DISTANCE EDUCATION M.Phil. (Mathematics) DEGREE
... Let R S be domains and S integral over R. Prove that R is a field if and only if S is a field. Let R be an integrally closed domain with quotient field K and S a normal extension of R with Galois group G = G(L/K). Prove that (i) G is the group of R-automorphisms of S (a) ...
... Let R S be domains and S integral over R. Prove that R is a field if and only if S is a field. Let R be an integrally closed domain with quotient field K and S a normal extension of R with Galois group G = G(L/K). Prove that (i) G is the group of R-automorphisms of S (a) ...
The structure of reductive groups - UBC Math
... A split torus T is completely and simply characterized by X ∗ (T ). For one thing, we may describe the F rational points of the torus as Hom(X ∗ (T ), F × ). For another, if we are given an algebraic homomorphism of tori ϕ: S → T , there exists a corresponding map ϕ∗ : X ∗ (T ) → X ∗ (S). This is fu ...
... A split torus T is completely and simply characterized by X ∗ (T ). For one thing, we may describe the F rational points of the torus as Hom(X ∗ (T ), F × ). For another, if we are given an algebraic homomorphism of tori ϕ: S → T , there exists a corresponding map ϕ∗ : X ∗ (T ) → X ∗ (S). This is fu ...
groups and categories
... and join x ∨ y operations, with terminal object 1 and initial object 0 as units, respectively (assuming P has these structures), as well as the poset End(P ) of monotone maps f : P → P , ordered pointwise, with composition g ◦ f as ⊗ and 1P as unit. A discrete monoidal category, i.e. one with a disc ...
... and join x ∨ y operations, with terminal object 1 and initial object 0 as units, respectively (assuming P has these structures), as well as the poset End(P ) of monotone maps f : P → P , ordered pointwise, with composition g ◦ f as ⊗ and 1P as unit. A discrete monoidal category, i.e. one with a disc ...
What is a Dirac operator good for?
... (2) The “Canonical Line Bundle” or “Tautological Line Bundle” or “Hyperplane Bundle” H over CP 1 = S 2 . The canonical line bundle of any complex projective space CP n is the union of the set of all complex lines through the origin in Cn+1 , and the projection to CP n is given by projecting the elem ...
... (2) The “Canonical Line Bundle” or “Tautological Line Bundle” or “Hyperplane Bundle” H over CP 1 = S 2 . The canonical line bundle of any complex projective space CP n is the union of the set of all complex lines through the origin in Cn+1 , and the projection to CP n is given by projecting the elem ...
Connected covers and Neisendorfer`s localization theorem
... B-null spaces includes all finite-dimensional spaces as well as their iterated loop spaces. Thus in the following theorem the spaces are not necessarily finite-dimensional; nor do they necessarily have finite type. Theorem 7.1. Let X be a B-null space. Let Y be a 1-connected space such that Lp (ΩY ) ...
... B-null spaces includes all finite-dimensional spaces as well as their iterated loop spaces. Thus in the following theorem the spaces are not necessarily finite-dimensional; nor do they necessarily have finite type. Theorem 7.1. Let X be a B-null space. Let Y be a 1-connected space such that Lp (ΩY ) ...
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... Example 3. Let X be a smooth differentiable manifold. Let DX be the presheaf on X, with values in the category of real vector spaces, defined by setting DX (U ) to be the space of smooth real–valued functions on U , for each open set U , and with the restriction morphism given by restriction of func ...
... Example 3. Let X be a smooth differentiable manifold. Let DX be the presheaf on X, with values in the category of real vector spaces, defined by setting DX (U ) to be the space of smooth real–valued functions on U , for each open set U , and with the restriction morphism given by restriction of func ...
Grobner
... But, not every algebraic space curve can be defined as the intersection of 2 surfaces. Example where 3 are needed*: twisted cubic (in parametric form): ...
... But, not every algebraic space curve can be defined as the intersection of 2 surfaces. Example where 3 are needed*: twisted cubic (in parametric form): ...
here - Halfaya
... (r2 , s2 ) if r1 s2 − s1 r2 = 0. It’s not hard to show that this is an equivalence relation. So let’s call the representatives of our equivalence classes something familiar. For the equivalence class represented by (r1 , s1 ), we’ll call it sr11 . This should start looking like fractions! Note that ...
... (r2 , s2 ) if r1 s2 − s1 r2 = 0. It’s not hard to show that this is an equivalence relation. So let’s call the representatives of our equivalence classes something familiar. For the equivalence class represented by (r1 , s1 ), we’ll call it sr11 . This should start looking like fractions! Note that ...
LECTURES ON ALGEBRAIC VARIETIES OVER F1 Contents
... Thirty five years later, Smirnov [7], and then Kapranov and Manin, wrote about F1 , viewed as the missing ground field over which number rings are defined. Since then several people studied F1 and tried to define algebraic geometry over it. Today, there are at least seven different definitions of su ...
... Thirty five years later, Smirnov [7], and then Kapranov and Manin, wrote about F1 , viewed as the missing ground field over which number rings are defined. Since then several people studied F1 and tried to define algebraic geometry over it. Today, there are at least seven different definitions of su ...
