Topics in Algebraic Geometry (Math 825)
Introduction to Schemes
Outline with Comments

Spring Semester, 2006

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1.  Outline

Fri., May. 5:
A 1949 paper by André Weil gave evidence for the existence of “topological cohomology” in algebraic geometry linked to the notion of zeta function for a non-singular projective algebraic variety defined over a finite field.

Let X be a scheme of finite type over Z. For each element xX 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 ζXs=x closed in X11Nxs where Nx=κx is the number of elements of the residue field of X at x. (Ignore questions of convergence for now.) When X=SpecZ, ζXs is Riemann's 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, Nx=qdx where dx is the extension degree. With t=qs one writes ζXs=ZXt=x closed11tdx With the condition ZX0=1 the Z form of the zeta function is determined by its logarithmic derivative  ddtlogZXt=x closeddxtdx11tdx =1tr1x closeddx=rrtr1tr =1tr1rcrtr1tr =1tr1rcrs1trs =1tν1r divides νrcrtν =ν1Nνtν1 where cr denotes the number of closed elements in X with dx=r and Nν denotes the number of points of X with values in the degree ν extension of Fq.

For a beginning example, when X=An, one has Nν=qnν, and, therefore, ZAnt=11qnt. Of course, An is not a projective variety for n1.

When F is a field, the set of F-valued points of Pn is the disjoint union of A0F,A1F,,AnF. Therefore, DlogZPnt (over Fq) is the sum of DlogZAjt for 0jn. Hence, ZPnt=11t1qt1qnt.

For P1×P1, one has Nν=1+qν2, and, therefore ZP1×P1t=11t1qt21q2t.

For curves of genus 1 defined over finite fields, the shape of its Z function was established before the time of Weil's conjectures. For example, in the case of the curve E given by the Weierstrass equation y2=x32x over the field F5, simply by counting points to see that EF5=10, it is a consequence of the theoretical framework that Zt is the rational function ZEt=1+4t+5t21t15t.

For each of these last examples Pn, P1×P1, and E one may observe that ZXt, relative to the field Fq is a rational function in one variable and that:

  1. the denominator is the product of polynomials whose degrees are the classical topological Betti numbers of the base extension XC of X for even dimensions.

  2. the numerator is the product of polynomials whose degrees are the classical topological Betti numbers of the base extension XC of X for odd dimensions.

  3. the polynomial factor corresponding to classical cohomology in dimension j has the form of the characteristic polynomial of a linear endomorphism φ of the form det1tφ with complex reciprocal roots all of absolute value qj2.

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 non-singular varieties in general is studying the group DivXDivX. For curves one has DivXDiv0XDivX where the quotient for the second step is the discrete group Z when Div0X 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 Div0X such that the first step is a complete irreducible variety and the second step a finitely-generated 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 DivXDivXZ×Z.

For the purpose of gaining insight about DivXDivX in the theory of curves while at the same time beginning to understand what might be required for defining Div0X when dimX>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: 0ZOholeOhol*0 where ef=e2πif. In the long cohomology sequence the 0 stage splits off since H0X,OholC. GAGA tells us that coherent module cohomology matches, and although O* is certainly not an O-module, its H1 in both algebraic and transcendental theories viewed through Czech theory classifies isomorphism classes of invertible coherent modules. One has the exact sequence: 0H1X,ZH1X,OholH1X,Ohol*H2X,Z. If dimX=1, then H2X,ZZ, and one finds that the last map in this sequence, a “connecting homomorphism”, sends the isomorphism class of an invertible Ohol-module to its degree. Therefore, remembering that DivXDivXH1X,O*, one has H1X,OholH1X,ZDiv0XDivX, and, in fact, the left side is the quotient of a g-dimensional vector space over C by a lattice. Thus, Div0XDivX is a g-dimensional complex torus; it is, moreover, a complete group variety over C.

For dimX>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 Zariski-based cohomology cannot be used.

