Real Algebraic Sets
... Theorem 1.12 Let P and Q be two compact polyhedra. If P and Q are semialgebraically homeomorphic, then they are PL homeomorphic. Let us explain in which sense this result means the uniqueness of semialgebraic triangulation. If h : |K| → A and h0 : |K 0 | → A are semialgebraic triangulation of the co ...
... Theorem 1.12 Let P and Q be two compact polyhedra. If P and Q are semialgebraically homeomorphic, then they are PL homeomorphic. Let us explain in which sense this result means the uniqueness of semialgebraic triangulation. If h : |K| → A and h0 : |K 0 | → A are semialgebraic triangulation of the co ...
Projective ideals in rings of continuous functions
... to show the existence of projective and nonprojective ideals having the same 3-filter. Such examples indicate that the topology of a space is not rich enough to distinguish between the projective and nonprojective ideals in the general setting. In § 3 projectivity within the class of z-ideals is top ...
... to show the existence of projective and nonprojective ideals having the same 3-filter. Such examples indicate that the topology of a space is not rich enough to distinguish between the projective and nonprojective ideals in the general setting. In § 3 projectivity within the class of z-ideals is top ...
On different notions of tameness in arithmetic geometry
... Let X be a normal, noetherian scheme and let X 0 ⊂ X be a dense open subscheme. Assume we are given an an étale covering Y 0 → X 0 . Definition. Let x ∈ X r X 0 be a point. We say that Y 0 → X 0 is unramified along x if it extends to an étale covering of some open subscheme U ⊂ X which contains X 0 ...
... Let X be a normal, noetherian scheme and let X 0 ⊂ X be a dense open subscheme. Assume we are given an an étale covering Y 0 → X 0 . Definition. Let x ∈ X r X 0 be a point. We say that Y 0 → X 0 is unramified along x if it extends to an étale covering of some open subscheme U ⊂ X which contains X 0 ...
Some topics in the theory of finite groups
... G. Thus, if X is a non-empty subset of G, then there exists the smallest subgroup of G containing X. It is denoted by hXi and called the subgroup generated by X. We say that a group G is finitely generated if there exists a finite set X of its elements such that G = hXi. Let G1 and G2 be groups. A m ...
... G. Thus, if X is a non-empty subset of G, then there exists the smallest subgroup of G containing X. It is denoted by hXi and called the subgroup generated by X. We say that a group G is finitely generated if there exists a finite set X of its elements such that G = hXi. Let G1 and G2 be groups. A m ...
FROM COMMUTATIVE TO NONCOMMUTATIVE SETTINGS 1
... normalizing case (aS = Sa holds for all a in the Krull monoid S) many results from the commutative setting generalize. In [Sme13] this approach is, by means of divisorial one-sided ideal theory, extended to a class of semigroups that includes commutative and normalizing Krull monoids as special case ...
... normalizing case (aS = Sa holds for all a in the Krull monoid S) many results from the commutative setting generalize. In [Sme13] this approach is, by means of divisorial one-sided ideal theory, extended to a class of semigroups that includes commutative and normalizing Krull monoids as special case ...
A refinement of the Artin conductor and the base change conductor
... A refinement of the Artin conductor without making any assumptions on the dimension or the ramification behavior. Let us assume that K has positive residue characteristic p. To generalize (1.1), we extend the classes of objects on both sides of that equation. On the left-hand side, we consider anal ...
... A refinement of the Artin conductor without making any assumptions on the dimension or the ramification behavior. Let us assume that K has positive residue characteristic p. To generalize (1.1), we extend the classes of objects on both sides of that equation. On the left-hand side, we consider anal ...
Course Notes (Gross
... the left hand side of (1.1) has an even number of 2’s in it. What about the right-hand side? The prime factorization of b2 has an even number of 2’s in it, and so the prime factorization of 2b2 has an odd number of 2’s in it. This shows that the prime factorization of the right hand side of (1.1) ha ...
... the left hand side of (1.1) has an even number of 2’s in it. What about the right-hand side? The prime factorization of b2 has an even number of 2’s in it, and so the prime factorization of 2b2 has an odd number of 2’s in it. This shows that the prime factorization of the right hand side of (1.1) ha ...
abstract algebra: a study guide for beginners - IME-USP
... ad (mod n), then c ≡ d (mod n). Just be very careful when you cancel! One of the important techniques to understand is how to switch between congruences and ordinary equations. First, any equation involving integers can be converted into a congruence by just reducing modulo n. This works because if ...
... ad (mod n), then c ≡ d (mod n). Just be very careful when you cancel! One of the important techniques to understand is how to switch between congruences and ordinary equations. First, any equation involving integers can be converted into a congruence by just reducing modulo n. This works because if ...
Modular functions and modular forms
... Arithmetic of Modular Curves. The theorem shows that we can regard X.N / as an algebraic curve, defined by some homogeneous polynomial(s) with coefficients in C. The central fact underlying the arithmetic of the modular curves (and hence of modular functions and modular forms) is that this algebraic ...
... Arithmetic of Modular Curves. The theorem shows that we can regard X.N / as an algebraic curve, defined by some homogeneous polynomial(s) with coefficients in C. The central fact underlying the arithmetic of the modular curves (and hence of modular functions and modular forms) is that this algebraic ...
Haar Measure on LCH Groups
... The above four propositions are all direct results of the continuity of group operations. For part 1, fix x ∈ G, we know y 7→ xy is a homeomorphism. Hence U open ⇐⇒ xU open. And similar arguments work for U x. For part 2, WLOG we assume U is open (otherwise just work on the interior of U ). Then U − ...
... The above four propositions are all direct results of the continuity of group operations. For part 1, fix x ∈ G, we know y 7→ xy is a homeomorphism. Hence U open ⇐⇒ xU open. And similar arguments work for U x. For part 2, WLOG we assume U is open (otherwise just work on the interior of U ). Then U − ...
Birkhoff's representation theorem
This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and Birkhoff's HSP theorem representing algebras as products of irreducible algebras. Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices.