Derived splinters in positive characteristic
... independent of singularity considerations. In fact, we can prove ‘up to finite cover’ analogues in characteristic p of many vanishing theorems known in characteristic 0. All these results fit naturally in the study of F-singularities, and are motivated by a desire to understand the direct summand co ...
... independent of singularity considerations. In fact, we can prove ‘up to finite cover’ analogues in characteristic p of many vanishing theorems known in characteristic 0. All these results fit naturally in the study of F-singularities, and are motivated by a desire to understand the direct summand co ...
A gentle introduction to von Neumann algebras for model theorists
... von Neumann algebra theory and one often proves facts about arbitrary von Neumann algebras by first proving the result for factors. We end this section with one of the most difficult open problems in von Neumann algebra theory: Question 2.9. If m, n ≥ 2 are distinct, is L(Fm ) ∼ = L(Fn )? Even thoug ...
... von Neumann algebra theory and one often proves facts about arbitrary von Neumann algebras by first proving the result for factors. We end this section with one of the most difficult open problems in von Neumann algebra theory: Question 2.9. If m, n ≥ 2 are distinct, is L(Fm ) ∼ = L(Fn )? Even thoug ...
Sample pages 2 PDF
... k ∈ N, let Ik denote the set of all matrices in I with the property that their nonzero entries appear only in the first k rows. It is easy to check that Ik is a nilpotent ideal of A; in fact, Ikk+1 = 0. If A had a maximal nilpotent ideal N, then, by Lemma 2.11, N would contain each Ik , and hence ...
... k ∈ N, let Ik denote the set of all matrices in I with the property that their nonzero entries appear only in the first k rows. It is easy to check that Ik is a nilpotent ideal of A; in fact, Ikk+1 = 0. If A had a maximal nilpotent ideal N, then, by Lemma 2.11, N would contain each Ik , and hence ...
Brauer groups of abelian schemes
... beautiful construction of the dual abelian variety in the spirit of Grothendieck style algebraic geometry by using the theorem of the square, its corollaries, and cohomology theory. Since the /c-points of Pic^n is H1 (A, G^), it is natural to ask how much of this work carries over to higher cohomolo ...
... beautiful construction of the dual abelian variety in the spirit of Grothendieck style algebraic geometry by using the theorem of the square, its corollaries, and cohomology theory. Since the /c-points of Pic^n is H1 (A, G^), it is natural to ask how much of this work carries over to higher cohomolo ...
EVERY CONNECTED SUM OF LENS SPACES IS A REAL
... work of Dovermann, Masuda and Suh [2], that would have been useful in realizing algebraically the equivariant set-up above. However, the results of Doverman et al. apply only to semi-free actions of a group, whereas here, the action of G is, more or less, arbitrary, in any case, not necessarily semi ...
... work of Dovermann, Masuda and Suh [2], that would have been useful in realizing algebraically the equivariant set-up above. However, the results of Doverman et al. apply only to semi-free actions of a group, whereas here, the action of G is, more or less, arbitrary, in any case, not necessarily semi ...
Elliptic Curves Lecture Notes
... Proof. (sketch) This result is not difficult if the characteristic of k is not 2. We have checked that the unique point at infinity is nonsingular, so we work with the corresponding affine curve. The change of variables y 0 = 21 (y − a1 x − a3 ) reduces the equation to the simpler form y 2 = 4x3 + b ...
... Proof. (sketch) This result is not difficult if the characteristic of k is not 2. We have checked that the unique point at infinity is nonsingular, so we work with the corresponding affine curve. The change of variables y 0 = 21 (y − a1 x − a3 ) reduces the equation to the simpler form y 2 = 4x3 + b ...
On continuous images of ultra-arcs
... subcontinua of XD . For µ ∈ XD , R(µ) is the family of all regular subcontinua containing µ, and we define µ, ν ∈ XD to be R-equivalent if R(µ) = R(ν). Clearly any R-class containing a regular point is degenerate, so there are generally lots of R R-classes. The associated quotient map is denoted rX, ...
... subcontinua of XD . For µ ∈ XD , R(µ) is the family of all regular subcontinua containing µ, and we define µ, ν ∈ XD to be R-equivalent if R(µ) = R(ν). Clearly any R-class containing a regular point is degenerate, so there are generally lots of R R-classes. The associated quotient map is denoted rX, ...
MATH 436 Notes: Finitely generated Abelian groups.
... free Abelian of rank 1 with basis {d}. This led to the classification of all one-generated Abelian groups as the cyclic groups Z or Z/dZ for d ≥ 1. Now picture the group Z2 as the subgroup of the Euclidean plane (R2 , +) consisting of vectors with integer entries. Then H = {(2s, 3t)|s, t ∈ Z} is a s ...
... free Abelian of rank 1 with basis {d}. This led to the classification of all one-generated Abelian groups as the cyclic groups Z or Z/dZ for d ≥ 1. Now picture the group Z2 as the subgroup of the Euclidean plane (R2 , +) consisting of vectors with integer entries. Then H = {(2s, 3t)|s, t ∈ Z} is a s ...
LECTURES ON ERGODIC THEORY OF GROUP ACTIONS (A VON
... If the group Γ in 1.3.1.1◦ above is finitely generated, say by g1 , ..., gk ∈ Γ, and we denote by T the Laplacian k −1 Σh σgi ∈ B(H), then by von Neumann’s ergodic mean value theorem (for the semigoup H = {n | n ≥ 1}) it follows that for each ξ ∈ H, ε > 0 there exists n large enough such that kT n ξ ...
... If the group Γ in 1.3.1.1◦ above is finitely generated, say by g1 , ..., gk ∈ Γ, and we denote by T the Laplacian k −1 Σh σgi ∈ B(H), then by von Neumann’s ergodic mean value theorem (for the semigoup H = {n | n ≥ 1}) it follows that for each ξ ∈ H, ε > 0 there exists n large enough such that kT n ξ ...
Slides
... appears in an equation of the form xi = c (delete those) or can be replaced by some appropriate value in T. The resulting system E’ is over T. For a solution x = (x1, ..., xn) of E, define weight(x) = i (# of ghosts of xi) Summing the weights of all solutions to E’ is essentially equivalent to coun ...
... appears in an equation of the form xi = c (delete those) or can be replaced by some appropriate value in T. The resulting system E’ is over T. For a solution x = (x1, ..., xn) of E, define weight(x) = i (# of ghosts of xi) Summing the weights of all solutions to E’ is essentially equivalent to coun ...