Math 548: Algebraic geometry II John Lesieutre September 4, 2020 Announcements

Note to self: start recording! • HW due Friday. • Office hours? • Functions on Spec

LetR be a ring. There is a (slightly awkward) way of • setting up a sheaf onX = SpecR with the O X properties that: What’s the stalk? It’s =R .(P prime ideal) • O X,P P What’s it on opens? (D(f )) =R . • O X f (What are the global functions?) • Very briefly

LetR be a ring andU an open subset. • A regular function onU is given by(φ P)P U where • ∈ φ R . P ∈ P We require local compatibility: for anyP U there • ∈ is an openU P andf,g R such that if P � ∈ Q U : a)f Q and b)φ = g (as elts ofR ) ∈ P �∈ Q f Q The formulag/f holds for the function on an open, • but we don’t require there to be one formula that works at every point. Affine schemes

An affine scheme is a locally ringed space isomorphic • to the locally ringed space SpecR for a ringR. Note: a morphismR S gives rise to a scheme • → map SpecS SpecR. → (I think we already know this in the case of varieties.) • In fact, every scheme map between them comes from • a ring map. Things to contemplate

What are the functions on Speck[x]/(x 2)? • LetX = Spec . What’sD(6)? • Z What are the maps • SpecQ SpecQ[x,y]/(x 2 y 2 7)? → − − Schemes

Def: a scheme is a locally ringed space that has an • open cover by affine schemes. Think: projective variety can be covered by affine • slices. “Proj construction” takes the (graded) coordinate • ring of a projective variety and spits out a scheme that stands in for the variety. It’s a bit annoying. • The Proj construction

SupposeX P n is a projective variety. • ⊂ Add in a non-closed point to topology for an • irreducible closed subset, just like we did for affines using Spec. Define a sheaf using regular functions in the • O X sense of Shafarevich. That’s a sheaf! • More detail

S aZ-graded ring, meaningS= n Sn and • • S S S . m · n ⊂ m+n � Whole thing is an algebra overS . • 0 Homogenous ideal = generated by homogeneous • elements. S = S • + i>0 i Ex: • � Build up Proj as a set, then a topological space, then • a scheme. cont Iff S is homogeneous, bijection between: • ∈ + Homogenous primes in(S )f (it’s graded: • • deg(1/f)= degf) − Primes in the 0-graded piece Spec((S )f )0 (which isn’t • • graded)

(One way is easy:((S )f )0 (S )f , and a • • → • homogenous prime pull back to a prime.) Def: points of ProjS are homegenous primes not • • containing the irrelevant ideal. Topology

Easy to cook up a graded version of Zariski topology. • V(T) is all homogeneous primes containingT but • notS +. Take those as closed sets. • D(f ) = ProjS V(f). • • \ cont

ThenD(f) corresponds to homogenous primes in • (S ) not containingf. • Which is homogenous primes in(S )f • • Which is Spec((S )f )0. • • This gives our cover by affines, one for each • homogeneous elementf. The structure sheaf

Look at ratios of homogeneous elements of the same • degree, choosing one such ratio in each stalk, plus local compatibility.

On coverD(f), check it restricts to Spec((S )f )0. • • Example

S=k[X,Y,Z] (which gives 2) • P A point is(X 2Z,Y 3Z) (i.e.[2,3,1]) • − − Or(X Y). • − Example

Letf=X. • A point inD(f) is(X 2Z,Y) (ie[2,0,1]) • − A point not inD(f) is(X,Z) (ie[0,1,0]) • The former gives a prime ink[X,Y,Z] . • X The latter doesn’t. • cont

D(f) identified with homogenous primes in • k[X,Y,Z] X . Theorem says those are the same as primes in • (k[X,Y,Z] X )0. Elements look like g(X,Y,Z) ,g homogeneous degd. • X d But that’s just iso to [˛Y,Z]. • So we’ve recovered a standard affine slice. •