Algebraic K-theory of rings from a topological viewpoint
... paper is to describe some of them. This will give us the opportunity to introduce the definition of the groups Ki (R) for all integers i ≥ 0 (in Sections 1, 2 and 3), to explore their structure (in Section 4) and to present classical results (in Sections 5 and 8). Moreover, the second part of the pa ...
... paper is to describe some of them. This will give us the opportunity to introduce the definition of the groups Ki (R) for all integers i ≥ 0 (in Sections 1, 2 and 3), to explore their structure (in Section 4) and to present classical results (in Sections 5 and 8). Moreover, the second part of the pa ...
Splitting of short exact sequences for modules
... where the map I ⊕ J → R is addition, whose kernel is {(x, −x) : x ∈ I ∩ J}, and the map I∩J → I⊕J is x 7→ (x, −x). This is not the short exact sequence 0 −→ I −→ I⊕J −→ J −→ 0 as in Example 1.2, even though the middle modules in both are I ⊕ J. Any short exact sequence that looks like the short exac ...
... where the map I ⊕ J → R is addition, whose kernel is {(x, −x) : x ∈ I ∩ J}, and the map I∩J → I⊕J is x 7→ (x, −x). This is not the short exact sequence 0 −→ I −→ I⊕J −→ J −→ 0 as in Example 1.2, even though the middle modules in both are I ⊕ J. Any short exact sequence that looks like the short exac ...
1 Lecture 13 Polynomial ideals
... vanishing ideal, I(V(f1 , . . . , fs )). How do these two ideals related to each other? Is it always the case that �f1 , . . . , fs � = I(V(f1 , . . . , fs )), and if it is not, what are the reasons? The answer to these questions (and more) will be given by another famous result by Hilbert, known as ...
... vanishing ideal, I(V(f1 , . . . , fs )). How do these two ideals related to each other? Is it always the case that �f1 , . . . , fs � = I(V(f1 , . . . , fs )), and if it is not, what are the reasons? The answer to these questions (and more) will be given by another famous result by Hilbert, known as ...
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000
... the homotopy groups of the sphere, while the range is, by definition, the AdamsNovikov E2 -term for formal A-modules. It is natural to ask whether this map of spectral sequences arises as a filtration of a map S → SA from the sphere spectrum to an algebraic extension of S. Unfortunately, the answer ...
... the homotopy groups of the sphere, while the range is, by definition, the AdamsNovikov E2 -term for formal A-modules. It is natural to ask whether this map of spectral sequences arises as a filtration of a map S → SA from the sphere spectrum to an algebraic extension of S. Unfortunately, the answer ...
Derived funcors, Lie algebra cohomology and some first applications
... Corollary 5.1. If M is a trivial g-modul H 1 (g, M ) ∼ = Der(g, M ) ∼ = Homk (gab , M ). Considering k as a trivial g-module yields, for semisimple and finite dimensional g and char(k) = 0: Corollary 5.2. If g is finite-dimensional and semisimple and char(k) = 0 then H 1 (g, k) = 0. Recall that a mo ...
... Corollary 5.1. If M is a trivial g-modul H 1 (g, M ) ∼ = Der(g, M ) ∼ = Homk (gab , M ). Considering k as a trivial g-module yields, for semisimple and finite dimensional g and char(k) = 0: Corollary 5.2. If g is finite-dimensional and semisimple and char(k) = 0 then H 1 (g, k) = 0. Recall that a mo ...
Final Exam
... and that (α, β) is exact if α and β are closed. d) If f and g are smooth functions on M , let ...
... and that (α, β) is exact if α and β are closed. d) If f and g are smooth functions on M , let ...
(pdf)
... We assume some familiarity with the theory of algebraic curves and algebraic number theory. As such, many definitions will be stated quickly and under the assumption that the reader has seen them before, and results that come purely from one of these fields will be stated without proof. Definition 1 ...
... We assume some familiarity with the theory of algebraic curves and algebraic number theory. As such, many definitions will be stated quickly and under the assumption that the reader has seen them before, and results that come purely from one of these fields will be stated without proof. Definition 1 ...
Algebraic closure
... and is therefore algebraic over F . We conclude that (K, +, · ) ∈ E and that E 4 K for all E in C . Zorn’s Lemma now guarantees the existence of a maximal element F of E . By definition of E , F is an algebraic field extension of F and F ⊆ Ω. It remains to be shown that F is algebraically closed. To ...
... and is therefore algebraic over F . We conclude that (K, +, · ) ∈ E and that E 4 K for all E in C . Zorn’s Lemma now guarantees the existence of a maximal element F of E . By definition of E , F is an algebraic field extension of F and F ⊆ Ω. It remains to be shown that F is algebraically closed. To ...
