A primer on homotopy colimits
... This is an expository paper on homotopy colimits and homotopy limits. These are constructions which should arguably be in the toolkit of every modern algebraic topologist, yet there does not seem to be a place in the literature where a graduate student can easily read about them. Certainly there are ...
... This is an expository paper on homotopy colimits and homotopy limits. These are constructions which should arguably be in the toolkit of every modern algebraic topologist, yet there does not seem to be a place in the literature where a graduate student can easily read about them. Certainly there are ...
Lecture Notes for Math 614, Fall, 2015
... equations? No algorithm is known for settling questions of this sort, and many are open, even for relatively small specific examples. In the example considered here, it turns out that 3 equations are needed. I do not know an elementary proof of this fact — perhaps you can find one! One of the themes ...
... equations? No algorithm is known for settling questions of this sort, and many are open, even for relatively small specific examples. In the example considered here, it turns out that 3 equations are needed. I do not know an elementary proof of this fact — perhaps you can find one! One of the themes ...
model categories of diagram spectra
... category of finite based sets, and W is the category of based spaces homeomorphic to finite CW complexes. We often use D generically to denote such a domain category for diagram spectra. When D = F or D = W , there is no distinction between Dspaces and D-spectra, DT = DS . The functors U are forgetf ...
... category of finite based sets, and W is the category of based spaces homeomorphic to finite CW complexes. We often use D generically to denote such a domain category for diagram spectra. When D = F or D = W , there is no distinction between Dspaces and D-spectra, DT = DS . The functors U are forgetf ...
A conjecture in Rational Homotopy
... How to kill the torsion? Let M be a Z-module, the structure theorem for finitely generated modules over a principal ideal domain gives a detailed description of the module M up to isomorphism. In particular, it claims that M ' F ⊕ T (M) where F is a free Z-module of finite rank (so without torsion) ...
... How to kill the torsion? Let M be a Z-module, the structure theorem for finitely generated modules over a principal ideal domain gives a detailed description of the module M up to isomorphism. In particular, it claims that M ' F ⊕ T (M) where F is a free Z-module of finite rank (so without torsion) ...
Model Categories and Simplicial Methods
... Model Categories and Simplicial Methods Paul Goerss and Kristen Schemmerhorn Abstract. There are many ways to present model categories, each with a different point of view. Here we’d like to treat model categories as a way to build and control resolutions. This an historical approach, as in his orig ...
... Model Categories and Simplicial Methods Paul Goerss and Kristen Schemmerhorn Abstract. There are many ways to present model categories, each with a different point of view. Here we’d like to treat model categories as a way to build and control resolutions. This an historical approach, as in his orig ...
Positivity for toric vectorbundles
... Remark 2.2. Note that if E is a vector bundle on an arbitrary complete variety X, then E is nef if and only if for every irreducible curve C ⊂ X, the restriction E|C is nef (this simply follows from the fact that every curve in P(E) is contained in some P(E|C )). The similar criterion for ampleness ...
... Remark 2.2. Note that if E is a vector bundle on an arbitrary complete variety X, then E is nef if and only if for every irreducible curve C ⊂ X, the restriction E|C is nef (this simply follows from the fact that every curve in P(E) is contained in some P(E|C )). The similar criterion for ampleness ...
Basic Modern Algebraic Geometry
... Condition 1.1.1.2 Composition of morphisms is associative, in the sense that whenever one side in the below equality is defined, so is the other and equality holds: (ϕ ◦ ψ) ◦ ξ = ϕ ◦ (ψ ◦ ξ) ...
... Condition 1.1.1.2 Composition of morphisms is associative, in the sense that whenever one side in the below equality is defined, so is the other and equality holds: (ϕ ◦ ψ) ◦ ξ = ϕ ◦ (ψ ◦ ξ) ...
Chapter IV. Quotients by group schemes. When we work with group
... (4.5) Definition. Let C be a category with finite products. Let G be a group object in C. Let X be an object of C. Throughout, we simply write X(T ) for hX (T ) = HomC (T, X). (i) A (left) action of G on X is a morphism ρ: G × X → X that induces, for every object T , a (left) action of the group G(T ...
... (4.5) Definition. Let C be a category with finite products. Let G be a group object in C. Let X be an object of C. Throughout, we simply write X(T ) for hX (T ) = HomC (T, X). (i) A (left) action of G on X is a morphism ρ: G × X → X that induces, for every object T , a (left) action of the group G(T ...
lecture notes
... Actually, a tendency to generalization has always been present in Algebra. Initially, this was the case with the successive generalizations of the concept of number: first from natural to positive rational, then to negative numbers, irrational numbers and complex numbers. In the XIX Century, mathema ...
