\u3ci\u3eAmerican Express\u3c/i\u3e, the Rule of Reason, and the Goals of Antitrust

I. Introduction II. The Debate over the Consumer Welfare Standard III. Applying the Rule of Reason ... A. The General Framework ... B. Step One ... C. Muddying the Waters ... D. The Bottom Line: An Analytical Mess IV. The Rule of Reason and Multi-Sided Platforms V. Missing in Action: The Consumer Welfare Standard VI. Conclusio

General Bilinear Forms

We introduce the new notion of general bilinear forms (generalizing sesquilinear forms) and prove that for every ring $R$ (not necessarily commutative, possibly without involution) and every right $R$-module $M$ which is a generator (i.e. $R_R$ is a summand of $M^n$ for some $n\in\N$), there is a one-to-one correspondence between the anti-automorphisms of \End(M) and the general regular bilinear forms on $M$, considered up to similarity. This generalizes a well-known similar correspondence in the case $R$ is a field. We also demonstrate that there is no such correspondence for arbitrary $R$-modules. We use the generalized correspondence to show that there is a canonical set isomorphism between the orbits of the left action of \Inn(R) on the anti-automorphisms of $R$ and the orbits of the left action of \Inn(M_n(R)) on the anti-automorphisms of $M_n(R)$, provided $R_R$ is the only right $R$-module $N$ satisfying $N^n\cong R^n$. We also prove a variant of a theorem of Osborn. Namely, we classify all semisimple rings with involution admitting no non-trivial idempotents that are invariant under the involution.Comment: 26 page

Rings That Are Morita Equivalent to Their Opposites

We consider the following problem: Under what assumptions do one or more of the following are equivalent for a ring $R$: (A) $R$ is Morita equivalent to a ring with involution, (B) $R$ is Morita equivalent to a ring with an anti-automorphism, (C) $R$ is Morita equivalent to its opposite ring. The problem is motivated by a theorem of Saltman which roughly states that all conditions are equivalent for Azumaya algebras. Basing on the recent "general bilinear forms", we present a general machinery to attack the problem, and use it to show that (C)$\iff$(B) when $R$ is semilocal or $\mathbb{Q}$-finite. Further results of similar flavor are also obtained, for example: If $R$ is a semilocal ring such that $\mathrm{M}_{n}(R)$ has an involution, then $\mathrm{M}_{2}(R)$ has an involution, and under further mild assumptions, $R$ itself has an involution. In contrast to that, we demonstrate that (B) does not imply (A). Our methods also give a new perspective on the Knus-Parimala-Srinivas proof of Saltman's Theorem. Finally, we give a method to test Azumaya algebras of exponent $2$ for the existence of involutions, and use it to construct explicit examples of such algebras.Comment: 28 pages; minor corrections form previous version, a mistake in Corollary 7.4 was correcte

Rationally Isomorphic Hermitian Forms and Torsors of Some Non-Reductive Groups

Let $R$ be a semilocal Dedekind domain. Under certain assumptions, we show that two (not necessarily unimodular) hermitian forms over an $R$-algebra with involution, which are rationally ismorphic and have isomorphic semisimple coradicals, are in fact isomorphic. The same result is also obtained for quadratic forms equipped with an action of a finite group. The results have cohomological restatements that resemble the Grothendieck--Serre conjecture, except the group schemes involved are not reductive. We show that these group schemes are closely related to group schemes arising in Bruhat--Tits theory.Comment: 27 pages. Changes from previous version: Section 5 was split into two sections, several proofs have been simplified, other mild modification

The Mozambican Miner: A study in the export of labour

Marc Wuyts' copy of the Mozambican Miner report. A report on Mozambican miners produced by the Centro de Estudos Africanos at Universidade Eduardo Mondlane in Maputo, 1977. The research was directed by Ruth First and conducted by up to 40 other researchers and activists

Semi-Invariant Subrings

We say that a subring $R_0$ of a ring $R$ is semi-invariant if $R_0$ is the ring of invariants in $R$ under some set of ring endomorphisms of some ring containing $R$. We show that $R_0$ is semi-invariant if and only if there is a ring $S\supseteq R$ and a set $X\subseteq S$ such that R_0=\Cent_R(X):={r\in R \suchthat xr=rx \forall x\in X}; in particular, centralizers of subsets of $R$ are semi-invariant subrings. We prove various properties of semi-invariant subrings and show how they can be used for various applications including: (1) The center of a semiprimary (resp. right perfect) ring is semiprimary (resp. right perfect). (2) If $M$ is a finitely presented module over a "good" semiperfect ring (e.g. an inverse limit of semiprimary rings), then $(M)$ is semiperfect, hence $M$ has a Krull-Schmidt decomposition. (This generalizes results of Bjork and Rowen). (3) If $\rho$ is a representation of a monoid or a ring over a module with a "good" semiperfect endomorphism ring (in the sense of (2)), then $\rho$ has a Krull-Schmidt decomposition. (4) If $S$ is a "good" commutative semiperfect ring and $R$ is an $S$-algebra that is f.p.\ as an $S$-module, then $R$ is semiperfect. (5) Let $R\subseteq S$ be rings and let $M$ be a right $S$-module. If $(M_R)$ is semiprimary (resp. right perfect), then $(M_S)$ is semiprimary (resp. right perfect).Comment: 31 page