THE ZEN OF ∞-CATEGORIES Contents 1. Derived categories
... understanding composition in BG amounts to understanding the multiplication law of G, but this is an intractable (in fact, computationally undecidable) task, closely related to the so-called “word problem” for generators and relations in abstract algebra. More generally, given a relative category (R ...
... understanding composition in BG amounts to understanding the multiplication law of G, but this is an intractable (in fact, computationally undecidable) task, closely related to the so-called “word problem” for generators and relations in abstract algebra. More generally, given a relative category (R ...
functors of artin ringso
... which is easily seen to determine, for each -qe F(A), a group action of tFI on
the subset F(p)'1(iq) of F(A') (provided that subset is not empty). (Hx) implies
that this action is "transitive," while (H4) is precisely the condition that this action
makes F(p)-1(^) a (formally) principal homogene ...
... which is easily seen to determine, for each -qe F(A), a group action of tF
Cohomology of Categorical Self-Distributivity
... A quandle, X, is a set with a binary operation (a, b) 7→ a / b such that (I) For any a ∈ X, a / a = a. (II) For any a, b ∈ X, there is a unique c ∈ X such that a = c / b. (III) For any a, b, c ∈ X, we have (a / b) / c = (a / c) / (b / c). A rack is a set with a binary operation that satisfies (II) a ...
... A quandle, X, is a set with a binary operation (a, b) 7→ a / b such that (I) For any a ∈ X, a / a = a. (II) For any a, b ∈ X, there is a unique c ∈ X such that a = c / b. (III) For any a, b, c ∈ X, we have (a / b) / c = (a / c) / (b / c). A rack is a set with a binary operation that satisfies (II) a ...
Nilpotence and Stable Homotopy Theory II
... from the sphere spectrum to a ring spectrum, is nilpotent if it is nilpotent when regarded as an element of the ring π∗ R. The main result of [7] is Theorem 2. In each of the above situations, the map f is nilpotent if the spectrum F is finite, and if M U∗ f = 0. In case the range of f is p-local, t ...
... from the sphere spectrum to a ring spectrum, is nilpotent if it is nilpotent when regarded as an element of the ring π∗ R. The main result of [7] is Theorem 2. In each of the above situations, the map f is nilpotent if the spectrum F is finite, and if M U∗ f = 0. In case the range of f is p-local, t ...
The Essential Dimension of Finite Group Schemes Corso di Laurea Magistrale in Matematica
... of linear homomorphisms {Φi : V ⊗ki → V ⊗hi }i∈I . The type of the algebraic structure is the triple (I, {ki }i∈I , {hi }i∈I ). Fixed a type there is an obvious category of algebraic structure of the given type, where an arrow (V, {Φi }) → (W, {Ψi }) is a linear map f : V → W such that for each i ∈ ...
... of linear homomorphisms {Φi : V ⊗ki → V ⊗hi }i∈I . The type of the algebraic structure is the triple (I, {ki }i∈I , {hi }i∈I ). Fixed a type there is an obvious category of algebraic structure of the given type, where an arrow (V, {Φi }) → (W, {Ψi }) is a linear map f : V → W such that for each i ∈ ...
THE ORBIFOLD CHOW RING OF TORIC DELIGNE
... Graber and Vistoli [2], is the algebraic version of the orbifold cohomology ring introduced by W. Chen and Ruan [7], [8]. By design, this ring incorporates numerical invariants, such as the orbifold Euler characteristic and the orbifold Hodge numbers, of the underlying variety. The product structure ...
... Graber and Vistoli [2], is the algebraic version of the orbifold cohomology ring introduced by W. Chen and Ruan [7], [8]. By design, this ring incorporates numerical invariants, such as the orbifold Euler characteristic and the orbifold Hodge numbers, of the underlying variety. The product structure ...
Continuous cohomology of groups and classifying spaces
... interplay between topology and algebra. The examples range through geometry and into the study of differential equations. Indeed when Sophus Lie began to look at continuous groups around 1870 [35], he was particularly interested in those respecting a geometric structure and those respecting the solu ...
... interplay between topology and algebra. The examples range through geometry and into the study of differential equations. Indeed when Sophus Lie began to look at continuous groups around 1870 [35], he was particularly interested in those respecting a geometric structure and those respecting the solu ...
Some results on the existence of division algebras over R
... 2.2 Reduced singular homology . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Definition of the reduced singular homology functors . . 13 2.2.2 The homology groups of ∅ and of { x } . . . . . . . . . . . 15 2.2.3 How the homology functors factor through homotopy . 16 2.2.4 The Meyer-Vietoris sequ ...
... 2.2 Reduced singular homology . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Definition of the reduced singular homology functors . . 13 2.2.2 The homology groups of ∅ and of { x } . . . . . . . . . . . 15 2.2.3 How the homology functors factor through homotopy . 16 2.2.4 The Meyer-Vietoris sequ ...
