LECTURE 13
MARIA VEGA
Abstract. From Dr. Homans lecture on October 3, 2005 transcribed by Maria Vega.
1. Projective V-I and The Affine Cone
Key Point: If F (x0 , ., xn ) k [x0 , ., xn ] where k is a eld, then
a = [a0 , ., an ] = [a0 , ., an ] Pn (k ) wher
Lecture 11: Proof of the NSZ
11.1 Theorem (NSZ): Let k be an algebraically closed eld. (k = k )
Let J k [x1 , . . . , xn ] be an ideal. If J = k [x1 , . . . , xn ], then V(J ) = .
Rather than prove this directly, we shall instead prove an equivalent
theor
LECTURE 10: INTEGRAL EXTENSIONS
GIRJA SHANKER TRIPATHI
In all the discussions that follow A is a commutative ring with identity.
1. Definitions
Denition 1.1. An Amodule M is said to be nitely generated, if there exist
elements m1 , ., mn M such that every
LECTURE 7
PIOTR MACIAK
Abstract. This lecture provides some examples that ilustrate the
theory we have developed so far.
1. Points in An
Proposition 1.1. Any nite set of points in An is an ane algebraic
set.
Proof. Let a = (a1 , . . . , an ) An and Ma = x
6. Zariski Topology
Denition 6.1. A topological space X is Noetherian if it satises the
descending chain condition for closed sets, that is for closed sets Yi if
Y1 Y2 , then for some R, we have Yr = Yr+1 for all r R.
Example 6.2. (i) An (k ) with the Zar
LECTURE 23
KARLI MORRIS
Sheaf Theory Part 3
Denition 0.1. Let B be a basis for the topology of X . We say that F is a
B -sheaf on X if for all U B , we have F (U ) Ab, and if for all V U B ,
we have F U : F (U ) F (V ) in Ab.
V
Now we will discuss transit
33. Functor of Points, Fiber Product
For x C (a category) we have a map hX : Co (Sets) = E
dened by T HomC (T, X ). We introduce the following notation:
X (T ) = HomC (T, X ) = hX (T ).
Denition 33.1. Call X (T ) the T -valued points of X . For us C =
Sch
Lecture 35: S -schemes II
35.1 Example: S = Spec(k [t]) = A1 ,
k=k
k
X = Spec (k [x, y ]/ < xy >) V (xy ) A2
k
< xy >=< x > < y >, so V (< xy >) = V < x > V < y >
X = Spec (k [x, y ]/ < xy >) A2
k
f
S = Spec (k [t])
A2 (R) = R2 (x, y )
k
g
1
Ak (R) = R x
Lecture 24: Sheaf Theory IV, the Finale
24.1 Denition: A morphism of sheaves on X , : F G, is
injective (resp. surjective) if for all x X , x : Fx Gx is injective
(resp. surjective).
24.2 Proposition: a) A morphism of sheaves is injective i for all open
s
LECTURE 36: MORE EXAMPLES
GIRJA SHANKER TRIPATHI
We continue our discussion of examples. In all these discussions we consider only algebraically closed elds. Well denote any such eld by k ; that
is, k = k , the closure of k .
k [x,y ]
Example 0.1. Let S =
LECTURE 37
MARIA VEGA
Abstract. From Dr. Homans lecture on December 2, 2005
transcribed by Maria Vega.
1. Affine and Projective Spaces over Z
Proposition 1.1. The prime ideals of Z[x] are:
(1) 0 , with height = 0.
(2) p where p Z is a prime. The height of
5. Morphisms
Denition 5.1. If X Am (k ), Y An (k ) are algebraic sets, a morphism is a
regular map f : X Y given by polynomials.
a = (a1 , . . . , am ) X Am (k )
b = (b1 , . . . , bm ) Y An (k )
There exists polynomials f1 , . . . , fm k [x1 , . . . , xm
2. Noetherian Rings and Modules
Denition 2.0. A ring R is Noetherian if one of the following equivalent conditions
are satised:
(1) Every chain of ideals in R
I1 I 2 I n
is eventually stationary. That is, there exists an n such that
In = In+1 = .
