S1-Equivariant K-Theory of CP1
... If X is a G -space and {Ex : x ∈ X } is a collection of finite dimensional vector spaces, we call E = tx∈X Ex a G -vector bundle if it is equipped with certain topological structure and a continuous action of G such that g .Ex = Eg .x . Example. (The Hopf Bundle) A canonical example of a S 1 -bundle ...
... If X is a G -space and {Ex : x ∈ X } is a collection of finite dimensional vector spaces, we call E = tx∈X Ex a G -vector bundle if it is equipped with certain topological structure and a continuous action of G such that g .Ex = Eg .x . Example. (The Hopf Bundle) A canonical example of a S 1 -bundle ...
Functors and natural transformations A covariant functor F : C → D is
... For W a topological space, let [B, W ] be the set of homotopy classes of maps f : B → W . Theorem For each topological group G, there is a space BG called the classifying space of G and an isomorphism of functors between FG and the functor B → [B, BG]. The homotopy class of the identity map BG → BG ...
... For W a topological space, let [B, W ] be the set of homotopy classes of maps f : B → W . Theorem For each topological group G, there is a space BG called the classifying space of G and an isomorphism of functors between FG and the functor B → [B, BG]. The homotopy class of the identity map BG → BG ...
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... 1. A subset A ⊂ V (K) is said to be of type C1 if there is a closed subset W ⊂ V , with W 6= V , such that A ⊂ W (K). In other words, A is not dense in V (with respect to the Zariski topology). 2. A subset A ⊂ V (K) is said to be of type C2 if there is an irreducible variety V 0 of the same dimensio ...
... 1. A subset A ⊂ V (K) is said to be of type C1 if there is a closed subset W ⊂ V , with W 6= V , such that A ⊂ W (K). In other words, A is not dense in V (with respect to the Zariski topology). 2. A subset A ⊂ V (K) is said to be of type C2 if there is an irreducible variety V 0 of the same dimensio ...
Exercise sheet 5
... 2. (a) You have 7 pieces of paper, and you apply the following procedure as many times as you want: Pick any one of your pieces of paper and cut it in 7. Show that you can never get 1997 pieces of paper. Hint: Think modulo 6. (b) Find the remainder in the division by 3 of each of the following numbe ...
... 2. (a) You have 7 pieces of paper, and you apply the following procedure as many times as you want: Pick any one of your pieces of paper and cut it in 7. Show that you can never get 1997 pieces of paper. Hint: Think modulo 6. (b) Find the remainder in the division by 3 of each of the following numbe ...
Categories and functors
... b. I will say as little as you let me about set theory. It suffices to make a naive distinction between small and large collections, which can be interpreted as meaning ‘sets’ and ‘proper classes’ respectively. A category A is locally small if A(A, B) is a small collection for each A and B. (Many a ...
... b. I will say as little as you let me about set theory. It suffices to make a naive distinction between small and large collections, which can be interpreted as meaning ‘sets’ and ‘proper classes’ respectively. A category A is locally small if A(A, B) is a small collection for each A and B. (Many a ...
RIGID RATIONAL HOMOTOPY THEORY AND
... Question. Is there a way to ’do algebraic topology on X’ ? More specifically, we can ask if there are cohomology functors H i pXq which is some way behave like the singular cohomology X ÞÑ H i pXpCq, Zq for C-varieties X? Famously, the answer to this question is yes, if one is prepared to work with ...
... Question. Is there a way to ’do algebraic topology on X’ ? More specifically, we can ask if there are cohomology functors H i pXq which is some way behave like the singular cohomology X ÞÑ H i pXpCq, Zq for C-varieties X? Famously, the answer to this question is yes, if one is prepared to work with ...
Algebraic Models for Homotopy Types EPFL July 2013 Exercises 1
... square, if one arrow is an acyclic cofibration, so is the parallel arrow. (Hint: Use the previous exercise.) 3. Let X be an object in a closed model category and (IX, ∂0 , ∂1 , σ) a cylinder object. Assume that X is cofibrant. (a) Prove that the maps ∂0 : X → IX and ∂1 : X → IX are cofibrations. (b) ...
... square, if one arrow is an acyclic cofibration, so is the parallel arrow. (Hint: Use the previous exercise.) 3. Let X be an object in a closed model category and (IX, ∂0 , ∂1 , σ) a cylinder object. Assume that X is cofibrant. (a) Prove that the maps ∂0 : X → IX and ∂1 : X → IX are cofibrations. (b) ...
7. A1 -homotopy theory 7.1. Closed model categories. We begin with
... which is the identity on objects and whose set of morphisms from X to Y equals the set of homotopy classes of morphisms from some fibrant/cofibrant replacement of X to some fibrant/cofibrant replacement of Y : HomHo(C) (X, Y ) = π(RQX, RQY ). If F : C → D if a functor with the property that F sends ...
... which is the identity on objects and whose set of morphisms from X to Y equals the set of homotopy classes of morphisms from some fibrant/cofibrant replacement of X to some fibrant/cofibrant replacement of Y : HomHo(C) (X, Y ) = π(RQX, RQY ). If F : C → D if a functor with the property that F sends ...
THE GEOMETRY OF FORMAL VARIETIES IN ALGEBRAIC
... geometry. We think of the nilpotent elements of a ring R as being infinitesimal. Relatedly, there’s a very useful category similar to our category of schemes. Let R be a topologized augmented k-algebra whose augmentation ideal I is topologically nilpotent — the category of such objects is called the ...
... geometry. We think of the nilpotent elements of a ring R as being infinitesimal. Relatedly, there’s a very useful category similar to our category of schemes. Let R be a topologized augmented k-algebra whose augmentation ideal I is topologically nilpotent — the category of such objects is called the ...
UIUC Math 347H Lecture 6: Discussion questions Equivalence
... If also cancellation (C) ALSO holds (ab = ac and a 6= 0 implies b = c) then R is an integral domain. Consider (M5): if a 6= 0 then there is an inverse a−1 such that aa−1 = a−1 a = 1. If (M5) ALSO holds, we say R is a field. 5. Let R = Z[x1 , . . . , xn ] be the set of polynomials with integer coeffi ...
... If also cancellation (C) ALSO holds (ab = ac and a 6= 0 implies b = c) then R is an integral domain. Consider (M5): if a 6= 0 then there is an inverse a−1 such that aa−1 = a−1 a = 1. If (M5) ALSO holds, we say R is a field. 5. Let R = Z[x1 , . . . , xn ] be the set of polynomials with integer coeffi ...
What Is...a Topos?, Volume 51, Number 9
... of toposes without points (he has also given criteria for the existence of “enough points”) [SGA 4 IV 7, VI 9]. Moreover, if x and y are points of a topos T , there may exist nontrivial morphisms (of functors) from x to y . In the case of BG , for example, the forgetful functor from BG to {pt} is th ...
... of toposes without points (he has also given criteria for the existence of “enough points”) [SGA 4 IV 7, VI 9]. Moreover, if x and y are points of a topos T , there may exist nontrivial morphisms (of functors) from x to y . In the case of BG , for example, the forgetful functor from BG to {pt} is th ...
Algebraic Number Theory
... multiplication and is a commutative division algebra • Other Useful information • Finite group theory • Commutative rings and quotient rings • Elementary number theory ...
... multiplication and is a commutative division algebra • Other Useful information • Finite group theory • Commutative rings and quotient rings • Elementary number theory ...