Mon., May. 1:
Continuing with the discussion of the previous hour: If p,q,r are any three points of Xk, then the triple sum p+q+r, like any point of Xk is characterized by the linear equivalence class of the associated one point divisor. One has the relation of linear equivalence p+q+rp+q+r2o. Therefore,  p+q+r=op+q+r3o p+q+r=divh+3o for some hL3o p+q+r=divs for some sH0X,O3o p+q+r=divaxu3+byu3+cu3, some a:b:cPk2ˇ p+q+r=f1D,D=divaX+bY+cZDivPk2 where f:XPk2 is the projective embedding of X given by the invertible O-module O3o. In other words, taking multiplicities into consideration, three points sum to o in the group law on Xk 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 Xk, 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

  1. Addition X×XX and negation XX are morphisms of varieties over k.

  2. If F is the field generated over the prime field by the coefficients a0,,a6 of the Weierstrass equation, then

    1. The Weierstrass equation defines a scheme XF of finite type over F whose base extension to k is X.

    2. For each extension E of F the set XFE is a group in a functorial way.

    3. XFkXk.

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 O3o,oXk. This notational change notwithstanding, o is an arbitrary point. Under the projective embedding given by O3o, one has fo=0:0:1, the unique point of fX on the line at infinity. We wish to show that there is a unique commutative group law on the set Xk for which the map φ:DivXXk D=pXknppφD=pXknpp, which is tautologically a group homomorphism, has the property that φD1=φD2 whenever D1D2 (linear equivalence), and further the property that o is the zero element in Xk. (This is not the strongest statement of this type that can be made.) Addition in Xk is defined by observing that since for given p,qXk the divisor p+qo has degree 1, its complete linear system consists of a single non-negative divisor of degree 1, i.e., r, and this unique rXk is defined to be p+q. Since p+qor, the properties specified for φ 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 2op. It is obvious that this group law on Xk is commutative and that φ is surjective. If Div0X denotes the group of divisors of degree 0, then since φD=φDdegDo, one sees that the restriction φ0 of φ to Div0X is a surjective homomorphism. Let DivX denote the group of divisors linearly equivalent to zero. It is trivial that the map DDdegDo defines a homomorphism DivXDiv0X which, when followed with reduction provides a homomorphism DivXDiv0XDivX. It is not difficult to verify that another homomorphism between this latter pair of groups is given by DφDomodDivX. (That this is a homomorphism follows from reviewing the definition of φD1+φD2.) Since these two homomorphisms agree on divisors of the form p - which generate the free abelian group DivX -, one has for all DDivX that DdegD0φDo. We know that degD depends only on the linear equivalence class of D as the first consequence of the Riemann-Roch Theorem. Since rXk is determined uniquely by the linear equivalence class of r, this formula tells us that φD 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 φD and degD. In particular, one has Div0XDivXXk.

