The Functor Category in Relation to the Model Theory of Modules
... pp-formulas = finitely generated subfunctors of representables: If φ is a pp-formula, Fφ is a finitely generated subfunctor of a representable functor in fp(mod(R), Ab). If F is a finitely generated subfunctor of a representable functor, then there exists pp-formula φ such that F ∼ = Fφ . ...
... pp-formulas = finitely generated subfunctors of representables: If φ is a pp-formula, Fφ is a finitely generated subfunctor of a representable functor in fp(mod(R), Ab). If F is a finitely generated subfunctor of a representable functor, then there exists pp-formula φ such that F ∼ = Fφ . ...
borisovChenSmith
... N , a simplicial fan Σ in Q ⊗Z N with n rays, and a map β : Zn → N where the image of the standard basis in Zn generates the rays in Σ. A rational simplicial fan Σ produces a canonical stacky fan Σ := (N, Σ, β) where N is the distinguished lattice and β is the map defined by the minimal lattice point ...
... N , a simplicial fan Σ in Q ⊗Z N with n rays, and a map β : Zn → N where the image of the standard basis in Zn generates the rays in Σ. A rational simplicial fan Σ produces a canonical stacky fan Σ := (N, Σ, β) where N is the distinguished lattice and β is the map defined by the minimal lattice point ...
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The Kazhdan-Lusztig polynomial of a matroid
... braid matroid. Polo [Pol99] has shown that any polynomial with non-negative coefficients and constant term 1 appears as a Kazhdan-Lusztig polynomial associated to some symmetric group, while Kazhdan-Lusztig polynomials of matroids are far more restrictive (see Proposition 2.14). The original work of ...
... braid matroid. Polo [Pol99] has shown that any polynomial with non-negative coefficients and constant term 1 appears as a Kazhdan-Lusztig polynomial associated to some symmetric group, while Kazhdan-Lusztig polynomials of matroids are far more restrictive (see Proposition 2.14). The original work of ...
HOW TO DO A p-DESCENT ON AN ELLIPTIC CURVE
... In order to implement this description, we need a practical representation of the a priori rather abstractly defined group H 1 (K, E[θ]; S) and the maps δv . Our approach (based on [25]) is to identify the cohomology group with a subgroup of D(S, p) for a suitable ´etale algebra D over K. It will tu ...
... In order to implement this description, we need a practical representation of the a priori rather abstractly defined group H 1 (K, E[θ]; S) and the maps δv . Our approach (based on [25]) is to identify the cohomology group with a subgroup of D(S, p) for a suitable ´etale algebra D over K. It will tu ...
Universal unramified cohomology of cubic fourfolds containing a plane
... from P . Its discriminant divisor D ⊂ P is a sextic curve. The rationality of a cubic fourfold containing a plane over C is also a well known problem. Expectation. The very general cubic fourfold containing a plane over C is irrational. Assuming that the discriminant divisor D is smooth, the discrim ...
... from P . Its discriminant divisor D ⊂ P is a sextic curve. The rationality of a cubic fourfold containing a plane over C is also a well known problem. Expectation. The very general cubic fourfold containing a plane over C is irrational. Assuming that the discriminant divisor D is smooth, the discrim ...
Motivic interpretation of Milnor K
... To unify the Moore exact sequence and the Bloch exact sequence, K. Kato defined the generalized Milnor K-groups attached to finite family of semi-abelian varieties over a base field k in [Som90]. (See also [Kah92]). Given semi-abelian varieties G1 , . . . , Gr over k, one defines K(k, {Gi }ri=1 ) = ...
... To unify the Moore exact sequence and the Bloch exact sequence, K. Kato defined the generalized Milnor K-groups attached to finite family of semi-abelian varieties over a base field k in [Som90]. (See also [Kah92]). Given semi-abelian varieties G1 , . . . , Gr over k, one defines K(k, {Gi }ri=1 ) = ...
