Morphisms of Algebraic Stacks

... then j is the structure morphism G → S. Hence the diagonal is not automatically separated itself (contrary to what happens in the case of schemes and algebraic spaces). To say that [S/G] is quasi-separated over S should certainly imply that G → S is quasi-compact, but we hesitate to say that [S/G] i ...

... then j is the structure morphism G → S. Hence the diagonal is not automatically separated itself (contrary to what happens in the case of schemes and algebraic spaces). To say that [S/G] is quasi-separated over S should certainly imply that G → S is quasi-compact, but we hesitate to say that [S/G] i ...

Abstract Algebra - UCLA Department of Mathematics

... multiply them by scalars. In abstract algebra, we attempt to provide lists of properties that common mathematical objects satisfy. Given such a list of properties, we impose them as “axioms”, and we study the properties of objects that satisfy these axioms. The objects that we deal with most in the ...

... multiply them by scalars. In abstract algebra, we attempt to provide lists of properties that common mathematical objects satisfy. Given such a list of properties, we impose them as “axioms”, and we study the properties of objects that satisfy these axioms. The objects that we deal with most in the ...

Commutative ideal theory without finiteness

... Moreover, every nonzero fractional R-ideal has a unique representation as an irredundant intersection of infinitely many completely Q-irreducible R-submodules of Q. It follows that R has no nonzero fractional ideal that is Q-irreducible. In Section 2 we establish basic properties of irreducible subm ...

... Moreover, every nonzero fractional R-ideal has a unique representation as an irredundant intersection of infinitely many completely Q-irreducible R-submodules of Q. It follows that R has no nonzero fractional ideal that is Q-irreducible. In Section 2 we establish basic properties of irreducible subm ...

An Introduction to Algebraic Number Theory, and the Class Number

... The extension E/F is algebraic (or E is algebraic over F ) if every element of E is algebraic over F . Let E/F be a field extension, and let α ∈ E be algebraic over F . The minimal polynomial of α over F , denoted irr(α, F ), is the unique monic irreducible polynomial f ∈ F [X] such that f (α) = 0. ...

... The extension E/F is algebraic (or E is algebraic over F ) if every element of E is algebraic over F . Let E/F be a field extension, and let α ∈ E be algebraic over F . The minimal polynomial of α over F , denoted irr(α, F ), is the unique monic irreducible polynomial f ∈ F [X] such that f (α) = 0. ...

STABLE CANONICAL RULES 1. Introduction It is a well

... let X∗ = (X ∗ , ♦), where X ∗ is the Boolean algebra of clopens of X and ♦(U ) = R−1 [U ]. For a bounded morphism f : X → Y , its dual f ∗ : Y ∗ → X ∗ is given by f −1 . Let A = (A, ♦) be a modal algebra and let X = (X, R) be its dual space. Then it is well known that R is reflexive iff a ≤ ♦a, and ...

... let X∗ = (X ∗ , ♦), where X ∗ is the Boolean algebra of clopens of X and ♦(U ) = R−1 [U ]. For a bounded morphism f : X → Y , its dual f ∗ : Y ∗ → X ∗ is given by f −1 . Let A = (A, ♦) be a modal algebra and let X = (X, R) be its dual space. Then it is well known that R is reflexive iff a ≤ ♦a, and ...

Algebra: Monomials and Polynomials

... • Sage 3.x and later[Ste08]; • Lyx [Lyx ] (and therefore LATEX [Lam86, Grä04] (and therefore TEX [Knu84])), along with the packages ...

... • Sage 3.x and later[Ste08]; • Lyx [Lyx ] (and therefore LATEX [Lam86, Grä04] (and therefore TEX [Knu84])), along with the packages ...

Ergodic theory lecture notes

... However, one might expect (1.1.1) to hold for ‘typical’ points x ∈ X (where again we can make ‘typical’ precise using measure theory). One might also want to replace the function χ[a,b] with an arbitrary function f : X → R. In this case one would want to ask: for the doubling map T , when is it the ...

... However, one might expect (1.1.1) to hold for ‘typical’ points x ∈ X (where again we can make ‘typical’ precise using measure theory). One might also want to replace the function χ[a,b] with an arbitrary function f : X → R. In this case one would want to ask: for the doubling map T , when is it the ...

