X Pk2 is the projective embedding of X given by the invertible Omodule O(3 o) In other words, taking multiplicities into consideration, three points sum to o in the group law on X(k) if and only if the corresponding points of a Weierstrass model in Pk2, with o corresponding to the point on the line at infinity, are collinear From this description of the group law on X(k), in view of the fact that the third point of a cubic on the line through two given points (tangent if the two points coincide) depends rationally on the coordinates of the given points, it follows that Addition X X X and negation X X are morphisms of varieties over k If F is the field generated over the prime field by the coefficients a₀, , a₆ of the Weierstrass equation, then The Weierstrass equation defines a scheme X_F of finite type over F whose base extension to k is X For each extension E of F the set X_F(E) is a group in a functorial way X_F(k) X(k) Fri.,Apr.28: Continuing with curves of genus 1, we wish to change notation so that the projective embedding of the previous hour is given by the very ample invertible sheaf O(3o), o X(k) This notational change notwithstanding, o is an arbitrary point Under the projective embedding given by O(3o), one has f(o) (0: 0: 1), the unique point of f(X) on the line at infinity We wish to show that there is a unique commutative group law on the set X(k) for which the map : Div(X) X(k) D _{p X(k)} n_p p (D) _{p X(k)} n_p p which is tautologically a group homomorphism, has the property that (D₁) (D₂) whenever D₁ D₂ (linear equivalence), and further the property that o is the zero element in X(k) (This is not the strongest statement of this type that can be made.) Addition in X(k) is defined by observing that since for given p, q X(k) the divisor p q o has degree 1, its complete linear system consists of a single nonnegative divisor of degree 1, i.e., r, and this unique r X(k) is defined to be p q Since p q o r the properties specified for make this definition necessary if, indeed, it defines a group It is straightforward to verify that the addition is associative, that o is its identity, and that p is given by the unique member of the complete linear system 2o p It is obvious that this group law on X(k) is commutative and that is surjective If Div₀(X) denotes the group of divisors of degree 0, then since (D) (D (deg D)o), one sees that the restriction ₀ of to Div₀(X) is a surjective homomorphism Let Div(X) denote the group of divisors linearly equivalent to zero It is trivial that the map D D (deg D)o defines a homomorphism Div(X) Div₀(X) which, when followed with reduction provides a homomorphism Div(X) Div₀(X)Div(X) It is not difficult to verify that another homomorphism between this latter pair of groups is given by D (D) o Div(X) (That this is a homomorphism follows from reviewing the definition of (D₁) (D₂).) Since these two homomorphisms agree on divisors of the form p which generate the free abelian group Div(X) , one has for all D Div(X) that D (deg D)0 (D) o We know that deg D depends only on the linear equivalence class of D as the first consequence of the iref="RRThm"RiemannRoch Theorem Since r X(k) is determined uniquely by the linear equivalence class of r, this formula tells us that (D) depends only on the linear equivalence class of D However, the formula also tells us that the linear equivalence class of D depends only on (D) and deg(D) In particular, one has Div₀(X)Div(X) X(k) Wed.,Apr.26: Suppose that X is a complete nonsingular curve over an algebraically closed field k of genus 1 The range of degrees where a divisor D has H¹(O(D)) (0) is deg(D) 1, while we have dim H¹(O) 1 For each a X(k) the invertible module O(2a) has no base point, and, therefore, defines a morphism to Pk1 One has a two step filtration of the 3dimensional linear subspace L(3a) of k(X): k L(0) L(a) L(2a) L(3a) Choosing x L(2a)L(0) and y L(3a) L(2a) one obtains a filtrationcompatible basis 1, x, y of L(3a), and if u is a rational section of O(a) with div(u) a, the morphism f : X Pk2 given by f (Z: X: Y), Z u³, X x u³, Y y u³ provides a projective embedding of X by the theorem of the last hour Extending the filtration inside k(X) by the L(m a), one sees that 1, x, y, x², xy, x³ is a filtrationcompatible basis of L(6 a) Since y² L(6a) L(5a), one has a linear relation among monomials of degree 3 Y² Z a₁ X Y Z a₃ Y Z² a₀ X³ a₂ X² Z a₄ X Z² a₆ Z³ with a₀ 0 that characterizes f(X) as a nonsingular hypersurface in Pk2 One says that f(X) is in generalized Weierstrass form One regards Z 0 as the line at infinity in Pk2, while one calls affine a point (X,Y) (1: X: Y) The intersection of f(X) with the line at infinity reduces