Minimal Resolution of Relatively Compressed Level Algebras

A relatively compressed algebra with given socle degrees is an Artinian quotient $A$ of a given graded algebra R/\fc, whose Hilbert function is maximal among such quotients with the given socle degrees. For us \fc is usually a ``general'' complete intersection and we usually require that $A$ be level. The precise value of the Hilbert function of a relatively compressed algebra is open, and we show that finding this value is equivalent to the Fr\"oberg Conjecture. We then turn to the minimal free resolution of a level algebra relatively compressed with respect to a general complete intersection. When the algebra is Gorenstein of even socle degree we give the precise resolution. When it is of odd socle degree we give good bounds on the graded Betti numbers. We also relate this case to the Minimal Resolution Conjecture of Mustata for points on a projective variety. Finding the graded Betti numbers is essentially equivalent to determining to what extent there can be redundant summands (i.e. ``ghost terms'') in the minimal free resolution, i.e. when copies of the same $R(-t)$ can occur in two consecutive free modules. This is easy to arrange using Koszul syzygies; we show that it can also occur in more surprising situations that are not Koszul. Using the equivalence to the Fr\"oberg Conjecture, we show that in a polynomial ring where that conjecture holds (e.g. in three variables), the possible non-Koszul ghost terms are extremely limited. Finally, we use the connection to the Fr\"oberg Conjecture, as well as the calculation of the minimal free resolution for relatively compressed Gorenstein algebras, to find the minimal free resolution of general Artinian almost complete intersections in many new cases. This greatly extends previous work of the first two authors.Comment: 31 page

Fixed Point Algebras for Easy Quantum Groups

Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their $K$-groups. Building on prior work by the second author, we prove that free easy quantum groups satisfy these conditions and we compute the $K$-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group $S_n^+$, the free orthogonal quantum group $O_n^+$ and the quantum reflection groups $H_n^{s+}$. Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups, which are related to Hopf-Galois extensions

The Principle of Locality. Effectiveness, fate and challenges

The Special Theory of Relativity and Quantum Mechanics merge in the key principle of Quantum Field Theory, the Principle of Locality. We review some examples of its ``unreasonable effectiveness'' (which shows up best in the formulation of Quantum Field Theory in terms of operator algebras of local observables) in digging out the roots of Global Gauge Invariance in the structure of the local observable quantities alone, at least for purely massive theories; but to deal with the Principle of Local Gauge Invariance is still a problem in this frame. This problem emerges also if one attempts to figure out the fate of the Principle of Locality in theories describing the gravitational forces between elementary particles as well. Spacetime should then acquire a quantum structure at the Planck scale, and the Principle of Locality is lost. It is a crucial open problem to unravel a replacement in such theories which is equally mathematically sharp and reduces to the Principle of Locality at larger scales. Besides exploring its fate, many challenges for the Principle of Locality remain; among them, the analysis of Superselection Structure and Statistics also in presence of massless particles, and to give a precise mathematical formulation to the Measurement Process in local and relativistic terms; for which we outline a qualitative scenario which avoids the EPR Paradox.Comment: 36 pages. Survey partially based on a talk delivered at the Meeting "Algebraic Quantum Field Theory: 50 years", Goettingen, July 29-31, 2009, in honor of Detlev Buchholz. Submitted to Journal of Mathematical Physic

On the Logic of Belief and Propositional Quantification

We consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as “everything I believe is true” or “there is some-thing that I neither believe nor disbelieve.” Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator (BAOs) validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. Hence, by duality, we also cover the usual method of adding propositional quantifiers to normal modal logics by considering their classes of Kripke frames. In addition, we obtain decidability for all the concrete logics we discuss

Resolution of Stringy Singularities by Non-commutative Algebras

In this paper we propose a unified approach to (topological) string theory on certain singular spaces in their large volume limit. The approach exploits the non-commutative structure of D-branes, so the space is described by an algebraic geometry of non-commutative rings. The paper is devoted to the study of examples of these algebras. In our study there is an auxiliary commutative algebraic geometry of the center of the (local) algebras which plays an important role as the target space geometry where closed strings propagate. The singularities that are resolved will be the singularities of this auxiliary geometry. The singularities are resolved by the non-commutative algebra if the local non-commutative rings are regular. This definition guarantees that D-branes have a well defined K-theory class. Homological functors also play an important role. They describe the intersection theory of D-branes and lead to a formal definition of local quivers at singularities, which can be computed explicitly for many types of singularities. These results can be interpreted in terms of the derived category of coherent sheaves over the non-commutative rings, giving a non-commutative version of recent work by M. Douglas. We also describe global features like the Betti numbers of compact singular Calabi-Yau threefolds via global holomorphic sections of cyclic homology classes.Comment: 36 pages, Latex, 5 figures. v2:Reference adde

Singularities in positive characteristic, stratification and simplification of the singular locus

We introduce an upper semi-continuous function that stratifies the highest multiplicity locus of a hypersurface in arbitrary characteristic (over a perfect field). The blow-up along the maximum stratum defined by this function leads to a form of simplification of the singularities, also known as a reduction to the monomial case.Comment: Several typos corrected. Minor improvements on the presentation of the published pape