LOCAL ALGEBRA IN ALGEBRAIC GEOMETRY Contents
... (2) Since codim supp(M ) ≤ codim supp(Mp ), it suffices to prove the claim for a regular local ring (A, m, k). We proceed by induction on n = dim(A). If n = 0, there is nothing to prove. Let i be the smallest integer such that ExtiA (M, A) 6= 0, and set N = ExtiA (M, A). By the induction hypothesis, ...
... (2) Since codim supp(M ) ≤ codim supp(Mp ), it suffices to prove the claim for a regular local ring (A, m, k). We proceed by induction on n = dim(A). If n = 0, there is nothing to prove. Let i be the smallest integer such that ExtiA (M, A) 6= 0, and set N = ExtiA (M, A). By the induction hypothesis, ...
13 Lecture 13: Uniformity and sheaf properties
... Example 13.3.2 The adic space Spa(Z, Z) associated to the noetherian scheme Spec(Z) (where A+ = Z is given the discrete topology) is the final object in the category of adic spaces. There is exactly one valuation in Spa(Z, Z) above the closed point (p) ∈ Spec(Z) (corresponding to the trivial valuati ...
... Example 13.3.2 The adic space Spa(Z, Z) associated to the noetherian scheme Spec(Z) (where A+ = Z is given the discrete topology) is the final object in the category of adic spaces. There is exactly one valuation in Spa(Z, Z) above the closed point (p) ∈ Spec(Z) (corresponding to the trivial valuati ...
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... Monads can also be thought of as theories. As we will see, there is a notion of an algebra of a monad. These are the structured objects produced by the monad-asprocess. In the ultrafilter example above, an algebra would be a set X together with a map from the set of ultrafilters on X to X (subject t ...
... Monads can also be thought of as theories. As we will see, there is a notion of an algebra of a monad. These are the structured objects produced by the monad-asprocess. In the ultrafilter example above, an algebra would be a set X together with a map from the set of ultrafilters on X to X (subject t ...
SHAPIRO`S LEMMA FOR TOPOLOGICAL K
... proper if and only if the underlying group action of G on Z is proper, i.e., if and only if the structural map G × Z → Z × Z; (g, z) 7→ (g · z, z) is proper in the sense that inverse images of compact sets are compact. A map φ : Y → Z between two X o G-spaces Y and Z is X o G-equivariant if φ is G-e ...
... proper if and only if the underlying group action of G on Z is proper, i.e., if and only if the structural map G × Z → Z × Z; (g, z) 7→ (g · z, z) is proper in the sense that inverse images of compact sets are compact. A map φ : Y → Z between two X o G-spaces Y and Z is X o G-equivariant if φ is G-e ...
Cobordism of pairs
... 4. Application of THOMtheory I t follows from the work of T~oM [13], (which we shall suppose known), t h a t cobordism groups for the structural group G, are given by the homotopy groups of M(Gn). Now in all the cases with which we shall be concerned, {Gn} satisfies a certain stability condition. We ...
... 4. Application of THOMtheory I t follows from the work of T~oM [13], (which we shall suppose known), t h a t cobordism groups for the structural group G, are given by the homotopy groups of M(Gn). Now in all the cases with which we shall be concerned, {Gn} satisfies a certain stability condition. We ...
equivariant homotopy and cohomology theory
... There is a great deal of literature on this subject. The original construction of the nonequivariant stable homotopy category was due to Mike Boardman. One must make a sharp distinction between the stable homotopy category, which is xed and unique up to equivalence, and any particular point-set lev ...
... There is a great deal of literature on this subject. The original construction of the nonequivariant stable homotopy category was due to Mike Boardman. One must make a sharp distinction between the stable homotopy category, which is xed and unique up to equivalence, and any particular point-set lev ...
Solutions
... Solutions to selected problems from Homework # 1 2. Let E/F and F/K be separable (algebraic) extensions (but not necessarily finite). Prove that E/K is separable. Solution: We first prove this in the case E/K is a finite extension. As E/F and F/K are separable, we have that [E : F ] = [E : F ]s and ...
... Solutions to selected problems from Homework # 1 2. Let E/F and F/K be separable (algebraic) extensions (but not necessarily finite). Prove that E/K is separable. Solution: We first prove this in the case E/K is a finite extension. As E/F and F/K are separable, we have that [E : F ] = [E : F ]s and ...
Generalized Cohomology
... In the 1950s, several examples of generalized (co)homology theories were discovered. Each of them has its own geometric origin but it turns out that they can be expressed as homotopy sets by using the notion of spectrum. Before we list the axioms for generalized homology and cohomology, let us take ...
... In the 1950s, several examples of generalized (co)homology theories were discovered. Each of them has its own geometric origin but it turns out that they can be expressed as homotopy sets by using the notion of spectrum. Before we list the axioms for generalized homology and cohomology, let us take ...
Introduction to derived algebraic geometry
... spaces with sheaves of cdgas has a model structure, on which dSch happens to be closed under taking holims. On the other hand, h0 (X ×hS Y ) = h0 (X) ×h0 (S) h0 (Y ); thus, homotopy fiber product is a strengthening of the usual notion of fiber product. Remark 3. The π0 map always induces a map h0 (X ...
... spaces with sheaves of cdgas has a model structure, on which dSch happens to be closed under taking holims. On the other hand, h0 (X ×hS Y ) = h0 (X) ×h0 (S) h0 (Y ); thus, homotopy fiber product is a strengthening of the usual notion of fiber product. Remark 3. The π0 map always induces a map h0 (X ...
Completion of rings and modules Let (Λ, ≤) be a directed set. This is
... Xλ ∩ Xµ . The construction for the inverse limit given above yields all functions on these sets with a constant value in the intersection of all of them. This set evidently may be T identified with λ Xλ . We are particularly interested in inverse limit systems indexed by N. To give such a system one ...
... Xλ ∩ Xµ . The construction for the inverse limit given above yields all functions on these sets with a constant value in the intersection of all of them. This set evidently may be T identified with λ Xλ . We are particularly interested in inverse limit systems indexed by N. To give such a system one ...
On the field of definition of superspecial polarized
... Many algebro-geometrical phenomena on an abelian variety A which is the product of supersingular elliptic curves over the field of characteristic p > 0 are related to the arithmetic theory on quaternion hermitian lattices, as was shown, for example, in [3], [10], [11], [8]. Now, in this paper, ...
... Many algebro-geometrical phenomena on an abelian variety A which is the product of supersingular elliptic curves over the field of characteristic p > 0 are related to the arithmetic theory on quaternion hermitian lattices, as was shown, for example, in [3], [10], [11], [8]. Now, in this paper, ...
Cyclic Homology Theory, Part II
... B Twisted homology and Koszul B.1 Quantum plane . . . . . . . . B.1.1 General strategy . . . B.1.2 Step 1 a . . . . . . . . B.1.3 Step 1 b . . . . . . . . B.1.4 Step 1 c . . . . . . . . B.1.5 Step 2 a . . . . . . . . B.1.6 Step 2b . . . . . . . . B.1.7 Step 2c . . . . . . . . B.1.8 Step 2d . . . . . ...
... B Twisted homology and Koszul B.1 Quantum plane . . . . . . . . B.1.1 General strategy . . . B.1.2 Step 1 a . . . . . . . . B.1.3 Step 1 b . . . . . . . . B.1.4 Step 1 c . . . . . . . . B.1.5 Step 2 a . . . . . . . . B.1.6 Step 2b . . . . . . . . B.1.7 Step 2c . . . . . . . . B.1.8 Step 2d . . . . . ...
STRUCTURE THEOREMS OVER POLYNOMIAL RINGS 1
... become projective. This could be done using functor categories, but we choose to present a module-theoretic approach. As is customary, given any ring A (perhaps without an identity element) we let A Mod denote the category of left A-modules and let A mod denote the full subcategory of finitely gener ...
... become projective. This could be done using functor categories, but we choose to present a module-theoretic approach. As is customary, given any ring A (perhaps without an identity element) we let A Mod denote the category of left A-modules and let A mod denote the full subcategory of finitely gener ...
A very brief introduction to étale homotopy
... space K(πn (X), n). Recall that for a group G a connected CW-complex X is called an Eilenberg–MacLane space K(G, n) if the only non-trivial homotopy group of X is πn (X) = G; all such CW-complexes are homotopy equivalent. An important fact in homotopy theory is Whitehead’s theorem that states that a ...
... space K(πn (X), n). Recall that for a group G a connected CW-complex X is called an Eilenberg–MacLane space K(G, n) if the only non-trivial homotopy group of X is πn (X) = G; all such CW-complexes are homotopy equivalent. An important fact in homotopy theory is Whitehead’s theorem that states that a ...
On Gromov`s theory of rigid transformation groups: a dual approach
... ‘evaluation plane field’, G(x) = {X(x) : X ∈ G } (this is not necessarily a continuous plane field). Frobenius’ theorem states that in the ‘degenerate case’ where P is involutive, that is G = P , then through each point of N passes a leaf of P that is a submanifold of dimension d (the same as that o ...
... ‘evaluation plane field’, G(x) = {X(x) : X ∈ G } (this is not necessarily a continuous plane field). Frobenius’ theorem states that in the ‘degenerate case’ where P is involutive, that is G = P , then through each point of N passes a leaf of P that is a submanifold of dimension d (the same as that o ...
Homology and Cohomology
... All closed surfaces can be constructed from triangles by identifying edges. Using only triangles we can construct a large class of 2-dimensional spaces that are not surfaces in strict sense, by identifying more than two edges together. The idea of a ∆-complex is to generalise such consrtuctions in h ...
... All closed surfaces can be constructed from triangles by identifying edges. Using only triangles we can construct a large class of 2-dimensional spaces that are not surfaces in strict sense, by identifying more than two edges together. The idea of a ∆-complex is to generalise such consrtuctions in h ...