Homology of artinian and Matlis reflexive modules, I

Let R be a commutative local noetherian ring, and let L and L' be R-modules. We investigate the properties of the functors Tor_i^R(L,-) and Ext^i_R(L,-). For instance, we show the following: (a) if L is artinian and L' is noetherian, then Hom_R(L,L') has finite length; (b) if L and L' are artinian, then the tensor product L \otimes_R L' has finite length; (c) if L and L' are artinian, then Tor_i^R(L,L') is artinian, and Ext^i_R(L,L') is noetherian over the completion \hat R; and (d) if L is artinian and L' is Matlis reflexive, then Ext^i_R(L,L'), Ext^i_R(L',L), and Tor_i^R(L,L') are Matlis reflexive. Also, we study the vanishing behavior of these functors, and we include computations demonstrating the sharpness of our results.Comment: 24 page

Ascent of module structures, vanishing of Ext, and extended modules

Let (R,\m) and (S,\n) be commutative Noetherian local rings, and let $\phi:R\to S$ be a flat local homomorphism such that \m S = \n and the induced map on residue fields R/\m \to S/\n is an isomorphism. Given a finitely generated $R$-module $M$, we show that $M$ has an $S$-module structure compatible with the given $R$-module structure if and only if \Ext^i_R(S,M)=0 for each $i\ge 1$. We say that an $S$-module $N$ is {\it extended} if there is a finitely generated $R$-module $M$ such that $N\cong S\otimes_RM$. Given a short exact sequence $0 \to N_1\to N \to N_2\to 0$ of finitely generated $S$-modules, with two of the three modules $N_1,N,N_2$ extended, we obtain conditions forcing the third module to be extended. We show that every finitely generated module over the Henselization of $R$ is a direct summand of an extended module, but that the analogous result fails for the \m-adic completion.Comment: 16 pages, AMS-TeX; final version to appear in Michigan Math. J.; corrected proof of Main Theorem and made minor editorial changes; v3 has dedication to Mel Hochste

Presentations of rings with non-trivial semidualizing modules

Let R be a commutative noetherian local ring. A finitely generated R-module C is semidualizing if it is self-orthogonal and satisfies the condition Hom_R(C,C) \cong R. We prove that a Cohen-Macaulay ring R with dualizing module D admits a semidualizing module C satisfying R\ncong C \ncong D if and only if it is a homomorphic image of a Gorenstein ring in which the defining ideal decomposes in a cohomologically independent way. This expands on a well-known result of Foxby, Reiten and Sharp saying that R admits a dualizing module if and only if R is Cohen--Macaulay and a homomorphic image of a local Gorenstein ring.Comment: 16 pages, uses XY-pic; v.2 reorganized, main theorem revised, examples adde

Asymptotic Behavior of Ext functors for modules of finite complete intersection dimension

Let $R$ be a local ring, and let $M$ and $N$ be finitely generated $R$-modules such that $M$ has finite complete intersection dimension. In this paper we define and study, under certain conditions, a pairing using the modules \Ext_R^i(M,N) which generalizes Buchweitz's notion of the Herbrand diference. We exploit this pairing to examine the number of consecutive vanishing of \Ext_R^i(M,N) needed to ensure that \Ext_R^i(M,N)=0 for all $i\gg 0$. Our results recover and improve on most of the known bounds in the literature, especially when $R$ has dimension at most two

Dark Heritage

Peer reviewe

Tourism and toponymy: Commodifying and Consuming Place Names

Academic geographers have a long history of studying both tourism and place names, but have rarely made linkages between the two. Within critical toponymic studies there is increasing debate about the commodification of place names, but to date the role of tourism in this process has been almost completely overlooked. In some circumstances, toponyms can become tourist sights based on their extraordinary properties, their broader associations within popular culture, or their role as metanyms for some other aspect of a place. Place names may be sights in their own right or ‘markers’ of a sight and, in some cases, the marker may be more significant than the sight to which it refers. The appropriation of place names through tourism also includes the production and consumption of a broad range of souvenirs based on reproductions or replicas of the material signage that denote place names. Place names as attractions are also associated with a range of performances by tourists, and in some cases visiting a place name can be a significant expression of fandom. In some circumstances, place names can be embraced and promoted by tourism marketing strategies and are, in turn, drawn into broader circuits of the production and consumption of tourist space