Commutative ring objects in pro-categories and generalized Moore
... These include being able to define either derived functors or a homotopy category with the same underlying object set. We refer the reader to [DHKS04] (e.g. §8) for one solution, which involves employing a larger universe in which one constructs equivalence relations and produces canonical definitio ...
... These include being able to define either derived functors or a homotopy category with the same underlying object set. We refer the reader to [DHKS04] (e.g. §8) for one solution, which involves employing a larger universe in which one constructs equivalence relations and produces canonical definitio ...
2 Lecture 2: Spaces of valuations
... It is not at all true that, in general, v(A) ⊂ Γ≤1 ∪ {0}, as shown by the following example: Example 2.4.2 Let A = K a field. Then, we recover our old notion of valuation, as in Definition 2.2.1. If v is nontrivial, then K = A * Rv . Every field algebraic over a finite field admits only the trivial ...
... It is not at all true that, in general, v(A) ⊂ Γ≤1 ∪ {0}, as shown by the following example: Example 2.4.2 Let A = K a field. Then, we recover our old notion of valuation, as in Definition 2.2.1. If v is nontrivial, then K = A * Rv . Every field algebraic over a finite field admits only the trivial ...
A. m - TeacherWeb
... of 65° with the ground. If they are both standing at the same elevation, which kite is higher in the air? A. Meena’s kite B. Rita’s kite ...
... of 65° with the ground. If they are both standing at the same elevation, which kite is higher in the air? A. Meena’s kite B. Rita’s kite ...
My notes - Harvard Mathematics Department
... takes values in Z. Using this theorem, let’s see (loosely) why the Jacobian J(X) admits the structure of an algebraic variety. The fourth statement of the quoted theorem is the easiest, so let’s try to find a hermitian form on H 0 (X, ΩX )∨ satisfying the above properties. So, we’ll need a lemma fro ...
... takes values in Z. Using this theorem, let’s see (loosely) why the Jacobian J(X) admits the structure of an algebraic variety. The fourth statement of the quoted theorem is the easiest, so let’s try to find a hermitian form on H 0 (X, ΩX )∨ satisfying the above properties. So, we’ll need a lemma fro ...
Math 850 Algebra - San Francisco State University
... Algebra is one of the three main divisions of modern mathematics, the others being analysis and topology. Algebra studies algebraic structures, or sets with operations. Algebra abstracts the operations of mathematics–addition, multiplication, differentiation, integration–insofar as possible from thei ...
... Algebra is one of the three main divisions of modern mathematics, the others being analysis and topology. Algebra studies algebraic structures, or sets with operations. Algebra abstracts the operations of mathematics–addition, multiplication, differentiation, integration–insofar as possible from thei ...
Contents - Harvard Mathematics Department
... infinity” are included. Moreover, when endowed with the complex topology, (complex) projective varieties are compact, unlike all but degenerate affine varieties (i.e. finite sets). It is when defining the notion of a “variety” in projective space that one encounters gradedness. Now a variety in Pn m ...
... infinity” are included. Moreover, when endowed with the complex topology, (complex) projective varieties are compact, unlike all but degenerate affine varieties (i.e. finite sets). It is when defining the notion of a “variety” in projective space that one encounters gradedness. Now a variety in Pn m ...
The symplectic Verlinde algebras and string K e
... twisted K-theory, we could mimic, using string topology constructions, some of the properties of the Verlinde algebra, and in particular, perhaps, construct a self-dual topological field theory in some sense? The purpose of this paper was to investigate this question. What we found was partially sati ...
... twisted K-theory, we could mimic, using string topology constructions, some of the properties of the Verlinde algebra, and in particular, perhaps, construct a self-dual topological field theory in some sense? The purpose of this paper was to investigate this question. What we found was partially sati ...
VISIBLE EVIDENCE FOR THE BIRCH AND SWINNERTON
... Unless otherwise stated, f is a newform, If its annihilator, and A = Af is the corresponding optimal quotient of J0 (N ). 3.1. Modular symbols. Modular symbols are crucial to many algorithms for computing with modular abelian varieties, because they can be used to construct a finite presentation for ...
... Unless otherwise stated, f is a newform, If its annihilator, and A = Af is the corresponding optimal quotient of J0 (N ). 3.1. Modular symbols. Modular symbols are crucial to many algorithms for computing with modular abelian varieties, because they can be used to construct a finite presentation for ...
Invariants and Algebraic Quotients
... In the last sections we turn to some examples and applications. First we prove a geometric version of the so-called first fundamental theorem for GLn . This form of the fundamental theorem for classical groups is due to Th. Vust [Vus76]. As well we describe the “method of the associated cone”. Rough ...