Wed., Apr. 26:
Suppose that X is a complete non-singular curve over an algebraically closed field k of genus 1. The range of degrees where a divisor D has H1OD0 is degD1, while we have dimH1O=1. For each aXk the invertible module O2a has no base point, and, therefore, defines a morphism to Pk1. One has a two step filtration of the 3-dimensional linear subspace L3a of kX: k=L0=LaL2aL3a. Choosing xL2aL0 and yL3aL2a one obtains a filtration-compatible basis 1,x,y of L3a, and if u is a “rational section” of Oa with divu=a, the morphism f:XPk2 given by f=Z:X:Y,Z=u3,X=xu3,Y=yu3 provides a projective embedding of X by the theorem of the last hour. Extending the filtration inside kX by the Lma, one sees that 1,x,y,x2,xy,x3 is a filtration-compatible basis of L6a. Since y2L6aL5a, one has a linear relation among monomials of degree 3 Y2Z+a1XYZ+a3YZ2=a0X3+a2X2Z+a4XZ2+a6Z3 with a00 that characterizes fX as a non-singular hypersurface in Pk2. One says that fX 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 fX with the line at infinity reduces to the equation a0X3=0. Therefore, the point 0:0:1 is the only point of fX on the line at infinity, and as the point of intersection of the line at infinity with fX 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 degD2g, then for each aXk one has degDa2g1, and, therefore, LDa is a hyperplane in LD. Otherwise, said OD has no base point. A coordinate-free interpretation of the morphism f:XPkN, where N=degDg, given by a basis of H0X,OD is that fa is the hyperplane H0X,ODa regarded as a point in the projective space of hyperplanes through the origin in H0X,OD. If, moreover, degD2g+1, then for ab in Xk it follows that H0X,ODab has codimension 2 in H0X,OD so that fa and fb must be different points, i.e., f is injective. Since X is complete, fX must be a closed subvariety of dimension 1 in PkN. The fact that H0X,OD2a also has codimension 2 in H0X,OD guarantees that daf:TaXTfaPkN has rank 1 for each a, and, therefore, that fX is itself a complete non-singular curve. Since morphisms of complete non-singular curves are dual to the contravariant function field extensions, f must be an isomorphism, i.e., OD is very ample when degD2g+1. As first example, when g=0 and D=a, the morphism f given by H0X,Oa 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 OX-module L is called very ample if there is an integer N0 and a closed immersion f:XPkN such that Lf*OPkN1. (Recall the earlier description of the functor of points over k of PkN.) If L is very ample, then Lm is also very ample for each m1. One says that L is ample if there exists m1 such that Lm is very ample. Finally, if there is an integer N0 and a morphism f:XPkN such that Lf*OPkN1, 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 H0PkN,OPkN1, then the sj=f*zj are elements of H0X,L that do not vanish simultaneously at any point of Xk. It follows that the members of any basis of H0X,L also have no common zero, but it does not follow that the sj form a basis.

When dimX=1, recall that for a divisor D of negative degree one has dimkH0X,OD=0. If K is a canonical divisor and D a divisor with degD>degK=2g2, then KD is a divisor of negative degree, and, consequently, by Serre duality dimkH1X,OD=0 for any divisor D with degD2g1. When the genus g=1, this means that dimkH1X,OD=0 for any divisor D of degree at least 1. The Riemann Roch formula then implies that dimH0X,OD=degD. In particular if D=a for aXk, one sees that LaL0k while both have dimension 1. Hence, there can be no fkX* with only a single simple pole. The same type of reasoning shows that kX* contains an element whose only pole is a double pole at a given point aXk.

Wed., Apr. 19:
When A is a ring and B an A-algebra, the module ΩBA is the B-module receiving an A-derivation from B that is initially universal for derivations from B to B-modules. When f:XY is a morphism of schemes there is an OX-module ΩXY that globalizes the module of differentials from commutative algebra. A morphism f:XY of irreducible varieties over an algebraically closed field k is called smooth if (i) f is dominant, i.e., fX¯=Y, and (ii) ΩXY is a locally-free OX-module of rank dimXdimY. A non-singular variety over k is a variety X that is smooth over k. (An irreducible variety of dimension 1 is non-singular if and only if it is normal.) When X is a non-singular variety, one defines ΩXp to be the p-th exterior power pΩXk. For n=dimX the top exterior power ωX=ΩXn is a locally-free OX-module that is called the canonical OX-module.

A form of Serre duality, which could be the subject of an entire course, is this:

Theorem.   If X is a complete non-singular variety of dimension n and F a coherent OX-module, then HpX,F and ExtOnpF,ωX are dual vector spaces over k.

An important special case is that when F is a locally-free O-module. Then ExtOnpF,ωXExtOnpO,ωXFHnpX,ωXF where F denotes the O dual of F. In the case of a complete normal curve a canonical divisor is any divisor K for which OKωX. When F=OD for an arbitary divisor D, the vector spaces HpX,OD and H1pX,OKD have the same dimension for p=0,1. In particular one has g=dimH1X,OX=dimH0X,ωX, and application of the Riemann-Roch formula to a canonical divisor leads to the conclusion that any canonical divisor must have degree 2g2.