INTEGRABILITY CRITERION FOR ABELIAN EXTENSIONS OF LIE
... where Lg∗ denotes the pushforward map induced by the diffeomorphism Lg : G → G, h 7→ gh. By definition, X is completely determined by its value at the identity and g is therefore identified with T1 G as topological vector spaces endowed with the continuous Lie bracket of vector fields. The most stri ...
... where Lg∗ denotes the pushforward map induced by the diffeomorphism Lg : G → G, h 7→ gh. By definition, X is completely determined by its value at the identity and g is therefore identified with T1 G as topological vector spaces endowed with the continuous Lie bracket of vector fields. The most stri ...
The structure of Coh(P1) 1 Coherent sheaves
... It follows that si0 i1 cannot have any terms with the xi0 and xi1 exponents negative. In general, in s, there are thus no terms with more than one exponent negative. Next, consider the xi0 -exponent-negative terms in si0 and si1 . Because si0 i1 i2 = 0, these terms must be equal (recall the powers o ...
... It follows that si0 i1 cannot have any terms with the xi0 and xi1 exponents negative. In general, in s, there are thus no terms with more than one exponent negative. Next, consider the xi0 -exponent-negative terms in si0 and si1 . Because si0 i1 i2 = 0, these terms must be equal (recall the powers o ...
MILNOR K-THEORY OF LOCAL RINGS WITH FINITE RESIDUE
... x NB/A (y) = NB/A (i∗ (x) y) ∈ Kn+m (A) . (3) If A contains a field the transfer does not depend on the presentation of B over A chosen. (4) Let j : A → A0 be a homomorphism of local rings and let i0 : A0 → B 0 = B ⊗A A0 be the induced inclusion, for which we fix the induced presentation. Assume B 0 ...
... x NB/A (y) = NB/A (i∗ (x) y) ∈ Kn+m (A) . (3) If A contains a field the transfer does not depend on the presentation of B over A chosen. (4) Let j : A → A0 be a homomorphism of local rings and let i0 : A0 → B 0 = B ⊗A A0 be the induced inclusion, for which we fix the induced presentation. Assume B 0 ...
Section 3.2 - Cohomology of Sheaves
... the canonical natural equivalence HA,I (X, −) = HA,I 0 (X, −). This means that the A-module structure induced on H i (X, F ) is independent of the choice of resolutions on Mod(X). Definition 5. Let (X, OX ) be a ringed space and A = Γ(X, OX ). Fix an assignment of injective resolutions J to the obje ...
... the canonical natural equivalence HA,I (X, −) = HA,I 0 (X, −). This means that the A-module structure induced on H i (X, F ) is independent of the choice of resolutions on Mod(X). Definition 5. Let (X, OX ) be a ringed space and A = Γ(X, OX ). Fix an assignment of injective resolutions J to the obje ...
THE COHOMOLOGY RING OF FREE LOOP SPACES 1. Introduction
... is also a differential graded Hopf algebra up to homotopy (notion defined page 14). In section 5, using the Perturbation Lemma, we construct a small algebra up to homotopy whose homology is isomorphic to HH ∗ (A). In section 6, we make explicit this algebra up to homotopy in the following particular ...
... is also a differential graded Hopf algebra up to homotopy (notion defined page 14). In section 5, using the Perturbation Lemma, we construct a small algebra up to homotopy whose homology is isomorphic to HH ∗ (A). In section 6, we make explicit this algebra up to homotopy in the following particular ...
Noncommutative Lp-spaces of W*-categories and their applications
... the matrix whose entries are all zero except for the diagonal entry indexed by X, which is idX . We have X pX = 1 and pX pY = 0 for all X ̸= Y , hence (A, p) is indeed a linking algebra. 2.12 Given a linking algebra (A, p) we construct from it a small W*-category C as follows. The set of objects of ...
