ZQRCHPvpsetlengthheadheight0bpsetlengthheadsep0bpsetlengthtopmargin-36bpsetlengthtextheight704bpUniv at Albany: Math: W. F. Hammond: Courses: Math 825Spring Semester, 20061Note: If you found this document through a web search engine, you may not be aware of its href="http://math.albany.edu/math/pers/hammond/course/mat825s2006/"other presentation formats

&SecRef-1;SU-1OutlineFri.,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 Fq, each residue field at a closed element is a finite extension field of Fq, and, therefore, N(x) qd(x) where d(x) is the extension degree With t qs one writes X(s) ZX(t) x closed 11 td(x) With the condition ZX(0) 1 the Z form of the zeta function is determined by its logarithmic derivative ddt log ZX(t) x closed d(x) td(x)11 td(x) 1t r 1 x closedd(x) r r tr1 tr 1t r 1 r cr tr1 tr 1t r 1 r cr s 1 trs 1t 1 r divides r cr t 1 N t1 where cr 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 FqFor a beginning example, when X An, one has N qn, and, therefore, ZAn(t) 11qnt Of course, An is not a projective variety for n 1When F is a field, the set of Fvalued points of Pn is the disjoint union of A0(F), A1(F), , An(F) Therefore, Dlog ZPn(t) (over Fq) is the sum of Dlog ZAj(t) for 0 j n Hence, ZPn(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 ZE(t) 1 4 t 5t2(1 t) (1 5t) For each of these last examples Pn, P1 P1, and E one may observe that ZX(t), relative to the field Fq 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 XC of X for even dimensionsthe numerator is the product of polynomials whose degrees are the classical topological Betti numbers of the base extension XC of X for odd dimensionsthe 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 qj2Wed.,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 ZFor 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 Ohol e Ohol 0 where e(f) e2 i f In the long cohomology sequence the 0 stage splits off since H0(X, Ohol) 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, Ohol) H1(X, Ohol) 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 Oholmodule to its degree Therefore, remembering that Div(X)Div(X) H1(X, O), one has H1(X, Ohol)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 CFor dim(X) 1 the kernel of the connecting homomorphism will provide a correct notion of degree 0For 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 usedMon.,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 op 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) Pk2p 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 kIf 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 XF of finite type over F whose base extension to k is X For each extension E of F the set XF(E) is a group in a functorial way XF(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) np p (D) p X(k) np 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 3Mon.,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 da(f) : Ta(X) Tf(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 Pk1Fri.,Apr.21:In the context of a complete normal variety X over an algebraically closed field k an invertible OXmodule L is called very ample if there is an integer N 0 and a closed immersion f : X PkN such that L fOPkN(1) (Recall the iref="funcptsprojsp"earlier description of the functor of points over k of PkN.) If L is very ample, then Lm is also very ample for each m 1 One says that L is ample if there exists m 1 such that Lm is very ample Finally, if there is an integer N 0 and a morphism f : X PkN such that L fOPkN(1), one says that L has no base point For a particular value of N if z0, zN are homogeneous coordinates in PkN, hence, a basis of H0(PkN, OPkN(1)), then the sj f zj 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 sj form a basis When dim(X) 1, recall that for a divisor D of negative degree iref="nonnegdeg"one has dimk(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 dimk H1(X, O(D)) 0 for any divisor D with deg(D) 2 g 1 When the genus g 1, this means that dimk 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 OXmodule 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 OXmodule 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 pXk For n dim(X) the top exterior power X Xn is a locallyfree OXmodule that is called the canonical OXmodule A form of Serre duality, which could be the subject of an entire course, is this: AssertLabel-1AssertLabel-1AssertDefaultTheoremIf X is a complete nonsingular variety of dimension n and F a coherent OXmodule, then Hp(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) Hnp(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 Hp(X, O(D)) and H1p(X, O(K D)) have the same dimension for p 0, 1 In particular one has g dim H1(X, OX) 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 2Mon.,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 bothTo go further with complete normal curves we want to talk about Serre dualityFri.,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 Ia O iOa 0 