... In the last sections we turn to some examples and applications. First we prove a geometric version of the so-called first fundamental theorem for GLn . This form of the fundamental theorem for classical groups is due to Th. Vust [Vus76]. As well we describe the “method of the associated cone”. Rough ...
Groups with exponents I. Fundamentals of the theory and tensor
... In applications # will mostly be an embedding of rings. However, even in this case, the homomorphism A : G ---* G B'" is not always an embedding. Since in the abelian case the group G B results from by tensoring the A-module G by the ring B, appropriate examples can be found in many articles on comm ...
... In applications # will mostly be an embedding of rings. However, even in this case, the homomorphism A : G ---* G B'" is not always an embedding. Since in the abelian case the group G B results from by tensoring the A-module G by the ring B, appropriate examples can be found in many articles on comm ...
Structure theory of manifolds
... uniquely defined only up to homotopy types of spaces and homotopy of maps. For the orthogonal group we have BOn = BO(n) and hence On = O(n), because principal On -bundles correspond bijectively to orthogonal bundles with fiber Rn . Similarly BTopn = BTop(n) and Topn = Top(n). There is a subgroup P L ...
... uniquely defined only up to homotopy types of spaces and homotopy of maps. For the orthogonal group we have BOn = BO(n) and hence On = O(n), because principal On -bundles correspond bijectively to orthogonal bundles with fiber Rn . Similarly BTopn = BTop(n) and Topn = Top(n). There is a subgroup P L ...
A May-type spectral sequence for higher topological Hochschild
... We recover spectral sequence 1.0.1 as a special case of 1.0.2 by letting E˚ “ π˚ and letting X‚ be a simplicial model for the circle S 1 . A major part of the work we do in this paper is to formulate a definition (see Definition 3.1.1) of a “filtered E8 -ring spectrum” which is sufficiently well-beh ...
... We recover spectral sequence 1.0.1 as a special case of 1.0.2 by letting E˚ “ π˚ and letting X‚ be a simplicial model for the circle S 1 . A major part of the work we do in this paper is to formulate a definition (see Definition 3.1.1) of a “filtered E8 -ring spectrum” which is sufficiently well-beh ...
ON THE TATE AND MUMFORD-TATE CONJECTURES IN
... influenced my understanding of the notions that play a central role in this paper. Further, I thank J. Commelin, W. Goldring, C. Peters, R. Pignatelli and Q. Yin for inspiring discussions and helpful comments. 0.10 Notation and conventions. (a) By a Hodge structure of K3 type we mean a polarizable Q ...
... influenced my understanding of the notions that play a central role in this paper. Further, I thank J. Commelin, W. Goldring, C. Peters, R. Pignatelli and Q. Yin for inspiring discussions and helpful comments. 0.10 Notation and conventions. (a) By a Hodge structure of K3 type we mean a polarizable Q ...
Homology With Local Coefficients
... Since this new homologytheory(whichincludesthe old) seems to have such wide applicability,a completereviewof the oldertheoryis needed to determine to what extentand in what formits theoremsgeneralize. The object of this paper is to make such a survey. The general conclusion is that all major parts o ...
... Since this new homologytheory(whichincludesthe old) seems to have such wide applicability,a completereviewof the oldertheoryis needed to determine to what extentand in what formits theoremsgeneralize. The object of this paper is to make such a survey. The general conclusion is that all major parts o ...
Flatness
... Using the LES of Tor, this immediately implies: Proposition 1.2 If 0 → M 0 → M → M 00 → 0 is an exact sequence of A-modules, and M 0 and M 00 are flat, then so is M . If M and M 00 are both flat, so is M 0 . Another useful consequence is the following, which again is immediate upon passing to the LE ...
... Using the LES of Tor, this immediately implies: Proposition 1.2 If 0 → M 0 → M → M 00 → 0 is an exact sequence of A-modules, and M 0 and M 00 are flat, then so is M . If M and M 00 are both flat, so is M 0 . Another useful consequence is the following, which again is immediate upon passing to the LE ...
Hochschild cohomology: some methods for computations
... and H 1 (A, A) = 0, see [18]. The importance of the simply connected algebras follows from the fact that usually we may reduce the study of indecomposable modules over an algebra to that for the corresponding simply connected algebras, using Galois coverings. Despite this very little is known about ...
... and H 1 (A, A) = 0, see [18]. The importance of the simply connected algebras follows from the fact that usually we may reduce the study of indecomposable modules over an algebra to that for the corresponding simply connected algebras, using Galois coverings. Despite this very little is known about ...
PM 464
... 2. Properties (3), (4), and (5) show that the collection of algebraic sets form the closed sets of a topology on An . This topology is known as the Zariski topology 3. The Zariski topology is strictly weaker than the metric topology (for | = Q, R, C), i.e. Zariski closed subsets are metically closed ...