Two proofs of the infinitude of primes Ben Chastek
... without basis, but many times they can be helpful in group theory. Some clear examples of groups (abelian in this case) are the usual integers, Z, under addition, the rational numbers Q under addition, and the positive real numbers, R+ under multiplication. It is important to note that while Z forms ...
... without basis, but many times they can be helpful in group theory. Some clear examples of groups (abelian in this case) are the usual integers, Z, under addition, the rational numbers Q under addition, and the positive real numbers, R+ under multiplication. It is important to note that while Z forms ...
CW-complexes (some old notes of mine).
... to homotopy-equivalence” is always good enough anyway.) Example 10. Here is a typical example of an infinite-dimensional CWcomplex: let RP ∞ denote the set of lines through the origin in R∞ (a real vector space of countably infinite dimension, with basis e1 , e2 , ...). Regarding R∞ as the ascending ...
... to homotopy-equivalence” is always good enough anyway.) Example 10. Here is a typical example of an infinite-dimensional CWcomplex: let RP ∞ denote the set of lines through the origin in R∞ (a real vector space of countably infinite dimension, with basis e1 , e2 , ...). Regarding R∞ as the ascending ...
Document
... If i is the largest integer greater than 0 for which ai 0, the polynomial is of degree i. If no such i exists, the polynomial is of zero degree. Terms with zero coefficients are generally not written. The set of all polynomials in x over is denoted by R[x]. Algebraic Structures ...
... If i is the largest integer greater than 0 for which ai 0, the polynomial is of degree i. If no such i exists, the polynomial is of zero degree. Terms with zero coefficients are generally not written. The set of all polynomials in x over is denoted by R[x]. Algebraic Structures ...
MATH 176: ALGEBRAIC GEOMETRY HW 3 (1) (Reid 3.5) Let J = (xy
... Find Z(J 0 ) and calculate J 0 . (2) (Reid 3.6) Let J = (x2 + y2 − 1, y − 1). Find f ∈ I(Z(J)) \ J. (3) Let (X, TX ) and (Y, TY ) be topological spaces. The product topology on X × Y is defined by the basis B = {U × V ⊆ X × Y : U ∈ TX , V ∈ TY }. (a) Prove that the product topology is indeed a topol ...
... Find Z(J 0 ) and calculate J 0 . (2) (Reid 3.6) Let J = (x2 + y2 − 1, y − 1). Find f ∈ I(Z(J)) \ J. (3) Let (X, TX ) and (Y, TY ) be topological spaces. The product topology on X × Y is defined by the basis B = {U × V ⊆ X × Y : U ∈ TX , V ∈ TY }. (a) Prove that the product topology is indeed a topol ...
1 Jenia Tevelev
... for any fan supported on trop(Y ). If the structure map is smooth for a toric variety then it will be for any refinement, but these guys proved it for coarsenings as well! Definition 3. A smooth variety Y is called log minimal if for one (and hence any) compactification Y with SNC boundary B, the li ...
... for any fan supported on trop(Y ). If the structure map is smooth for a toric variety then it will be for any refinement, but these guys proved it for coarsenings as well! Definition 3. A smooth variety Y is called log minimal if for one (and hence any) compactification Y with SNC boundary B, the li ...
Algebraic Topology
... ∪K(C,1) K(B,1) is a K(A*CB, 1). Give a counterexample if they are not injections. 4. Show that any complex line bundle over Sn is trivial for n>2. (Change the definition from 2.4 from an R to a C.) 5. Show that if f:S1→ S1 satisfies f(x) = -f(-x), then the element of π1(S1) ≂ Z is odd. If f(x) = f(- ...
... ∪K(C,1) K(B,1) is a K(A*CB, 1). Give a counterexample if they are not injections. 4. Show that any complex line bundle over Sn is trivial for n>2. (Change the definition from 2.4 from an R to a C.) 5. Show that if f:S1→ S1 satisfies f(x) = -f(-x), then the element of π1(S1) ≂ Z is odd. If f(x) = f(- ...
Finally, we need to prove that HomR(M,R∧ ∼ = HomZ(M,Q/Z) To do
... (3) D is an injective Z-module. Proof. It is easy to see that the first two conditions are equivalent. Suppose that x ∈ D and n ≥ 0. Then, A = nZ is a subgroup of the cyclic group B = Z and f : nZ → D can be given by sending the generator n to x. The homomorphism f : nZ → D can be extended to Z if a ...
