CLASSIFICATION OF SEMISIMPLE ALGEBRAIC MONOIDS
... unipotent subgroup RU(G) [12, §19.5]. G is reductive if RU(G) = {e}. An algebraic group G is called a torus if T = A:* X ■■■ X k*; equivalently, T is connected and consists entirely of semisimple (i.e. diagonalizable) elements [12, §15.3]. A maximal torus T C G is a torus which is contained properly ...
... unipotent subgroup RU(G) [12, §19.5]. G is reductive if RU(G) = {e}. An algebraic group G is called a torus if T = A:* X ■■■ X k*; equivalently, T is connected and consists entirely of semisimple (i.e. diagonalizable) elements [12, §15.3]. A maximal torus T C G is a torus which is contained properly ...
Homology and topological full groups of etale groupoids on totally
... For almost finite groupoids, we study how their homology groups reflect dynamical properties of their topological full groups. It is shown that two clopen subsets of the unit space has the same class in H0 if and only if there exists an element in the topological full group which maps one to the other ...
... For almost finite groupoids, we study how their homology groups reflect dynamical properties of their topological full groups. It is shown that two clopen subsets of the unit space has the same class in H0 if and only if there exists an element in the topological full group which maps one to the other ...
Higher regulators and values of L
... Let X be a complex algebraic manifold, and let Kj(X), I @ ~ ( X , Q ) be its algebraic K-groups and singulary cohomology, respectively. We consider the Chern character ch: Kj(X)| H~-J(x,Q). It is easy to see that there are the Hodge conditions on the image of ch: we have ch (Kj(X))c~(~V2~I-f~-J(X,Q) ...
... Let X be a complex algebraic manifold, and let Kj(X), I @ ~ ( X , Q ) be its algebraic K-groups and singulary cohomology, respectively. We consider the Chern character ch: Kj(X)| H~-J(x,Q). It is easy to see that there are the Hodge conditions on the image of ch: we have ch (Kj(X))c~(~V2~I-f~-J(X,Q) ...
Dualizing DG modules and Gorenstein DG algebras
... central results in the theory of dualizing complexes for rings; see also [10], [12]. Section 3 delves deeper into this subject, focusing on finite commutative local DG algebras: a class of DG algebras which is slightly larger than that considered in Theorem I. For instance, one finds there the follo ...
... central results in the theory of dualizing complexes for rings; see also [10], [12]. Section 3 delves deeper into this subject, focusing on finite commutative local DG algebras: a class of DG algebras which is slightly larger than that considered in Theorem I. For instance, one finds there the follo ...
Representations of GL_2(A_Q^\infty)
... One thing that we neglected to discuss in the last note is the notion of Hecke algebras which are an invaluable tool in the study of smooth representations of TD groups. The idea is simple through the following analogy: Hecke algebras are to TD groups as group algebras are to finite groups. Namely, ...
... One thing that we neglected to discuss in the last note is the notion of Hecke algebras which are an invaluable tool in the study of smooth representations of TD groups. The idea is simple through the following analogy: Hecke algebras are to TD groups as group algebras are to finite groups. Namely, ...
M07/08
... explicit formula for C. Since we were not able to guess G from this formula, we applied a greedy algorithm (whose purpose was to maximize the number of associating triples) that resulted in another loop C with similar properties. The explicit formula for C is simple enough to connect it with a group ...
... explicit formula for C. Since we were not able to guess G from this formula, we applied a greedy algorithm (whose purpose was to maximize the number of associating triples) that resulted in another loop C with similar properties. The explicit formula for C is simple enough to connect it with a group ...
Closed sets and the Zariski topology
... Exercise 2.4 (20 push ups). A k-algebra A is a k-vector space which is also a commutative ring with identity. An example of a k-algebra is k[f1 , f2 , . . . , fs ], where each fi ∈ k[x]. A k-algebra A is said to be finitely generated if there exists a surjective ring homomorphism φ : k[x] → A. Show ...
... Exercise 2.4 (20 push ups). A k-algebra A is a k-vector space which is also a commutative ring with identity. An example of a k-algebra is k[f1 , f2 , . . . , fs ], where each fi ∈ k[x]. A k-algebra A is said to be finitely generated if there exists a surjective ring homomorphism φ : k[x] → A. Show ...
Rational points on Shimura curves and Galois representations Carlos de Vera Piquero
... where GQ = Gal (Q̄/Q) denotes the absolute Galois group of Q, τgd : τYd → XD stands (m) for the twisted form of the étale covering gd by the cohomology class τ , and we identify (m) (m) (m) (m) AutQ (Yd /XD ) = Aut(Yd /XD ) with the cyclic group Z/dZ of d elements. In particular, since GQ acts triv ...
