Projectively simple rings

We introduce the notion of a projectively simple ring, which is an infinite-dimensional graded k-algebra A such that every 2-sided ideal has finite codimension in A (over the base field k). Under some (relatively mild) additional assumptions on A, we reduce the problem of classifying such rings (in the sense explained in the paper) to the following geometric question, which we believe to be of independent interest. Let X is a smooth irreducible projective variety. An automorphism f: X -> X is called wild if it X has no proper f-invariant subvarieties. We conjecture that if X admits a wild automorphism then X is an abelian variety. We prove several results in support of this conjecture; in particular, we show that the conjecture is true if X is a curve or a surface. In the case where X is an abelian variety, we describe all wild automorphisms of X. In the last two sections we show that if A is projectively simple and admits a balanced dualizing complex, then Proj(A) is Cohen-Macaulay and Gorenstein.Comment: Some new material has been added in Section 1; to appear in Advances in Mathematic

Teaching linear algebra at university

Linear algebra represents, with calculus, the two main mathematical subjects taught in science universities. However this teaching has always been difficult. In the last two decades, it became an active area for research works in mathematics education in several countries. Our goal is to give a synthetic overview of the main results of these works focusing on the most recent developments. The main issues we will address concern: the epistemological specificity of linear algebra and the interaction with research in history of mathematics; the cognitive flexibility at stake in learning linear algebra; three principles for the teaching of linear algebra as postulated by G. Harel; the relation between geometry and linear algebra; an original teaching design experimented by M. Rogalsk

Idealizer Rings and Noncommutative Projective Geometry

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian property have very different behavior on the left and right sides.Comment: 17 Pages, revised version: significant changes--introduction rewritten, new section on tensor products added, main theorem restated at en

Generic Noncommutative Surfaces

We study a class of noncommutative surfaces and their higher dimensional analogues which provide answers to several open questions in noncommutative projective geometry. Specifically, we give the first known graded algebras which are noetherian but not strongly noetherian, answering a question of Artin, Small, and Zhang. In addition, these examples are maximal orders and satisfy the $\chi_1$ condition but not $\chi_i$ for $i \geq 2$, answering a questions of Stafford and Zhang and a question of Stafford and Van den Bergh. Finally, we show that these algebras have finite cohomological dimension.Comment: 43 pages, Latex, to appear in Advances in Math. Result on finite global dimension added, other minor change

Innovativeness of the US economy. Permanent or weakening dominance?

The paper is divided into three parts. In the first one, main causes of American leadership in the field of technology are explained. In the second part, innovation performance of the US economy in comparison with the EU and Asian economies is presented. Finally, there is an analysis of innovation capacity of US economy in the context of challenges resulting from the financial and economic crisis.Strukturę opracowania można przedstawić następująco: po wprowadzeniu dokonano charakterystyki głównych czynników i procesów, które przyczyniły się do powstania dominacji gospodarki Stanów Zjednoczonych w dziedzinie innowacyjności, następnie poddano analizie zjawisko zmniejszania się przewagi innowacyjnej tej gospodarki nad resztą świata, a w dalszej kolejności skoncentrowano uwagę na zagadnieniu wpływu współczesnego kryzysu gospodarczego na perspektywy utrzymania przewagi technologicznej Stanów Zjednoczonych

Skew Calabi-Yau Algebras and Homological Identities

A skew Calabi-Yau algebra is a generalization of a Calabi-Yau algebra which allows for a non-trivial Nakayama automorphism. We prove three homological identities about the Nakayama automorphism and give several applications. The identities we prove show (i) how the Nakayama automorphism of a smash product algebra A # H is related to the Nakayama automorphisms of a graded skew Calabi-Yau algebra A and a finite-dimensional Hopf algebra H that acts on it; (ii) how the Nakayama automorphism of a graded twist of A is related to the Nakayama automorphism of A; and (iii) that Nakayama automorphism of a skew Calabi-Yau algebra A has trivial homological determinant in case A is noetherian, connected graded, and Koszul.Comment: 39 pages; minor changes, mostly in the Introductio

Naive Noncommutative Blowing Up

Let B(X,L,s) be the twisted homogeneous coordinate ring of an irreducible variety X over an algebraically closed field k with dim X > 1. Assume that c in X and s in Aut(X) are in sufficiently general position. We show that if one follows the commutative prescription for blowing up X at c, but in this noncommutative setting, one obtains a noncommutative ring R=R(X,c,L,s) with surprising properties. In particular: (1) R is always noetherian but never strongly noetherian. (2) If R is generated in degree one then the images of the R-point modules in qgr(R) are naturally in (1-1) correspondence with the closed points of X. However, both in qgr(R) and in gr(R), the R-point modules are not parametrized by a projective scheme. (3) qgr R has finite cohomological dimension yet H^1(R) is infinite dimensional. This gives a more geometric approach to results of the second author who proved similar results for X=P^n by algebraic methods.Comment: Latex, 42 page