and, remembering that Ia 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 Oa(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 nz when D z nz 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 dimk 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 0Wed.,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 sf 0 is the section of O(D) corresponding biuniquely with f L(D), one has div(sf) div(f) D One sees that dimk H0(X, O(D)) 0 if and only if D is linearly equivalent to some nonnegative divisor A nonnegative divisor D determines an Oideal ID that is locally the principal ideal generated by a local equation for D It follows that ID 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 Hq(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) q0dim(X) (1)q dimk Hq(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 Ui of X that principalizes D, say, DUi divUi(fi), it follows from the computation of the kernel of div on the open subvariety Uij Ui Uj that fi uij fj (all elements of k(X)) where uij O(Uij) The Cech 1cocycle uij 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 ordZ(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 OZ is the generating divisor corresponding to Z Thus one sees that each divisor on X is locally principalWed.,Mar.29:If f : X Y is an affine morphism of algebraic varieties over an algebraically closed field k, then for each quasicoherent OXmodule F one has an isomorphism of Hq(X, F) with Hq(Y, fF) Finite morphisms and closed immersions present important special cases To know the cohomology of every coherent OPmodule on each projective space P PkN is to know the cohomology of every coherent OXmodule on every projective variety XMon.,Mar.27:On a Noetherian space the cohomological functor Hq 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 flasqueFri.,Mar.24:On a Noetherian space (descending chain condition for closed sets) each of the sheaf cohomology functors Hq on the category of abelian sheaves commutes with direct limitsWed.,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 ZmodulesMon.,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 Gf(x) Bf(x) Ax 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 OPN1 (x0, , xN) OP(1) 0 given by (f0, , fN) f0 x0 fN xN spawns, via pullback, the funcptsprojsp functor of points of PkN over k: a morphism : X PkN is the same thing as an invertible OXmodule L and an N1tuple of sections s0, sN of L that do not vanish simultaneously, i.e., that provide the exact sequence OXN1 (s0, , sN) 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): : sN(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 Hq(X,M) is defined as the qth right derived functor of Sideline example: the short exact sequence 0 Z Ohol e Ohol 0 of Zmodules in complex analytic geometry, where e(f) e2 i f is the complex exponentialWed.,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 1Mon.,Mar.13:Class cancelledFri.,Mar.10:Properties and significance of the OP modules OP(d) on P Pkn for d Z where k is an algebraically closed fieldWed.,Mar.8:The concept of sheaf of modules on a ringed space Quasicoherent and coherent modules on a scheme ExamplesMon.,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 morphismsFri.,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 affineWed.,Mar.1:If x is an element of X, the scheme underlying an irreducible algebraic variety, the Krull dimension of the local ring Ox 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 fieldMon.,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 closedFri.,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 extensionWed.,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 Xk 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 schemeFri.,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 SchemesSop Sets given by T HomS(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 KWed.,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 fieldMon.,Feb.6:The category of schemes Locally closed subschemes Morphisms; schemes over a base schemeFri.,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 ringsWed.,Feb.1:The notion of an affine scheme as a topological space equipped with a sheaf of rings; morphisms between affine schemesMon.,Jan.30:The sheaf of rings associated with the spectrum of a commutative ring; the initial ring is the ring of global sectionsFri.,Jan.27:The spectrum of a commutative ring and its Zariski topologyWed.,Jan.25:Presheaves and sheaves; examplesMon.,Jan.23:Overview

&SecRef-2;SU-2CommentsSU-2.1Things Spotted on the WebWikipedia There are a number of ways to enter href="http://en.wikipedia.org/wiki/AlgebraicGeometry"Algebraic Geometryhref="http://en.wikipedia.org/wiki/Scheme28mathematics29"Schemeshref="http://en.wikipedia.org/wiki/Special:Search?search=22algebraic+geometry22fulltext=fulltext"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.edu/AG.html Href="../"UP Href="/math/pers/hammond/"TOP Href="http://math.albany.edu/"Department