to the equation a₀ X³ 0 Therefore, the point (0: 0: 1) is the only point of f(X) on the line at infinity, and as the point of intersection of the line at infinity with f(X) it has multiplicity 3 Mon.,Apr.24: Continuing with the case of a complete normal curve over an algebraically closed field k When D is a divisor with deg(D) 2g, then for each a X(k) one has deg(D a) 2g 1, and, therefore, L(D a) is a hyperplane in L(D) Otherwise, said O(D) has no base point A coordinatefree interpretation of the morphism f : X PkN, where N deg(D) g, given by a basis of H⁰(X, O(D)) is that f(a) is the hyperplane H⁰(X, O(Da)) regarded as a point in the projective space of hyperplanes through the origin in H⁰(X, O(D)) If, moreover, deg(D) 2g 1, then for a b in X(k) it follows that H⁰(X, O(Dab)) has codimension 2 in H⁰(X, O(D)) so that f(a) and f(b) must be different points, i.e., f is injective Since X is complete, f(X) must be a closed subvariety of dimension 1 in PkN The fact that H⁰(X, O(D 2a)) also has codimension 2 in H⁰(X, O(D)) guarantees that d_a(f) : T_a(X) T_f(a)(PkN) has rank 1 for each a, and, therefore, that f(X) is itself a complete nonsingular curve Since morphisms of complete nonsingular curves are dual to the contravariant function field extensions, f must be an isomorphism, i.e., O(D) is very ample when deg(D) 2 g 1 As first example, when g 0 and D a, the morphism f given by H⁰(X, O(a)) is an isomorphism of X with Pk1 Fri.,Apr.21: In the context of a complete normal variety X over an algebraically closed field k an invertible O_Xmodule L is called very ample if there is an integer N 0 and a closed immersion f : X PkN such that L fO_PkN(1) (Recall the iref="funcptsprojsp"earlier description of the functor of points over k of PkN.) If L is very ample, then L^m is also very ample for each m 1 One says that L is ample if there exists m 1 such that L^m is very ample Finally, if there is an integer N 0 and a morphism f : X PkN such that L fO_PkN(1), one says that L has no base point For a particular value of N if z₀, z_N are homogeneous coordinates in PkN, hence, a basis of H⁰(PkN, O_PkN(1)), then the s_j f z_j are elements of H⁰(X, L) that do not vanish simultaneously at any point of X(k) It follows that the members of any basis of H⁰(X, L) also have no common zero, but it does not follow that the s_j form a basis When dim(X) 1, recall that for a divisor D of negative degree iref="nonnegdeg"one has dim_k(H⁰(X, O(D))) 0 If K is a canonical divisor and D a divisor with deg(D) deg(K) 2 g 2 then K D is a divisor of negative degree, and, consequently, by iref="serredimone"Serre duality dim_k H¹(X, O(D)) 0 for any divisor D with deg(D) 2 g 1 When the genus g 1, this means that dim_k H¹(X, O(D)) 0 for any divisor D of degree at least 1 The Riemann Roch formula then implies that dim H⁰(X, O(D)) deg(D) In particular if D a for a X(k), one sees that L(a) L(0) k while both have dimension 1 Hence, there can be no f k(X) with only a single simple pole The same type of reasoning shows that k(X) contains an element whose only pole is a double pole at a given point a X(k) Wed.,Apr.19: When A is a ring and B an Aalgebra, the module _BA is the Bmodule receiving an Aderivation from B that is initially universal for derivations from B to Bmodules When f : X Y is a morphism of schemes there is an O_Xmodule _XY that globalizes the module of differentials from commutative algebra A morphism f : X Y of irreducible varieties over an algebraically closed field k is called smooth if (i) f is dominant, i.e., f(X) Y, and (ii) _XY is a locallyfree O_Xmodule of rank dim(X) dim(Y) A nonsingular variety over k is a variety X that is smooth over k (An irreducible variety of dimension 1 is nonsingular if and only if it is normal.) When X is a nonsingular variety, one defines Xp to be the pth exterior power ^p_Xk For n dim(X) the top exterior power _X Xn is a locallyfree O_Xmodule that is called the canonical O_Xmodule A form of Serre duality, which could be the subject of an entire course, is this: Theorem If X is a complete nonsingular variety of dimension n and F a coherent O_Xmodule, then H^p(X, F) and ExtOnp(F, _X) are dual vector spaces over k An important special case is that when F is a locallyfree Omodule Then ExtOnp(F, _X) ExtOnp(O, _X F) H^np(X, _X F) where F denotes the O dual of F serredimone In the case of a complete normal curve a canonical divisor is any divisor K for which O(K) _X When F O(D) for an arbitary divisor D, the vector spaces H^p(X, O(D)) and H^1p(X, O(K D)) have the same dimension for p 0, 1 In particular one has g dim H¹(X, O_X) dim H⁰(X, _X), and application of the RiemannRoch formula to a canonical divisor leads