Group actions on manifolds - Department of Mathematics, University
... complex structure is a Lie group, provided M is compact. (By contrast, the group of symplectomorphisms of a symplectic manifold Diff(M, ω) is of course infinite-dimensional!) The general setting for this type of problem is explained in detail in Kobayashi’s book on transformation groups. Let M be a ...
... complex structure is a Lie group, provided M is compact. (By contrast, the group of symplectomorphisms of a symplectic manifold Diff(M, ω) is of course infinite-dimensional!) The general setting for this type of problem is explained in detail in Kobayashi’s book on transformation groups. Let M be a ...
(pdf)
... Throughout, unless explicitly stated, R is assumed to be a commutative integral domain. As is customary, we write Frac(R) to denote the field of fractions of a domain. By Z, Q, R, C we denote the integers, rationals, reals, and complex numbers, respectively. For R a ring, R[x], read ‘R adjoin x’, is ...
... Throughout, unless explicitly stated, R is assumed to be a commutative integral domain. As is customary, we write Frac(R) to denote the field of fractions of a domain. By Z, Q, R, C we denote the integers, rationals, reals, and complex numbers, respectively. For R a ring, R[x], read ‘R adjoin x’, is ...
Modules - University of Oregon
... These are called the left regular and right regular modules, respectively. Observe that R R is actually a cyclic left R-module, because R = R1R . The left R-submodules of R R are precisely the sub-Abelian groups I of R such that RI = I. These were called left ideals of R in section 2.1. Similarly, t ...
... These are called the left regular and right regular modules, respectively. Observe that R R is actually a cyclic left R-module, because R = R1R . The left R-submodules of R R are precisely the sub-Abelian groups I of R such that RI = I. These were called left ideals of R in section 2.1. Similarly, t ...
1 - Evan Chen
... It’s common the relations are the orders of the generators. A presentation is not unique, however. This particular presentation is useful because each element can be written as rk s` for ` ∈ {0, 1} and k ∈ {0, 1, · · · , n−1}. Presentations are like barycentric coordinates: for stuff they’re good, t ...
... It’s common the relations are the orders of the generators. A presentation is not unique, however. This particular presentation is useful because each element can be written as rk s` for ` ∈ {0, 1} and k ∈ {0, 1, · · · , n−1}. Presentations are like barycentric coordinates: for stuff they’re good, t ...
Tannaka Duality for Geometric Stacks
... It is natural to ask when φ is an equivalence. In the case where X and S are projective schemes, a satisfactory answer was obtained long ago. In this case, both algebraic and analytic maps may be classified by their graphs, which are closed in the product X × S. One may then deduce that any analytic ...
... It is natural to ask when φ is an equivalence. In the case where X and S are projective schemes, a satisfactory answer was obtained long ago. In this case, both algebraic and analytic maps may be classified by their graphs, which are closed in the product X × S. One may then deduce that any analytic ...
borisovChenSmith
... Graber and Vistoli [2], is the algebraic version of the orbifold cohomology ring introduced by W. Chen and Ruan [7], [8]. By design, this ring incorporates numerical invariants, such as the orbifold Euler characteristic and the orbifold Hodge numbers, of the underlying variety. The product structure ...
... Graber and Vistoli [2], is the algebraic version of the orbifold cohomology ring introduced by W. Chen and Ruan [7], [8]. By design, this ring incorporates numerical invariants, such as the orbifold Euler characteristic and the orbifold Hodge numbers, of the underlying variety. The product structure ...
Remedial topology
... all unmarked and !-problems or to find the solution online. It’s better to do it in order starting from the beginning, because the solutions are often contained in previous problems. The problems with * are harder, and ** are very hard; don’t be disappointed if you can’t solve them, but feel free to ...
... all unmarked and !-problems or to find the solution online. It’s better to do it in order starting from the beginning, because the solutions are often contained in previous problems. The problems with * are harder, and ** are very hard; don’t be disappointed if you can’t solve them, but feel free to ...
ABELIAN VARIETIES A canonical reference for the subject is
... Proof. Choose a nonempty open affine U ⊂ X and let D = X \ U with the reduced structure. We claim that D is a Cartier divisor (that is, its coherent ideal sheaf is invertible). Since X is regular, it is equivalent to say that all generic points of D have codimension 1. Mumford omits this explanation ...
... Proof. Choose a nonempty open affine U ⊂ X and let D = X \ U with the reduced structure. We claim that D is a Cartier divisor (that is, its coherent ideal sheaf is invertible). Since X is regular, it is equivalent to say that all generic points of D have codimension 1. Mumford omits this explanation ...
A convenient category for directed homotopy
... U-initial lift of a cone (fi : X → UAi )i∈I is given by putting a ≤ b on X if and only if fi (a) ≤ fi (b) for each i ∈ I. (2) An ordered set is a preordered set (A, ≤) where ≤ is also antisymmetric, i.e., if it satisfies (∀x, y)(x ≤ y ∧ y ≤ x → x = y). The category of ordered sets is not topological ...