... Actually, a tendency to generalization has always been present in Algebra. Initially, this was the case with the successive generalizations of the concept of number: first from natural to positive rational, then to negative numbers, irrational numbers and complex numbers. In the XIX Century, mathema ...
HIGHER HOMOTOPY OF GROUPS DEFINABLE IN O
... would be easy to prove if one could show (by analogy with the torus) that G factors definably into one-dimensional subgroups (as πn (G) would also factor), but in general this is not the case (here we measure the effect of the lack of the exponential maps). Since πn (G) can be proved to be divisible ...
... would be easy to prove if one could show (by analogy with the torus) that G factors definably into one-dimensional subgroups (as πn (G) would also factor), but in general this is not the case (here we measure the effect of the lack of the exponential maps). Since πn (G) can be proved to be divisible ...
Modules and Vector Spaces
... Modules are a generalization of the vector spaces of linear algebra in which the “scalars” are allowed to be from an arbitrary ring, rather than a field. This rather modest weakening of the axioms is quite far reaching, including, for example, the theory of rings and ideals and the theory of abelian ...
... Modules are a generalization of the vector spaces of linear algebra in which the “scalars” are allowed to be from an arbitrary ring, rather than a field. This rather modest weakening of the axioms is quite far reaching, including, for example, the theory of rings and ideals and the theory of abelian ...
Algebraic models for rational G
... Cohomology theories are very important in algebraic topology as they are invariants for topological spaces. A cohomology theory E ∗ is a functor on spaces to the category of graded abelian groups. Moreover, it has to satisfy Eilenberg- Steenrod axioms, except for the dimension axiom. Every cohomolog ...
... Cohomology theories are very important in algebraic topology as they are invariants for topological spaces. A cohomology theory E ∗ is a functor on spaces to the category of graded abelian groups. Moreover, it has to satisfy Eilenberg- Steenrod axioms, except for the dimension axiom. Every cohomolog ...
The local structure of algebraic K-theory
... The general plan of the book is as follows. In section I.1 we give some general background on algebraic K-theory. The length of this introductory section is defended by the fact that this book is primarily concerned with algebraic K-theory; the theories that fill the last chapters are just there in ...
... The general plan of the book is as follows. In section I.1 we give some general background on algebraic K-theory. The length of this introductory section is defended by the fact that this book is primarily concerned with algebraic K-theory; the theories that fill the last chapters are just there in ...
Hodge Cycles on Abelian Varieties
... violate the sign conventions for complexes.) R A number !, 2 Hn .X; Q/, is called a period of !. The map in (1.2) identifies H n .X; Q/ with the space of classes of closed forms whose periods are all rational. Theorem 1.2 can be restated as follows: a closed differential form is exact if all its ...
... violate the sign conventions for complexes.) R A number !, 2 Hn .X; Q/, is called a period of !. The map in (1.2) identifies H n .X; Q/ with the space of classes of closed forms whose periods are all rational. Theorem 1.2 can be restated as follows: a closed differential form is exact if all its ...
GEOMETRY FINAL EXAM MATERIAL
... o Be able to find the geometric mean of two numbers o Geometric Mean Triangle Altitude Conjecture, Geometric Mean Triangle Leg Conjecture Special Right Triangles o 30-60-90 , 45-45-90 Right Triangle Trigonometry o SohCahToa o Be able to use trigonometric functions (sine, cosine, tangent) to find ...
... o Be able to find the geometric mean of two numbers o Geometric Mean Triangle Altitude Conjecture, Geometric Mean Triangle Leg Conjecture Special Right Triangles o 30-60-90 , 45-45-90 Right Triangle Trigonometry o SohCahToa o Be able to use trigonometric functions (sine, cosine, tangent) to find ...
derived smooth manifolds
... This construction provides a correspondence which is homotopical in nature: one begins with a homotopy class of maps and receives a cobordism class. However, it is close to existing on the nose, in that a dense subset of all representing maps f : S n → MO is transverse to B and yields an imbedded ma ...
... This construction provides a correspondence which is homotopical in nature: one begins with a homotopy class of maps and receives a cobordism class. However, it is close to existing on the nose, in that a dense subset of all representing maps f : S n → MO is transverse to B and yields an imbedded ma ...
On the structure of triangulated categories with finitely many
... a) T is Hom-finite, i.e. the space HomT (X, Y ) is finite-dimensional for all objects X, Y of T . It follows that indecomposable objects of T have local endomorphism rings and that each object of T decomposes into a finite direct sum of indecomposables [9, 3.3]. We assume moreover that b) T is locally ...
