390 research outputs found
Reverse mathematics, well-quasi-orders, and Noetherian spaces
A quasi-order Q induces two natural quasi-orders on P(Q) P(Q) , but if Q is a well-quasi-order, then these quasi-orders need not necessarily be well-quasi-orders. Nevertheless, Goubault-Larrecq (Proceedings of the 22nd Annual IEEE Symposium 4 on Logic in Computer Science (LICSâ07), pp. 453â462, 2007) showed that moving from a well-quasi-order Q to the quasi-orders on P(Q) P(Q) preserves well-quasi-orderedness in a topological sense. Specifically, Goubault-Larrecq proved that the upper topologies of the induced quasi-orders on P(Q) P(Q) are Noetherian, which means that they contain no infinite strictly descending sequences of closed sets. We analyze various theorems of the form âif Q is a well-quasi-order then a certain topology on (a subset of) P(Q) P(Q) is Noetherianâ in the style of reverse mathematics, proving that these theorems are equivalent to ACA0 over RCA0. To state these theorems in RCA0 we introduce a new framework for dealing with second-countable topological spaces
Gr\"obner methods for representations of combinatorial categories
Given a category C of a combinatorial nature, we study the following
fundamental question: how does the combinatorial behavior of C affect the
algebraic behavior of representations of C? We prove two general results. The
first gives a combinatorial criterion for representations of C to admit a
theory of Gr\"obner bases. From this, we obtain a criterion for noetherianity
of representations. The second gives a combinatorial criterion for a general
"rationality" result for Hilbert series of representations of C. This criterion
connects to the theory of formal languages, and makes essential use of results
on the generating functions of languages, such as the transfer-matrix method
and the Chomsky-Sch\"utzenberger theorem.
Our work is motivated by recent work in the literature on representations of
various specific categories. Our general criteria recover many of the results
on these categories that had been proved by ad hoc means, and often yield
cleaner proofs and stronger statements. For example: we give a new, more
robust, proof that FI-modules (originally introduced by Church-Ellenberg-Farb),
and a family of natural generalizations, are noetherian; we give an easy proof
of a generalization of the Lannes-Schwartz artinian conjecture from the study
of generic representation theory of finite fields; we significantly improve the
theory of -modules, introduced by Snowden in connection to syzygies of
Segre embeddings; and we establish fundamental properties of twisted
commutative algebras in positive characteristic.Comment: 41 pages; v2: Moved old Sections 3.4, 10, 11, 13.2 and connected text
to arxiv:1410.6054v1, Section 13.1 removed and will appear elsewhere; v3:
substantial revision and reorganization of section
The telescope conjecture for algebraic stacks
Using Balmer--Favi's generalized idempotents, we establish the telescope
conjecture for many algebraic stacks. Along the way, we classify the thick
tensor ideals of perfect complexes of stacks.Comment: 20 page
Markov random fields and iterated toric fibre products
We prove that iterated toric fibre products from a finite collection of toric
varieties are defined by binomials of uniformly bounded degree. This implies
that Markov random fields built up from a finite collection of finite graphs
have uniformly bounded Markov degree.Comment: several improvements, final versio
Non-commutative crepant resolutions: scenes from categorical geometry
Non-commutative crepant resolutions are algebraic objects defined by Van den
Bergh to realize an equivalence of derived categories in birational geometry.
They are motivated by tilting theory, the McKay correspondence, and the minimal
model program, and have applications to string theory and representation
theory. In this expository article I situate Van den Bergh's definition within
these contexts and describe some of the current research in the area.Comment: 57 pages; final version, to appear in "Progress in Commutative
Algebra: Ring Theory, Homology, and Decompositions" (Sean Sather-Wagstaff,
Christopher Francisco, Lee Klingler, and Janet Vassilev, eds.), De Gruyter.
Incorporates many small bugfixes and adjustments addressing comments from the
referee and other
Homological dimensions for co-rank one idempotent subalgebras
Let be an algebraically closed field and be a (left and right)
Noetherian associative -algebra. Assume further that is either
positively graded or semiperfect (this includes the class of finite dimensional
-algebras, and -algebras that are finitely generated modules over a
Noetherian central Henselian ring). Let be a primitive idempotent of ,
which we assume is of degree if is positively graded. We consider the
idempotent subalgebra and the simple right
-module , where is the Jacobson radical
of , or the graded Jacobson radical of if is positively graded. In
this paper, we relate the homological dimensions of and , using the
homological properties of . First, if has no self-extensions of any
degree, then the global dimension of is finite if and only if that of
is. On the other hand, if the global dimensions of both and
are finite, then cannot have self-extensions of degree greater
than one, provided is finite dimensional.Comment: 24 page
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