... where GQ = Gal (Q̄/Q) denotes the absolute Galois group of Q, τgd : τYd → XD stands (m) for the twisted form of the étale covering gd by the cohomology class τ , and we identify (m) (m) (m) (m) AutQ (Yd /XD ) = Aut(Yd /XD ) with the cyclic group Z/dZ of d elements. In particular, since GQ acts triv ...
A somewhat gentle introduction to differential graded commutative
... DG commutative algebra provides a useful framework for proving theorems about rings and modules, the statements of which have no reference to the DG universe. For instance, a standard theorem says the following: Theorem 1.2 ([20, Corollary 1]) Let (R, m) → (S, n) be a flat local ring homomorphism, t ...
... DG commutative algebra provides a useful framework for proving theorems about rings and modules, the statements of which have no reference to the DG universe. For instance, a standard theorem says the following: Theorem 1.2 ([20, Corollary 1]) Let (R, m) → (S, n) be a flat local ring homomorphism, t ...
on the foundations of quasigroups
... It shall be treated in §§9, 10. The only left-distributive group is the trivial group. This serves to illustrate another of the attractions of quasigroups—they may satisfy identities which usually conflict with associativity. Also, in groups it is usually obvious whether a constraint is or is not sa ...
... It shall be treated in §§9, 10. The only left-distributive group is the trivial group. This serves to illustrate another of the attractions of quasigroups—they may satisfy identities which usually conflict with associativity. Also, in groups it is usually obvious whether a constraint is or is not sa ...
Algebraic Methods
... number up to addition by an integer looks like a number in [0, 1). Thus R = ∪0≤x<1 (x + Z), and the cosets of Z partition R. Furthermore, since the map h 7→ ha, h ∈ H, is a one-to-one correspondence, each coset has |H| elements. Definition 1.9. The index of a subgroup H in G is the number of right ( ...
... number up to addition by an integer looks like a number in [0, 1). Thus R = ∪0≤x<1 (x + Z), and the cosets of Z partition R. Furthermore, since the map h 7→ ha, h ∈ H, is a one-to-one correspondence, each coset has |H| elements. Definition 1.9. The index of a subgroup H in G is the number of right ( ...
slides
... Let M be a locally finite monoid. Then, its canonical filtration induces (functorially) a structure of an exhausted and separated filtered algebra on R[[M]]. It is given by In = { f ∈ R M : ∀x(`(x) < n ⇒ f (x) = 0) }. The associated (linear) topology is always stronger than the product topology (i.e ...
... Let M be a locally finite monoid. Then, its canonical filtration induces (functorially) a structure of an exhausted and separated filtered algebra on R[[M]]. It is given by In = { f ∈ R M : ∀x(`(x) < n ⇒ f (x) = 0) }. The associated (linear) topology is always stronger than the product topology (i.e ...
Coarse structures on groups - St. John`s University Unofficial faculty
... a Hausdorff topological group with a compact set of generators, then the asymptotic dimension of G with respect to the group-compact coarse structure is zero if and only if G is compact (Corollary 3.8). If H is a closed subgroup of G, then asdim H ≤ asdim G (Corollary 3.11). As a consequence, all di ...
... a Hausdorff topological group with a compact set of generators, then the asymptotic dimension of G with respect to the group-compact coarse structure is zero if and only if G is compact (Corollary 3.8). If H is a closed subgroup of G, then asdim H ≤ asdim G (Corollary 3.11). As a consequence, all di ...
Compactifications and Function Spaces
... An extreme example is the Cantor set. If X is the Cantor set, then there are no primitive idempotents in C ∗ (X) – every idempotent can be decomposed into orthogonal pieces. In fact, if X is compact and C ∗ (X) contains a closed subring R with no primitive idempotents, then there is a continuous sur ...
... An extreme example is the Cantor set. If X is the Cantor set, then there are no primitive idempotents in C ∗ (X) – every idempotent can be decomposed into orthogonal pieces. In fact, if X is compact and C ∗ (X) contains a closed subring R with no primitive idempotents, then there is a continuous sur ...
inductive limits of normed algebrasc1
... topology for the case where £ is the subalgebra of K[Xi, ■ ■ ■ , Xn] consisting of all polynomials in n indeterminants without constant term. If (ai, ■ ■ ■ , an) is any ra-tuple of positive real numbers, let V(ai, • ■ • , an) be the convex ...
... topology for the case where £ is the subalgebra of K[Xi, ■ ■ ■ , Xn] consisting of all polynomials in n indeterminants without constant term. If (ai, ■ ■ ■ , an) is any ra-tuple of positive real numbers, let V(ai, • ■ • , an) be the convex ...