... U-initial lift of a cone (fi : X → UAi )i∈I is given by putting a ≤ b on X if and only if fi (a) ≤ fi (b) for each i ∈ I. (2) An ordered set is a preordered set (A, ≤) where ≤ is also antisymmetric, i.e., if it satisfies (∀x, y)(x ≤ y ∧ y ≤ x → x = y). The category of ordered sets is not topological ...
EXAMPLES AND EXERCISES IN BASIC CATEGORY THEORY 1
... 1.8. Groupoids. A groupoid is a category in which every morphism is invertible. The notion of a groupoid is very important in algebraic topology. In particular, in §1.9 below I will define the fundamental groupoid of a topological space, which provides the correct way of thinking about fundamental g ...
... 1.8. Groupoids. A groupoid is a category in which every morphism is invertible. The notion of a groupoid is very important in algebraic topology. In particular, in §1.9 below I will define the fundamental groupoid of a topological space, which provides the correct way of thinking about fundamental g ...
A conjecture on the Hall topology for the free group - LaCIM
... Proof. By Proposition 2.3, every element of 3F is a closed rational set. Conversely, if S is a closed rational set, then S = S belongs to J^, by Theorem 2.4. Thus Conjecture 1 implies that the closure of a rational set is rational and gives an algorithm to compute this closure. 3. A consequence of C ...
... Proof. By Proposition 2.3, every element of 3F is a closed rational set. Conversely, if S is a closed rational set, then S = S belongs to J^, by Theorem 2.4. Thus Conjecture 1 implies that the closure of a rational set is rational and gives an algorithm to compute this closure. 3. A consequence of C ...
Manifolds and Varieties via Sheaves
... Definition 1.2.1. Let R be a sheaf of k-valued functions on X. We say that R is a sheaf of algebras if each R(U ) ⊆ M apk (U ) is a subalgebra. We call the pair (X, R) a concrete ringed space over k, or simply a k-space. (Rn , CR ), (Rn , C ∞ ) and (Cn , O) are examples of R and C-spaces. Definition ...
... Definition 1.2.1. Let R be a sheaf of k-valued functions on X. We say that R is a sheaf of algebras if each R(U ) ⊆ M apk (U ) is a subalgebra. We call the pair (X, R) a concrete ringed space over k, or simply a k-space. (Rn , CR ), (Rn , C ∞ ) and (Cn , O) are examples of R and C-spaces. Definition ...
Hilbert`s Nullstellensatz and the Beginning of Algebraic Geometry
... set V (S), Then one can ask the converse question: Suppose one knows that some polynomial f(Xl, '" Xn) identically vanishes on V (S), Can one assert that I is somehow a combination of some Ii's in S ? A trivial example shows that this is too much to expect, To wit, consider the degree 2 polynomial F ...
... set V (S), Then one can ask the converse question: Suppose one knows that some polynomial f(Xl, '" Xn) identically vanishes on V (S), Can one assert that I is somehow a combination of some Ii's in S ? A trivial example shows that this is too much to expect, To wit, consider the degree 2 polynomial F ...
pdf
... if k = C, sheaves in the classical topology. Throughout E will denote the coefficient field for the sheaves. That is E = Q` or E = Q. The main example we will have in mind are local systems (otherwise known as locally constant sheaves). There is an equivalence of categories between local systems and ...
... if k = C, sheaves in the classical topology. Throughout E will denote the coefficient field for the sheaves. That is E = Q` or E = Q. The main example we will have in mind are local systems (otherwise known as locally constant sheaves). There is an equivalence of categories between local systems and ...
ON SOME DIFFERENTIALS IN THE MOTIVIC COHOMOLOGY
... The motivic cohomology spectral sequence is an algebraic-geometrical analogue of the Atiyah–Hirzebruch spectral sequence in topology. For smooth varieties, it has the second term consisting of motivic cohomology groups and converges to algebraic K-theory. The spectral sequence was initially construc ...
... The motivic cohomology spectral sequence is an algebraic-geometrical analogue of the Atiyah–Hirzebruch spectral sequence in topology. For smooth varieties, it has the second term consisting of motivic cohomology groups and converges to algebraic K-theory. The spectral sequence was initially construc ...
EXAMPLE SHEET 1 1. If k is a commutative ring, prove that b k
... (b) if G is a monoid, the coalgebra structure on k G induced by it. 17. Let A be an algebra over a field k. A right A-module M is rational if for each m P M the orbit m ¨ A “ tm ¨ x : x P Au is finite dimensional. Given a coalgebra C, prove that the functor ComodpCq Ñ Mod-C ˚ is an isomorphism into ...
... (b) if G is a monoid, the coalgebra structure on k G induced by it. 17. Let A be an algebra over a field k. A right A-module M is rational if for each m P M the orbit m ¨ A “ tm ¨ x : x P Au is finite dimensional. Given a coalgebra C, prove that the functor ComodpCq Ñ Mod-C ˚ is an isomorphism into ...
Geometry Fall 2012 Lesson 017 _Using postulates and theorems to
... 8. Substitution Postulate (6,7) ...
... 8. Substitution Postulate (6,7) ...