... 2. Properties (3), (4), and (5) show that the collection of algebraic sets form the closed sets of a topology on An . This topology is known as the Zariski topology 3. The Zariski topology is strictly weaker than the metric topology (for | = Q, R, C), i.e. Zariski closed subsets are metically closed ...
PROJECTIVITY AND FLATNESS OVER THE
... Example 2.3. Let Λ be a ring, and (C, ∆C , εC ) a Λ-coring. (this is a coalgebra in the monoidal category Λ MΛ ). Corings were introduced by Sweedler [15]. The language of corings allows us to generalize, unify and formulate more elegantly many results about generalized Hopf modules. This is due to ...
... Example 2.3. Let Λ be a ring, and (C, ∆C , εC ) a Λ-coring. (this is a coalgebra in the monoidal category Λ MΛ ). Corings were introduced by Sweedler [15]. The language of corings allows us to generalize, unify and formulate more elegantly many results about generalized Hopf modules. This is due to ...
FIELDS AND RINGS WITH FEW TYPES In
... small division ring of positive characteristic is locally finite dimensional over its centre. Recall that superstable division rings [3, Cherlin, Shelah] and even supersimple ones [18, Pillay, Scanlon, Wagner] are known to be fields. We then turn to small difference fields. Hrushowski proved that in ...
... small division ring of positive characteristic is locally finite dimensional over its centre. Recall that superstable division rings [3, Cherlin, Shelah] and even supersimple ones [18, Pillay, Scanlon, Wagner] are known to be fields. We then turn to small difference fields. Hrushowski proved that in ...
HIGHER CATEGORIES 4. Model categories, 2: Topological spaces
... 4.1.1. Simplicial enrichment of sSet and Top. For X, Y simplicial sets, one defines X Y as the simplicial set representing the functor Z 7→ Hom(Z × Y, X). Similarly, for a simplicial set S and for a topological space X we define X S as the topological space representing the functor Z 7→ Hom(|S| × Z, ...
... 4.1.1. Simplicial enrichment of sSet and Top. For X, Y simplicial sets, one defines X Y as the simplicial set representing the functor Z 7→ Hom(Z × Y, X). Similarly, for a simplicial set S and for a topological space X we define X S as the topological space representing the functor Z 7→ Hom(|S| × Z, ...
WHICH ARE THE SIMPLEST ALGEBRAIC VARIETIES? Contents 1
... An algebraic variety is a subset of Cn defined by polynomial equations. It is rather clear that the higher the degree of the defining equations, the more complicated the corresponding variety can be. There have been various approaches to define what the “most complicated” varieties are, but it is on ...
... An algebraic variety is a subset of Cn defined by polynomial equations. It is rather clear that the higher the degree of the defining equations, the more complicated the corresponding variety can be. There have been various approaches to define what the “most complicated” varieties are, but it is on ...
Moduli Problems for Ring Spectra - International Mathematical Union
... In order to adequately treat cases (b) through (d), it is important to note that for 0 ≤ n ≤ ∞, an En -ring is an essentially homotopy-theoretic object, and should therefore be treated using the formalism of higher category theory. In §2 we will give an overview of this formalism; in particular, we ...
... In order to adequately treat cases (b) through (d), it is important to note that for 0 ≤ n ≤ ∞, an En -ring is an essentially homotopy-theoretic object, and should therefore be treated using the formalism of higher category theory. In §2 we will give an overview of this formalism; in particular, we ...
The congruence subgroup problem
... We saw above that the answer is ‘yes’ when n = 2. In 1962, Bass–Lazard–Serre and independently Mennicke discovered that SL(2, Z) is exceptional. They proved the following theorem. Theorem. If n > 2, every subgroup of finite index in SL(n, Z) is a congruence subgroup. The problem can be generalized. ...
... We saw above that the answer is ‘yes’ when n = 2. In 1962, Bass–Lazard–Serre and independently Mennicke discovered that SL(2, Z) is exceptional. They proved the following theorem. Theorem. If n > 2, every subgroup of finite index in SL(n, Z) is a congruence subgroup. The problem can be generalized. ...
Covering Groupoids of Categorical Rings - PMF-a
... A group-groupoid is a group object in the category of groupoids [7]; equivalently, it is an internal category and hence an internal groupoid in the category of groups ([18], [1]) . An alternative name, quite generally used, is “2-group”, see for example [4]. Recently for example in [1], [12] and [13 ...
... A group-groupoid is a group object in the category of groupoids [7]; equivalently, it is an internal category and hence an internal groupoid in the category of groups ([18], [1]) . An alternative name, quite generally used, is “2-group”, see for example [4]. Recently for example in [1], [12] and [13 ...