A splinter is a notion of singularity that has seen numerous recent applications, especially in connection with the direct summand theorem, mixed characteristic minimal model program, Cohen–Macaulayness absolute integral closures and cohomology vanishing theorems. Nevertheless, many basic questions about these singularities remain elusive. One outstanding problem whether property spreads from p...

A bounded linear operator A on a Hilbert space H is called finite if \\AX-XA -l|| èl for each XEB(H). The class of finite operators is uniformly closed, contains every normal operator, every operator with a compact direct summand, and the entire C*-algebra generated by each of its members. These results imply that the set of operators with a finite dimensional reducing subspace is not uniformly...

The original article expressed the special values of the zeta function of a variety over a finite field in terms of the Ẑ-cohomology of the variety. As the article was being completed, Lichtenbaum conjectured the existence of certain motivic cohomology groups. Progress on his conjecture allows one to give a beautiful restatement of the main theorem of the article in terms of Z-cohomology groups...

Abstract We provide a proof of Jacobson’s theorem on derivations primitive rings with nonzero socle. Both and its formulation (in terms the so-called differential operators left vector spaces over division ring) underlie our paper. apply to describe standard operator real, complex, or quaternionic normed spaces. Indeed, when space is infinite-dimensional, every derivation such ring form $$A\rig...

In this handout we will briefly explore the topic of projective modules in a bit more detail than we covered in class. Throughout R is a commutative ring. Recall that, by definition, a projective module is an R-module that is a direct summand of a free R-module. As mentioned in class, if the ring R is decomposable, e.g., R = R1 ⊕R2 is a direct sum of rings, then there are many examples of non-f...

We prove a formula for the multiplicity of the irreducible representation V (n) of sl(2, C) as a direct summand of its own exterior cube ΛV (n). From this we determine that V (n) occurs exactly once as a summand of ΛV (n) if and only if n = 3, 5, 6, 7, 8, 10. These representations admit a unique sl(2)-invariant alternating ternary structure obtained from the projection ΛV (n) → V (n). We calcul...

For any central simple algebra, the Grothendieck Chow-motive of the corresponding Severi-Brauer variety is decomposed in a direct sum where each summand is a twisted motive of the Severi-Brauer variety corresponding to the underlying division algebra. It leads to decompositions in other theories (for instance, of K-cohomologies) because of the universal property of the Chow-motives. In the seco...