... the matrix whose entries are all zero except for the diagonal entry indexed by X, which is idX . We have X pX = 1 and pX pY = 0 for all X ̸= Y , hence (A, p) is indeed a linking algebra. 2.12 Given a linking algebra (A, p) we construct from it a small W*-category C as follows. The set of objects of ...
IDEAL BICOMBINGS FOR HYPERBOLIC GROUPS
... A worth noting feature of Theorem 7 is that it goes beyond the setting of CAT(−1) target spaces, to the more robust framework of hyperbolicity. For example, in [MSa] it is observed that if Γ is any Gromov-hyperbolic group and X is the Cayley graph associated to the free product of Γ and a free group ...
... A worth noting feature of Theorem 7 is that it goes beyond the setting of CAT(−1) target spaces, to the more robust framework of hyperbolicity. For example, in [MSa] it is observed that if Γ is any Gromov-hyperbolic group and X is the Cayley graph associated to the free product of Γ and a free group ...
Some results on the existence of division algebras over R
... of division algebra is defined as a (not necessarily associative) algebra in which left- and right-multiplication with a non-zero element is bijective. It is noted that the zero algebra, the Real numbers and the Complex numbers form division algebras of respective dimension 0, 1 and 2 over R. In the ...
... of division algebra is defined as a (not necessarily associative) algebra in which left- and right-multiplication with a non-zero element is bijective. It is noted that the zero algebra, the Real numbers and the Complex numbers form division algebras of respective dimension 0, 1 and 2 over R. In the ...
A basic note on group representations and Schur`s lemma
... Group representations allow one to study an abstract group in terms of linear transformations on vector spaces. The main point of studying group representations is to reduce group theoretic problems to those of linear algebra which are well understood. The inverse direction can also be fruitful; tha ...
... Group representations allow one to study an abstract group in terms of linear transformations on vector spaces. The main point of studying group representations is to reduce group theoretic problems to those of linear algebra which are well understood. The inverse direction can also be fruitful; tha ...
The Essential Dimension of Finite Group Schemes Corso di Laurea Magistrale in Matematica
... Now we are interested to classify all the forms over k that may come out from a given form over L. In some sense we already did it: these corresponds to the semilinear Γ-actions. However it may be given a more explicit classification. To do so we will need Galois cohomology. In the following we will ...
... Now we are interested to classify all the forms over k that may come out from a given form over L. In some sense we already did it: these corresponds to the semilinear Γ-actions. However it may be given a more explicit classification. To do so we will need Galois cohomology. In the following we will ...
on the shape of torus-like continua and compact connected
... This is because this is true for compact connected abelian topological groups by Theorem 2.5 of [6] and Theorem 1.4 of [8]. Of course, the Cech cohomology of a torus-like continuum Y will be the same as its associated compact connected abelian topological group. Thus one can apply the results of [3] ...
... This is because this is true for compact connected abelian topological groups by Theorem 2.5 of [6] and Theorem 1.4 of [8]. Of course, the Cech cohomology of a torus-like continuum Y will be the same as its associated compact connected abelian topological group. Thus one can apply the results of [3] ...
The congruence subgroup problem
... A subgroup ⊂ SL(n, Z) (n integer ≥ 2) is a congruence subgroup iff there is a proper non-zero ideal I ⊂ Z such that ⊃ SL(n, I ) = {g ∈ SL(n, Z)|g ≡ 1(mod I )}. Are there subgroups of finite index (note that SL(n, I ) has finite index in SL(n, Z)) which are not congruence subgroups? We saw abov ...
... A subgroup ⊂ SL(n, Z) (n integer ≥ 2) is a congruence subgroup iff there is a proper non-zero ideal I ⊂ Z such that ⊃ SL(n, I ) = {g ∈ SL(n, Z)|g ≡ 1(mod I )}. Are there subgroups of finite index (note that SL(n, I ) has finite index in SL(n, Z)) which are not congruence subgroups? We saw abov ...