to the conclusion that any canonical divisor must have degree 2g 2 Mon.,Apr.17: Continuing with the case of a complete normal curve X over an algebraically closed field, some observations: nonnegdeg If H⁰(X, O(D)) (0), then deg(D) 0 since D is linearly equivalent to a nonnegative divisor div(f) D for some f L(D) The set D E Div(X)E 0,E D is called the complete linear system determined by D It may be bijectively identified with the projective space of lines through the origin in the vector space L(D) H⁰(X, O(D)) A linear system is a projective subspace of a complete linear system. One has D div(s)s H⁰(X, O(D)) Looking at the cohomology sequence associated with the short exact sequence 0 O(Da) O(D) iiO(D) 0 one sees that in going from Da to D either the dimension of H⁰ goes up by 1 or the dimension of H¹ goes down by 1 but not both To go further with complete normal curves we want to talk about Serre duality Fri.,Apr.7: When X is a complete normal curve over an algebraically closed field k, a X a closed point, a the corresponding divisor, and i : a X the corresponding closed immersion of a subvariety, one has the exact sequence of coherent Omodules 0 I_a O iO_a 0 and, remembering that I_a O(a), then tensoring this exact sequence with the invertible Omodule O(D), D an arbitary divisor on X, one obtains 0 O(Da) O(D) iiO(D) 0 The third term above is a skyscraper that is rank 1 on O_a(a) k The relation among Euler characteristics given by the last short exact sequence reduces to (X, D) (X, Da) 1 for every divisor D and every closed point a X, and, thus, the observation that (X, D) deg(D) is a constant depending only on X where deg(D) _z n_z when D _z n_z z This provides a substantial portion of the RRThmRiemannRoch Theorem: (X, D) deg(D) 1 g where g, the genus of X, is defined as dim_k H¹(X, O) As a corollary of this, together with the observation that (X, D) depends only on O(D), one sees that deg(D) depends only on O(D), and, therefore, deg(div(f)) 0 for each f k(X), a result that corresponds to the statement for compact Riemann surfaces that the number of zeroes of a meromorphic function equals the number of its poles For an initial understanding of the genus of a complete normal curve, consider the exact sequence of Omodules 0 O k(X) k(X)O 0 from which ensues the sequence of vector spaces over k 0 k k(X) H⁰(X, k(X)O) H¹(X, O) 0 where the last 0 is H¹ of the constant, hence flasque, sheaf k(X) and H⁰(X, k(X)O) is the vector space of principal part specifications Thus, g 0 if and only if every principal part specification is realized by an element of k(X) Thereby it is clear that the genus of Pk1 is 0 Wed.,Apr.5: For D Div(X), X a normal variety, one defines L(D) f k(X)div(f) D 0 0 L(D) is an O(X)module that is isomorphic to the module of global sections of O(D) While a (regular) section of a locallyfree Omodule of rank 1 is not represented by a single element of k(X), it does have local pieces that are unique up to multiplications from O and, consequently, has a globally welldefined divisor If s_f 0 is the section of O(D) corresponding biuniquely with f L(D), one has div(s_f) div(f) D One sees that dim_k H⁰(X, O(D)) 0 if and only if D is linearly equivalent to some nonnegative divisor A nonnegative divisor D determines an Oideal I_D that is locally the principal ideal generated by a local equation for D It follows that I_D is a rank 1 locallyfree Omodule, and one sees easily that it is isomorphic to O(D) When X is a complete variety over a field k and M a coherent Omodule the kmodules H^q(X,M) are finitedimensional over k for all q This is a consequence of the more general fact that direct images and higher direct images of a coherent module under a proper morphism are coherent (see the text) One defines the Euler characteristic of a coherent Omodule by (X, M) _q0^dim(X) (1)^q dim_k H^q(X, M) When 0 M M M 0 is an exact sequence of coherent Omodules on X, one has (X, M) (X, M) (X, M) Mon.,Apr.3: When X is a normal variety, the affine coordinate ring O(U) of an open affine subvariety U is the intersection of its localizations at the prime ideals corresponding to the irreducible closed sets in U of codimension 1 Hence O(X) is the kernel of the homomorphism div Given a divisor D Div(X) and an open covering U_i of X that principalizes D, say, DU_i div_Ui(f_i), it follows from the computation of the kernel of div on the open subvariety U_ij U_i U_j that f_i u_ij f_j (all elements of k(X)) where u_ij O(U_ij) The Cech 1cocycle u_ij determines an element O(D) of the group HCech1(X,O) of locallyfree Omodules of rank 1, the map D O(D) is a group homomorphism, and the sequence 1 O(X) k(X) Div(X) HCech1(X,O) 1 is exact One