Related papers: An Ext-Tor duality theorem, cohomological dimensio…

For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. As a…

Let (R,m) be a commutative Noetherian local ring. It is known that R is Cohen-Macaulay if there exists either a nonzero finitely generated R-module of finite injective dimension or a nonzero Cohen-Macaulay R-module of finite projective…

In this paper we study rigid modules over commutative Noetherian local rings, establish new freeness criteria for certain periodic rigid modules, and extend several results from the literature. Along the way, we prove general Ext vanishing…

We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincar\'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy…

The study of homological invariants such as Tor, Ext and local cohomology modules constitutes an important direction in commutative algebra. Explicit descriptions of these invariants are notoriously difficult to find and often involve…

Let $C_2$ be the cyclic group of order two. We show that the $RO(C_2)$-graded Bredon cohomology of a finite Rep($C_2$)-complex is free as a module over the cohomology of a point when using coefficients in the constant Mackey functor…

Let $k$ be a perfect field of characteristic $p>2$, $R := W(k)[\![t_1, \dots, t_d]\!]$ be the power series ring over the Witt vectors, and $X$ be a smooth proper scheme over $R$. The main goal of this article is to extend classical…

We prove various results in infinite-dimensional differential calculus which relate differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: 1. in the…

We generalize the completion theorem for equivariant MU-module spectra for finite groups or finite extensions of a torus to compact Lie groups using the splitting of global functors proved by Schwede. This proves a conjecture of Greenlees…

Let R be a complete intersection ring and let M and N be R-modules. It is shown that the vanishing of Ext^i_R(M,N) for a certain number of consecutive values of i starting at n forces the complete intersection dimension of M to be at most…

We introduce a general theory of homological Milnor-Witt cycle modules over an excellent base scheme equipped with a dimension function, extending both Rost's cycle modules and Feld's theory over fields. To any such module we associate a…

Let R be a Noetherian standard graded ring, and M and N two finitely generated graded R-modules. We introduce reg_R (M,N) by using the notion of generalized local cohomology instead of local cohomology, in the definition of regularity. We…

We study the Ext modules in the category of left modules over a twisted algebra of a finite quiver over a ringed space $(X,\mathcal O_X)$, allowing for the presence of relations. We introduce a spectral sequence which relates the Ext…

We show that if Auslander`s depth formula holds for non-zero Tor-independent modules over Cohen-Macaulay local rings of dimension 1, then it holds for such modules over any Cohen-Macaulay local ring. More generally, we show that the depth…

We describe some recent work concerning Gorenstein liaison of codimension two subschemes of a projective variety. Applications make use of the algebraic theory of maximal Cohen-Macaulay modules, which we review in an Appendix.

In this paper we study various convolution-type algebras associated with a locally compact quantum group from cohomological and geometrical points of view. The quantum group duality endows the space of trace class operators over a locally…

This chapter sets out preliminaries for the duality theory in later chapters. An underlying idea is that local cohomology functors are higher derived functors of colocalizations (a.k.a.~coreflections). Predominantly well-known facts about…

In this article, we prove that a free divisor in a three dimensional complex manifold must be Euler homogeneous in a strong sense if the cohomology of its complement is the hypercohomology of its logarithmic differential forms. F.J.…

Matlis duals of local cohomology modules are investigated with respect to many different topics (see section 0 - Introduction). One of these topics are complete intersections - see Corollary 1.1.4.

We construct a self-dual complete resolution of a module defined by a pair of embedded complete intersection ideals in a local ring. Our construction is based on a gluing construction of Herzog and Martsinkovsky and exploits the structure…