(2) Eve
LECTURE 15
MATTHEW BENNETT
1. Rational Functions and Regular Functions
Denition 1.1. A quasi-projective variety(algebraic set) is a Zariski open
subset X Y where Y is a projective variety(algebraic set).
Example 1.2. Examples of a quasi-projective varieti
Maiia Bakhova
Algebraic Geometry
Lecture 1.
ALGEBRAIC VARIETIES.
1. Notation
A eld with which we are working will be denoted by k . It is assumed that it is
algebraically closed: k = k , and it can be C, Fp , or Q. Well work with polynomial rings
k [x1 ,
16. Morphisms I
Denition 16.1. Let X and Y be varieties. A morphism is a continuous map
f : X Y with the following property: for every open set U Y , the map
f = f () sends any OY (U ) to f OX (f 1 (U ).
In this setup we are assuming Y Y Pn . Recall that
Lecture 17: Morphisms II
17.1 A Basis for the Zariski Topology
We begin with a proposition:
Proposition 1 Let X be an irreducible Zariski-closed subset of An , and f an element
of its coordinate ring k [X ]. Let Xf be the complement of the zero locus of f
LECTURE 21 - SHEAF THEORY II
PIOTR MACIAK
Abstract. This lecture develops the ideas introduced in Lecture
20. In particular, we dene a stalk of a sheaf and use this concept
along with the concept of a sheaf of continuous sections to show
that there is a n
20. Sheaf Theory
Let X be a topological space. Then dene Open(X ) = set of open
subsets of X . There is a natural partial ordering of open sets by inclusion. In fact, any partially ordered set is a category where the objects
are the sets and the morphisms
18. Birational Maps
These are notes taken October 14, 2005, transcribed by Charles
Egedy.
Denition 18.1. Rational, Dominant and Birational Maps
(1) If X, Y are varieties, a rational map is an equivalence class of
pairs (U, ), where = U X is open and : U Y
Maiia Bakhova
Algebraic Geometry
Lecture 14.
PROJECTIVE CLOSURE.
Consider an open set in the Zariski topology
Ui = Pn \ V (xi ) = cfw_[x0 , . . . , xn ] Pn |xi = 0.
Note that a conventional picture for a projective space (without a common zero for all var
4. Nullstellensatz
These are notes from September 9, 2005, transcribed by Charles
Egedy. The literal translation of Nullstellensatz, abreviated NSZ, is
theorem about location of zeroes
Theorem 4.1. Let k be an algebraically closed eld. The following are
e
LECTURE 3
MATTHEW BENNETT
1. V -I Correspondence
Recall that the following correspondence exists.
V
ideals
J k [x1 , . . . xm ]
I
subsets
X A(k )
Proposition 1.1. Fundamental Properties of the V -Correspondence
(i) V (0) = Am and V (A) = where A = k [x1 ,
Lecture 8: Examples
Today we look at some more examples of ane algebraic varieties.
8.1 Dimension
Denition 1 Let X be an irreducible algebraic set in An (that is, an ane algebraic
variety). We dene the dimension of X to be the Krull dimension of the coord
Applications of Homological Algebra Spring 2007
Introduction to Perverse Sheaves P. Achar
Problem Set 12
April 26, 2007 In all of the following problems, X is a stratied space with stratication S, and X has a unique open stratum U . All perverse sheaves a
Applications of Homological Algebra Spring 2007
Introduction to Perverse Sheaves P. Achar
Problem Set 11
April 12, 2007 In all of the following problems, X is a stratied space with stratication S, and p : S Z is a perversity function.
b 1. Prove that Dc (
Applications of Homological Algebra Spring 2007
Introduction to Perverse Sheaves P. Achar
Problem Set 10
March 29, 2007
1. A functor F : C1 C2 between two triangulated categories with t-structures is said to be t-exact if 0 0 0 0 F (C1 ) C2 and F (C1 ) C2
Applications of Homological Algebra Spring 2007
Introduction to Perverse Sheaves P. Achar
Problem Set 9
March 22, 2007 In this problem set, you will show that the derived category of the heart of a t-structure on a triangulated category need not be equiva