says that two divisors D₁ and D₂ are linearly equivalent (and one may write D₁ D₂) if D₂ D₁ div(f) for some f k(X) or, otherwise stated, if O(D₁) O(D₂) Fri.,Mar.31: For an irreducible variety X over an algebraically closed field k, a divisor is an element of the free abelian group Div(X) generated by the irreducible closed sets of codimension 1 When X is normal, the local ring at each irreducible closed set Z of codimension 1 is a principal valuation ring, and, therefore, each element f 0 in the function field k(X) gives rise to a divisor div(f) _Z ord_Z(f) which is called a principal divisor The map div : k(X) Div(X) is a homomorphism of abelian groups Since an open set U in X is also a variety, the functor U Div(U) defines an abelian sheaf Div on X that is easily seen to be flasque When X is normal and Z an irreducible closed set of codimension 1, the divisor in an open neighborhood of Z of the unique prime in O_Z is the generating divisor corresponding to Z Thus one sees that each divisor on X is locally principal Wed.,Mar.29: If f : X Y is an affine morphism of algebraic varieties over an algebraically closed field k, then for each quasicoherent O_Xmodule F one has an isomorphism of H^q(X, F) with H^q(Y, fF) Finite morphisms and closed immersions present important special cases To know the cohomology of every coherent O_Pmodule on each projective space P PkN is to know the cohomology of every coherent O_Xmodule on every projective variety X Mon.,Mar.27: On a Noetherian space the cohomological functor H^q for abelian sheaves vanishes when q dim(X) The E₂ spectral sequence for composite functors is operative when application of the first functor to an injective object in its domain yields an object that is acyclic for the second functor This applies to the direct image functor followed by the global sections functor on abelian sheaves since the direct image of an injective abelian sheaf is flasque Fri.,Mar.24: On a Noetherian space (descending chain condition for closed sets) each of the sheaf cohomology functors H^q on the category of abelian sheaves commutes with direct limits Wed.,Mar.22: More on cohomology: Every abelian sheaf on a topological space X may be regarded as a Zmodule (sheaf of modules over the constant sheaf Z) As base cohomology one uses the derived functors of the global sections functor in the category of Zmodules An abelian sheaf is flasque if its restrictions between open sets are all surjective Every flasque sheaf is acyclic for cohomology, and every injective Amodule, for any sheaf of rings A on X, is flasque Consequently, sheaf cohomology in the category of Amodules is consistent with that in the category of Zmodules Mon.,Mar.20: If f : (X,A) (Y,B) is a morphism of ringed spaces, for every Bmodule G there is an Amodule pullback f(G) which at stalk level satisfies f(G)_x G_{f(x) Bf(x)} A_x For a morphism of affine schemes pullback of quasicoherent modules on the target is the same thing as base extension For P PkN, k an algebraically closed field, the exact sequence O_P^N1 (x₀, , x_N) O_P(1) 0 given by (f₀, , f_N) f₀ x₀ f_N x_N spawns, via pullback, the funcptsprojsp functor of points of PkN over k: a morphism : X PkN is the same thing as an invertible O_Xmodule L and an N1tuple of sections s₀, s_N of L that do not vanish simultaneously, i.e., that provide the exact sequence O_X^N1 (s₀, , s_N) L 0 which is the pullback of the referenced exact sequence on PkN For a kvalued point x X(k) one has (x) (s₀(x): s₁(x): : s_N(x)) Fri.,Mar.17: The isomorphism classes of locallyfree Amodules of rank 1 form a group The notion of an exact sequence of Amodules Amodules form an abelian category in which every object admits an injective resolution The global sections functor (M) M(X) is left exact The qth cohomology functor X H^q(X,M) is defined as the qth right derived functor of Sideline example: the short exact sequence 0 Z O_hol e Ohol 0 of Zmodules in complex analytic geometry, where e(f) e^{2 i f} is the complex exponential Wed.,Mar.15: Homomorphisms of Amodules when A is a sheaf of rings on a topological space Locallyfree Amodules of rank r and transition matrices relative to a trivializing covering An invertible Amodule is a locallyfree Amodule of rank 1 Mon.,Mar.13: Class cancelled Fri.,Mar.10: Properties and significance of the O_P modules O_P(d) on P Pkn for d Z where k is an algebraically closed field Wed.,Mar.8: The concept of sheaf of modules on a ringed space Quasicoherent and coherent modules on a scheme Examples Mon.,Mar.6: If f : X Y is a morphism of schemes with Y separated, then f is universally closed if every split base extension of f is closed Proper morphisms Valuative