ON THE REPRESENTABILITY OF ACTIONS IN A SEMI

... exact since so is E (see [2] 5.11). It is protomodular by [8] 3.1.16. It is thus semi-abelian. One concludes by statement 3 and proposition 1.4. In this paper, we consider ﬁrst a certain number of other basic examples, where the functor Act(−, X) is representable by an easily describable object. And ...

... exact since so is E (see [2] 5.11). It is protomodular by [8] 3.1.16. It is thus semi-abelian. One concludes by statement 3 and proposition 1.4. In this paper, we consider ﬁrst a certain number of other basic examples, where the functor Act(−, X) is representable by an easily describable object. And ...

The constant term of tempered functions on a real spherical

... Then, by changing the element f f into aZ,E f f , we define a new choice W for which the polar decomposition (1.5) is valid and its elements satisfy (1.6). The elements of the original W satisfy aw ¨ z0 “ w ¨ z0 (cf. [11, Lemma 3.5 and its proof]). As the elements of the new set W are obtained by mu ...

... Then, by changing the element f f into aZ,E f f , we define a new choice W for which the polar decomposition (1.5) is valid and its elements satisfy (1.6). The elements of the original W satisfy aw ¨ z0 “ w ¨ z0 (cf. [11, Lemma 3.5 and its proof]). As the elements of the new set W are obtained by mu ...

IDEAL FACTORIZATION 1. Introduction We will prove here the

... for some nonzero primes pi . Use such a product where r is minimal. If r = 1 then p ⊃ (x) ⊃ p1 , so p = p1 since both ideals are maximal. Thus p = (x), so e p = (1/x)OK 6= OK , which is what we wanted (with y = 1). Thus we may suppose r ≥ 2. Since p ⊃ (x) ⊃ p1 · · · pr , p = pi for some i by Corolla ...

... for some nonzero primes pi . Use such a product where r is minimal. If r = 1 then p ⊃ (x) ⊃ p1 , so p = p1 since both ideals are maximal. Thus p = (x), so e p = (1/x)OK 6= OK , which is what we wanted (with y = 1). Thus we may suppose r ≥ 2. Since p ⊃ (x) ⊃ p1 · · · pr , p = pi for some i by Corolla ...

Form Methods for Evolution Equations, and Applications

... (c) First we show that, given x ∈ X, the orbit T (·)x is continuous. As the restriction of T to [0, ∞) is a C0 -semigroup it follows from (b) that T (·)x is continuous on [0, ∞). Let t 6 0. Then T (t + h)x − T (t)x = T (t − 1)(T (1 + h)x − T (1)x) → 0 (h → 0), and this implies that T (·)x is continu ...

... (c) First we show that, given x ∈ X, the orbit T (·)x is continuous. As the restriction of T to [0, ∞) is a C0 -semigroup it follows from (b) that T (·)x is continuous on [0, ∞). Let t 6 0. Then T (t + h)x − T (t)x = T (t − 1)(T (1 + h)x − T (1)x) → 0 (h → 0), and this implies that T (·)x is continu ...

Real Algebraic Sets

... map f : A → Rp is said to be semialgebraic if its graph Γ(f ) ⊂ Rn × Rp = Rn+p is semialgebraic. For instance, the polynomial maps and the regular maps (i.e. those maps whose coordinates are rational functions such√that the denominator does not vanish) are semialgebraic. The function x 7→ 1 − x2 for ...

... map f : A → Rp is said to be semialgebraic if its graph Γ(f ) ⊂ Rn × Rp = Rn+p is semialgebraic. For instance, the polynomial maps and the regular maps (i.e. those maps whose coordinates are rational functions such√that the denominator does not vanish) are semialgebraic. The function x 7→ 1 − x2 for ...

IDEAL FACTORIZATION 1. Introduction

... for some nonzero primes pi . Use such a product where r is minimal. If r = 1 then p ⊃ (x) ⊃ p1 , so p = p1 since both ideals are maximal. Thus p = (x), so e p = (1/x)OK 6= OK , which is what we wanted (with y = 1). Thus we may suppose r ≥ 2. Since p ⊃ (x) ⊃ p1 · · · pr , p = pi for some i by Corolla ...

... for some nonzero primes pi . Use such a product where r is minimal. If r = 1 then p ⊃ (x) ⊃ p1 , so p = p1 since both ideals are maximal. Thus p = (x), so e p = (1/x)OK 6= OK , which is what we wanted (with y = 1). Thus we may suppose r ≥ 2. Since p ⊃ (x) ⊃ p1 · · · pr , p = pi for some i by Corolla ...