Mon., Apr. 17:
Continuing with the case of a complete normal curve X over an algebraically closed field, some observations:
  1. If H0X,OD0, then degD0 since D is linearly equivalent to a non-negative divisor divf+D for some fLD.

  2. The set D=EDivXE0,ED 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 LDH0X,OD. A linear system is a projective subspace of a complete linear system. One has D=divssH0X,OD.

  3. Looking at the cohomology sequence associated with the short exact sequence 0ODaODi*i*OD0, 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.

  4. 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, aX a closed point, a the corresponding divisor, and i:aX the corresponding closed immersion of a subvariety, one has the exact sequence of coherent O-modules 0IaOi*Oa0, and, remembering that IaOa, then tensoring this exact sequence with the invertible O-module OD, D an arbitary divisor on X, one obtains 0ODaODi*i*OD0. The third term above is a skyscraper that is rank 1 on Oaak. The relation among Euler characteristics given by the last short exact sequence reduces to χX,D=χX,Da+1 for every divisor D and every closed point aX, and, thus, the observation that χX,DdegD is a constant depending only on X where degD=znzwhenD=znzz. This provides a substantial portion of the Riemann-Roch Theorem: χX,D=degD+1g where g, the genus of X, is defined as dimkH1X,O. As a corollary of this, together with the observation that χX,D depends only on OD, one sees that degD depends only on OD, and, therefore, degdivf=0 for each fkX*, 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 O-modules 0OkX¯kX¯O0 from which ensues the sequence of vector spaces over k 0kkXH0X,kX¯OH1X,O0 where the last 0 is H1 of the constant, hence flasque, sheaf kX¯ and H0X,kX¯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 kX. Thereby it is clear that the genus of Pk1 is 0.

Wed., Apr. 5:
For DDivX, X a normal variety, one defines LD=fkX*divf+D00. LD is an OX-module that is isomorphic to the module of global sections of OD. While a (regular) section of a locally-free O-module of rank 1 is not represented by a single element of kX*, it does have local pieces that are unique up to multiplications from O* and, consequently, has a globally well-defined divisor. If sf0 is the section of OD corresponding biuniquely with fLD, one has divsf=divf+D. One sees that dimkH0X,OD>0 if and only if D is linearly equivalent to some non-negative divisor.

A non-negative divisor D determines an O-ideal ID that is locally the principal ideal generated by a local equation for D. It follows that ID is a rank 1 locally-free O-module, and one sees easily that it is isomorphic to OD.

When X is a complete variety over a field k and M a coherent O-module the k-modules HqX,M are finite-dimensional 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 O-module by χX,M=q=0dimX1qdimkHqX,M. When 0MMM′′0 is an exact sequence of coherent O-modules on X, one has χX,M=χX,M+χX,M′′.

Mon., Apr. 3:
When X is a normal variety, the affine coordinate ring OU 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 OX* is the kernel of the homomorphism div. Given a divisor DDivX and an open covering Ui of X that principalizes D, say, D|Ui=divUifi, it follows from the computation of the kernel of div on the open subvariety Uij=UiUj that fi=uijfj (all elements of kX) where uijOUij*. The Cech 1-cocycle uij determines an element OD of the group HCech1X,O* of locally-free O-modules of rank 1, the map DOD is a group homomorphism, and the sequence 1OX*kX*DivXHCech1X,O*1 is exact. One says that two divisors D1 and D2 are linearly equivalent (and one may write D1D2) if D2D1=divf for some fkX* or, otherwise stated, if OD1OD2.
Fri., Mar. 31:
For an irreducible variety X over an algebraically closed field k, a divisor is an element of the free abelian group DivX 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 f0 in the function field kX gives rise to a divisor divf=ZordZf, which is called a principal divisor. The map div:kX*DivX is a homomorphism of abelian groups. Since an open set U in X is also a variety, the functor UDivU 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 principal.
Wed., Mar. 29:
If f:XY is an affine morphism of algebraic varieties over an algebraically closed field k, then for each quasi-coherent OX-module F one has an isomorphism of HqX,F with HqY,f*F. Finite morphisms and closed immersions present important special cases. To know the cohomology of every coherent OP-module on each projective space P=PkN is to know the cohomology of every coherent OX-module on every projective variety X.
Mon., Mar. 27:
On a Noetherian space the cohomological functor Hq for abelian sheaves vanishes when q>dimX. 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 Hq 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 Z-module (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 Z-modules. An abelian sheaf is flasque if its restrictions between open sets are all surjective. Every flasque sheaf is acyclic for cohomology, and every injective A-module, for any sheaf of rings A on X, is flasque. Consequently, sheaf cohomology in the category of A-modules is consistent with that in the category of Z-modules.
Mon., Mar. 20:
If f:X,AY,B is a morphism of ringed spaces, for every B-module G there is an A-module pullback f*G which at stalk level satisfies f*Gx=GfxBfxAx. For a morphism of affine schemes pullback of quasi-coherent modules on the target is the same thing as base extension. For P=PkN, k an algebraically closed field, the exact sequence OPN+1x0,,xNOP10 given by f0,,fNf0x0++fNxN spawns, via pullback, the functor of points of PkN over k: a morphism φ:XPkN is “the same thing” as an invertible OX-module L and an N+1-tuple of sections s0,sN of L that do not “vanish” simultaneously, i.e., that provide the exact sequence OXN+1s0,,sNL0, which is the φ-pullback of the referenced exact sequence on PkN. For a k-valued point xXk one has φx=s0x:s1x::sNx.
Fri., Mar. 17:
The isomorphism classes of locally-free A-modules of rank 1 form a group. The notion of an exact sequence of A-modules. A-modules form an abelian category in which every object admits an injective resolution. The global sections functor ΓM=MX is left exact. The q-th cohomology functor XHqX,M is defined as the q-th right derived functor of Γ. Sideline example: the short exact sequence 0ZOholeOhol*0 of Z-modules in complex analytic geometry, where ef=e2πif is the complex exponential.
Wed., Mar. 15:
Homomorphisms of A-modules when A is a sheaf of rings on a topological space. Locally-free A-modules of rank r and transition matrices relative to a trivializing covering. An invertible A-module is a locally-free A-module of rank 1.
Mon., Mar. 13:
Class cancelled.
Fri., Mar. 10:
Properties and significance of the OP modules OPd on P=Pkn for dZ where k is an algebraically closed field.
Wed., Mar. 8:
The concept of sheaf of modules on a ringed space. Quasi-coherent and coherent modules on a scheme. Examples.
Mon., Mar. 6:
If f:XY 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:XY is an S-morphism 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 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 non-trivial 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:XY of finite type over an element yY is a scheme of finite type over the residue field κy. Over its image a morphism may be viewed as providing a family of varieties, though not a well-behaved 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 Galk¯k on Xk¯ when X is a k-scheme (and k¯ is the algebraic closure of the field k).
Mon., Feb. 13:
Detailed examination of the functor of points for E=SpecZx,yFx,y where Fx,y is the polynomial Fx,y=y2xaxbxc, 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 SchemesSopSets given by THomST,X=XT is called the functor of points of X over S. X is determined as an S-scheme by its functor of points. If X is the scheme associated with a variety X0 over an algebraically closed field k, then Xk=XSpeck is the set underlying X0. If K is an extension field of k, a point ξXK determines an element xX (no longer called a “point”) that is called its center and a k-algebra homomorphism from the residue field at x to K. In the affine case XK is precisely the set of naive points of X in K.
Wed., Feb. 8:
A morphism from a scheme to the affine scheme SpecA is dual to a ring homomorphism from A to the ring of global sections of the scheme's 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 local-ringed 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:


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Notes on Lectures by Hartshorne
These are notes by William Stein of 1996 lectures given by Robin Hartshorne at UC Berkeley:

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