... (3) D is an injective Z-module. Proof. It is easy to see that the first two conditions are equivalent. Suppose that x ∈ D and n ≥ 0. Then, A = nZ is a subgroup of the cyclic group B = Z and f : nZ → D can be given by sending the generator n to x. The homomorphism f : nZ → D can be extended to Z if a ...
Modules Over Principal Ideal Domains
... Here we present many fundamental definitions to modules. Along the way, we will highlight analogous definitions from vector spaces. The first part of the paper will be full of definition, but once in place, we will have a language to generalize our important results from group theory. First though, ...
... Here we present many fundamental definitions to modules. Along the way, we will highlight analogous definitions from vector spaces. The first part of the paper will be full of definition, but once in place, we will have a language to generalize our important results from group theory. First though, ...
Sets with a Category Action Peter Webb 1. C-sets
... functor over R to be an R-linear functor BCat → R-mod. This notion evidently extends the usual notion of biset functors defined on groups, which are R-linear functors defined on the full subcategory BGroup of BCat whose objects are finite groups. The Burnside ring functor BR (C) := R ⊗Z B(C) is in f ...
... functor over R to be an R-linear functor BCat → R-mod. This notion evidently extends the usual notion of biset functors defined on groups, which are R-linear functors defined on the full subcategory BGroup of BCat whose objects are finite groups. The Burnside ring functor BR (C) := R ⊗Z B(C) is in f ...
Lecture 11
... but in general the last map is not surjective. For example, let X = C − { 0 }. Then z is a nowhere zero function which is not the exponential of any holomorphic function; the logarithm is not a globally well-defined function on the whole punctured plane. Sheaf cohomology is introduced exactly to fix ...
... but in general the last map is not surjective. For example, let X = C − { 0 }. Then z is a nowhere zero function which is not the exponential of any holomorphic function; the logarithm is not a globally well-defined function on the whole punctured plane. Sheaf cohomology is introduced exactly to fix ...
THE ADJUNCTION FORMULA FOR LINE BUNDLES Theorem 1. Let
... Abstract. We give a short report on the adjunction formula for line bundles as it can be found in Griffiths-Harris “Principles of Algebraic Geometry”. ...
... Abstract. We give a short report on the adjunction formula for line bundles as it can be found in Griffiths-Harris “Principles of Algebraic Geometry”. ...
Part C4: Tensor product
... to another exact sequence of the same form: M ⊗ A → M ⊗ B → M ⊗ C → 0. This statement appears stronger than the original statement since the hypothesis is weaker. But I explained that the first statement implies this second version. Suppose that we know that M ⊗ − sends short exact sequences to righ ...
... to another exact sequence of the same form: M ⊗ A → M ⊗ B → M ⊗ C → 0. This statement appears stronger than the original statement since the hypothesis is weaker. But I explained that the first statement implies this second version. Suppose that we know that M ⊗ − sends short exact sequences to righ ...
A remark on the group-completion theorem
... Abstract. Suppose M is a topological monoid satisfying π0 M = N to which the McDuffSegal group-completion theorem applies. We prove that if left- and right-stabilisation commute on H1 (M ), then the ‘McDuff-Segal comparison map’ M∞ → Ω0 BM is acyclic. For example, this always holds if π0 M lies in t ...
... Abstract. Suppose M is a topological monoid satisfying π0 M = N to which the McDuffSegal group-completion theorem applies. We prove that if left- and right-stabilisation commute on H1 (M ), then the ‘McDuff-Segal comparison map’ M∞ → Ω0 BM is acyclic. For example, this always holds if π0 M lies in t ...
(Less) Abstract Algebra
... In this section we introduce the formal notion of a group with the given axioms which a group satisfies. We continue with basic theory of groups but skip some of the more detailed proofs which are available throughout any textbook on abstract algebra. We will include the theorems which will be of im ...
... In this section we introduce the formal notion of a group with the given axioms which a group satisfies. We continue with basic theory of groups but skip some of the more detailed proofs which are available throughout any textbook on abstract algebra. We will include the theorems which will be of im ...
SHIMURA CURVES LECTURE 5: THE ADELIC PERSPECTIVE
... the case where the theory of quadratic forms tells us that any totally indefinite quaternary quadratic form over a number field F is universal, i.e., the map (character!) N : B × → F × is surjective. So certainly there exists some element of B of norm −1. A bit of classical number theory gives the f ...
... the case where the theory of quadratic forms tells us that any totally indefinite quaternary quadratic form over a number field F is universal, i.e., the map (character!) N : B × → F × is surjective. So certainly there exists some element of B of norm −1. A bit of classical number theory gives the f ...