criteria for separated morphisms and proper morphisms Fri.,Mar.3: Separated morphisms If f : X Y is an Smorphism and Y is separated over S, then the graph of f is closed in X Y and f is separated if and only if X is separated over S Henceforth, an algebraic variety will be assumed to be separated over its base field; consequently, all morphisms of varieties will be separated In a scheme that is separated over an affine base, the intersection of any two open affines is affine Wed.,Mar.1: If x is an element of X, the scheme underlying an irreducible algebraic variety, the Krull dimension of the local ring O_x is the codimension of x in X When X is normal, the local ring at an irreducible subvariety of codimension 1 in X is a discrete valuation ring The set of closed points of a complete and normal irreducible algebraic curve correspond biuniquely with the nontrivial discrete valuation rings in its function field that contain the ground field, and the entire structure of such a curve as a scheme may be recovered from its function field Mon.,Feb.27: Finite morphisms yet another class closed under composition and base extension The normalization of an irreducible variety Universally closed morphisms Finite morphisms are universally closed Fri.,Feb.17: Any base extension of a morphism of finite type is also a morphism of finite type Case in point: the fibre of a morphism f : X Y of finite type over an element y Y is a scheme of finite type over the residue field (y) Over its image a morphism may be viewed as providing a family of varieties, though not a wellbehaved one without assumptions on the morphism The notion of affine morphism: another class of morphisms that is closed under compostion and base extension Wed.,Feb.15: The join of two Cartesian squares is another Cartesian squares provide shelter for both the geometric notion of product and the algebraic notion of base extension The notion of base extension of a morphism Example: The action of Gal(kk) on X_k when X is a kscheme (and k is the algebraic closure of the field k) Mon.,Feb.13: Detailed examination of the functor of points for E SpecZx,yF(x,y) where F(x, y) is the polynomial F(x, y) y² (x a)(x b)(x c), particularly in relation to base extensions of the coordinate ring Existence and uniqueness of products in the category of schemes over a given scheme Fri.,Feb.10: The notion of morphism of a scheme over a base scheme globalizes the notion of homomorphism for algebras over a base ring If S is a scheme, the functor SchemesS^op Sets given by T Hom_S(T, X) X(T) is called the functor of points of X over S X is determined as an Sscheme by its functor of points If X is the scheme associated with a variety X₀ over an algebraically closed field k, then X(k) X(Spec(k)) is the set underlying X₀ If K is an extension field of k, a fieldpoint point X(K) determines an element x X (no longer called a point) that is called its center and a kalgebra homomorphism from the residue field at x to K In the affine case X(K) is precisely the set of naive points of X in K Wed.,Feb.8: A morphism from a scheme to the affine scheme Spec(A) is dual to a ring homomorphism from A to the ring of global sections of the schemes structure sheaf The scheme associated with an affine variety over an algebraically closed field is characterized as a reduced scheme of finite type over (the spectrum of) the field Mon.,Feb.6: The category of schemes Locally closed subschemes Morphisms; schemes over a base scheme Fri.,Feb.3: The category of affine schemes as (1) a fully faithful subcategory of the category of localringed spaces and (2) as the opposite category of the category of commutative rings Wed.,Feb.1: The notion of an affine scheme as a topological space equipped with a sheaf of rings; morphisms between affine schemes Mon.,Jan.30: The sheaf of rings associated with the spectrum of a commutative ring; the initial ring is the ring of global sections Fri.,Jan.27: The spectrum of a commutative ring and its Zariski topology Wed.,Jan.25: Presheaves and sheaves; examples Mon.,Jan.23: Overview

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Things Spotted on the Web Wikipedia There are a number of ways to enter href="http:en.wikipedia.orgwikiAlgebraicGeometry"Algebraic Geometry href="http:en.wikipedia.orgwikiScheme28mathematics29"Schemes href="http:en.wikipedia.orgwikiSpecial:Search?search22algebraicgeometry22fulltextfulltext"Search Wikipedia for algebraic geometry Notes on Lectures by Hartshorne These are notes by William Stein of 1996 lectures given by Robin Hartshorne at UC Berkeley: http:modular.ucsd.eduAG.html Href=".."UP Href="mathpershammond"TOP Href="http:math.albany.edu"Department