This paper applies$K$-homology to solve the index problem for a class of hypoelliptic (but not elliptic) operators on contact manifolds.$K$-homology is the dual theory to$K$-theory. We explicitly calculate the$K$-cycle (i.e., the element in geometric$K$-homology) determined by any hypoelliptic Fredholm operator in the Heisenberg calculus.

We study the classes of modules which are generated by a silting module. In the case of either hereditary or perfect rings, it is proved that these are exactly the torsion T such that the regular module has a special T-preenvelope. In particular, every torsion-enveloping class in Mod-R are of the form Gen(T) for a minimal silting module T. For the dual case, we obtain for general rings that the covering torsion-free classes of modules are exactly the classes of the form Cogen(T), where T is a cosilting module.

In this paper we study representation theory of the category FI^m introduced by Gadish which is a product of copies of the category FI, and show that quite a few interesting representational and homological properties of FI can be generalized to FI^m in a natural way. In particular, we prove the representation stability property of finitely generated FI^m -modules over fields of characteristic 0.

Let $D$ be a Dedekind scheme with the characteristic of all residue fields not equal to 2. To every tame cover $C\to D$ with only odd ramification we associate a second Stiefel-Whitney class in the second cohomology with mod 2 coefficients of a certain tame orbicurve $[D]$ associated to $D$ . This class is then related to the pull-back of the second Stiefel-Whitney class of the push-forward of the line bundle of half of the ramification divisor. This shows (indirectly) that our Stiefel-Whitney class is the pull-back of a sum of cohomology classes considered by Esnault, Kahn and Viehweg in ‘Coverings with odd ramification and Stiefel-Whitney classes’. Perhaps more importantly, in the case of a proper and smooth curve over an algebraically closed field, our Stiefel-Whitney class is shown to be the pull-back of an invariant considered by Serre in ‘Revêtements à ramification impaire et thêta-caractéristiques’, and in this case our arguments give a new proof of the main result of that article.

We show that the FI-homology of an FI-module can be computed via a Koszul complex. As an application, we prove that the Castelnuovo-Mumford regularity of a finitely generated torsion FI-module is equal to its degree.

We use techniques from both real and complex algebraic geometry to study$K$-theoretic and related invariants of the algebra$C$($X$) of continuous complex-valued functions on a compact Hausdorff topological space$X$. For example, we prove a parameterized version of a theorem by Joseph Gubeladze; we show that if$M$is a countable, abelian, cancellative, torsion-free, semi-normal monoid, and$X$is contractible, then every finitely generated projective module over$C$($X$)[$M$] is free. The particular case $$ M = \mathbb{N}_0^n $$ gives a parameterized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic$K$-theory of$C$($X$) follows from the particular case $$ M = {\mathbb{Z}^n} $$ . We also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on$C$^{*}-algebras, and for a homology theory of commutative algebras to vanish on$C$^{*}-algebras. These criteria have numerous applications. For example, the vanishing criterion applied to nil$K$-theory implies that commutative$C$^{*}-algebras are$K$-regular. As another application, we show that the familiar formulas of Hochschild–Kostant–Rosenberg and Loday–Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of$C$($X$). Applications to the conjectures of Beĭlinson-Soulé and Farrell–Jones are also given.

We study the coinduction functor on the category of FI-modules and its variants. Using the coinduction functor, we give new and simpler proofs of (generalizations of) various results on homological properties of FI-modules. We also prove that any finitely generated projective VI-module over a field of characteristic 0 is injective.

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In this paper we study the singularities of holomorphic functions of bicomplex variables introduced by G. B. Price ($An Introduction to Multicomplex Spaces and Functions$, Dekker, New York,1991). In particular, we use computational algebra techniques to show that even in the case of one bicomplex variable, there cannot be compact singularities. The same techniques allow us to prove a duality theorem for such functions.

Let k be a commutative Noetherian ring and C−− be a locally finite k-linear category equipped with a self-embedding functor of degree 1. We show under a moderate condition that finitely generated torsion representations of C−− are super finitely presented (that is, they have projective resolutions each term of which is finitely generated). In the situation that these self-embedding functors are genetic functors, we give upper bounds for homological degrees of finitely generated torsion modules. These results apply to quite a few categories recently appearing in representation stability theory. In particular, when k is a field of characteristic 0, we obtain another upper bound for homological degrees of finitely generated FI-modules.