Topology MA Comprehensive Exam
... 4. Prove that the closed interval [0, 1] considered as a subset of R in the usual topology is compact. 5. Let G be a topological group that acts on a topological space X. Prove that if both G and the quotient space are X/G connected then X is connected. 6. A topological space X is said to be locally ...
... 4. Prove that the closed interval [0, 1] considered as a subset of R in the usual topology is compact. 5. Let G be a topological group that acts on a topological space X. Prove that if both G and the quotient space are X/G connected then X is connected. 6. A topological space X is said to be locally ...
1. Lecture 4, February 21 1.1. Open immersion. Let (X,O X) be a
... isomorphism fi] to pass over to the structure sheaf on W . Concretely, we have the isomorphism (fi , fi] ) : (W, OW ) −→ (Wi , OX|Wi ), where Wi = fi (W ) is the open subset of Ui . Use these isomorphisms to identify the sections over W1 with sections over W2 . Example 1.3. Consider U = Spec(k[x]) t ...
... isomorphism fi] to pass over to the structure sheaf on W . Concretely, we have the isomorphism (fi , fi] ) : (W, OW ) −→ (Wi , OX|Wi ), where Wi = fi (W ) is the open subset of Ui . Use these isomorphisms to identify the sections over W1 with sections over W2 . Example 1.3. Consider U = Spec(k[x]) t ...
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... Using the notion of ordered pair, we soon get the very important structure called the product A × B of two sets A and B. Next, we can get such things as equivalence relations and order relations on a set A, for they are subsets of A×A. And we get the critical notion of a function AQ→ B, as a subset ...
... Using the notion of ordered pair, we soon get the very important structure called the product A × B of two sets A and B. Next, we can get such things as equivalence relations and order relations on a set A, for they are subsets of A×A. And we get the critical notion of a function AQ→ B, as a subset ...
Problems for Category theory and homological algebra
... 2. (a) Show that the embedding Z → Q is an epimorphism in Ring. (b) An (additively written) abelian group A is called divisible if for each a ∈ A and each n ∈ Z>0 there exists an element b ∈ A such that nb = a. The category of divisible abelian groups (with group homomorphisms as morphisms) is denot ...
... 2. (a) Show that the embedding Z → Q is an epimorphism in Ring. (b) An (additively written) abelian group A is called divisible if for each a ∈ A and each n ∈ Z>0 there exists an element b ∈ A such that nb = a. The category of divisible abelian groups (with group homomorphisms as morphisms) is denot ...
A quasi-coherent sheaf of notes
... Remark. The inverse image can be defined from the category of presheaves to sheaves. An important case is when f is the identity; this is sheafification. This can be described explicitly when one is working with the category of abelian groups (or rings, say). Let G be a presheaf on X in such a categ ...
... Remark. The inverse image can be defined from the category of presheaves to sheaves. An important case is when f is the identity; this is sheafification. This can be described explicitly when one is working with the category of abelian groups (or rings, say). Let G be a presheaf on X in such a categ ...