... a) T is Hom-finite, i.e. the space HomT (X, Y ) is finite-dimensional for all objects X, Y of T . It follows that indecomposable objects of T have local endomorphism rings and that each object of T decomposes into a finite direct sum of indecomposables [9, 3.3]. We assume moreover that b) T is locally ...
CLASSIFICATION OF SEMISIMPLE ALGEBRAIC MONOIDS
... exists a morphism x: E -» k such that x"'(^*) = G [19, I, Theorem 1.1]. 1(E) = {e e E\e2 = e) is the set of indempotents of E. If x e E and H c G is a subgroup, then C\H(x) c £ is the H-conjugacy class of x in £. A D-monoid Z is an irreducible, algebraic monoid such that G(Z) = T is a torus (2.2). T ...
... exists a morphism x: E -» k such that x"'(^*) = G [19, I, Theorem 1.1]. 1(E) = {e e E\e2 = e) is the set of indempotents of E. If x e E and H c G is a subgroup, then C\H(x) c £ is the H-conjugacy class of x in £. A D-monoid Z is an irreducible, algebraic monoid such that G(Z) = T is a torus (2.2). T ...
Chapter 7 Duality
... DM(S) can be constructed from the “naive” version Amot (SmS )0 , i.e., we may replace all the homotopy identities in the construction of the motivic DG tensor category A mot (SmS ) with strict identities. Combining this with (3.2.6), we arrive at a construction of Dbmot (S) as a localization of the ...
... DM(S) can be constructed from the “naive” version Amot (SmS )0 , i.e., we may replace all the homotopy identities in the construction of the motivic DG tensor category A mot (SmS ) with strict identities. Combining this with (3.2.6), we arrive at a construction of Dbmot (S) as a localization of the ...
pdf
... K-theory held in Sedano, Spain, during the week January 22–27 of 2007. It intends to be an introduction to K-theory, both algebraic and topological, with emphasis on their interconnections. While a wide range of topics is covered, an effort has been made to keep the exposition as elementary and self ...
... K-theory held in Sedano, Spain, during the week January 22–27 of 2007. It intends to be an introduction to K-theory, both algebraic and topological, with emphasis on their interconnections. While a wide range of topics is covered, an effort has been made to keep the exposition as elementary and self ...
finitely generated powerful pro-p groups
... Building upon the work of Bruce King [Kin74], Alexander Lubotzky and Avinoam Mann developed the notion of a powerful finite p-group and extended it to pro-p groups (in [LM87a] and [LM87b]). This allows us to restate Lazard’s result as follows. Theorem A A topological group G is a p-adic analytic gro ...
... Building upon the work of Bruce King [Kin74], Alexander Lubotzky and Avinoam Mann developed the notion of a powerful finite p-group and extended it to pro-p groups (in [LM87a] and [LM87b]). This allows us to restate Lazard’s result as follows. Theorem A A topological group G is a p-adic analytic gro ...
Solution 1 - D-MATH
... field Qp of p-adic numbers is the quotient field of Zp . For every element x of Qp there exists an n such that pn x ∈ Zp . Using the projections Z → Z/pn Z we can view Z as the subset of Zp of constant sequences, and similarly Q ⊂ Qp . Prove that there exist isomorphisms Hom(Q, Z(p∞ )) ∼ = Hom(Z[p−1 ...
... field Qp of p-adic numbers is the quotient field of Zp . For every element x of Qp there exists an n such that pn x ∈ Zp . Using the projections Z → Z/pn Z we can view Z as the subset of Zp of constant sequences, and similarly Q ⊂ Qp . Prove that there exist isomorphisms Hom(Q, Z(p∞ )) ∼ = Hom(Z[p−1 ...
Algebraic Number Theory, a Computational Approach
... hope, but you will have to do some additional reading and exercises. We will briefly review the basics of the Galois theory of number fields. Some of the homework problems involve using a computer, but there are examples which you can build on. We will not assume that you have a programming backgrou ...
... hope, but you will have to do some additional reading and exercises. We will briefly review the basics of the Galois theory of number fields. Some of the homework problems involve using a computer, but there are examples which you can build on. We will not assume that you have a programming backgrou ...
Lectures on Etale Cohomology
... The conventions concerning varieties are the same as those in my notes on Algebraic Geometry. For example, an affine algebra over a field k is a finitely generated k-algebra A such that A ˝k k al is has no nonzero nilpotents for one (hence every) algebraic closure k al of k — this implies that A its ...
... The conventions concerning varieties are the same as those in my notes on Algebraic Geometry. For example, an affine algebra over a field k is a finitely generated k-algebra A such that A ˝k k al is has no nonzero nilpotents for one (hence every) algebraic closure k al of k — this implies that A its ...