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Register a new userHomotopy categories of unbounded complexes of projective modules
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Authors
Yuji Yoshino
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We develop in this paper a stable theory for projective complexes, by which we mean to consider a chain complex of finitely generated projective modules as an object of the factor category of the homotopy category modulo split complexes. As a result of the stable theory we are able to prove that a complex of finitely generated projective modules over a generically Gorenstein ring is exact if and only if its dual complex is exact. This shows the dependence of total reflexivity conditions for modules over a generically Gorenstein ring.
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In this paper, we define and study a notion of Ding projective dimension for complexes of left modules over associative rings. In particular, we consider the class of homologically bounded below complexes of left R-modules, and show that Ding projective dimension has a nice functorial description.
The Buchsbaum-Eisenbud-Horrocks Conjecture predicts that if M is a non-zero module of finite length and finite projective dimension over a local ring R of dimension d, then the i-th Betti number of M is at least d choose i. This conjecture implies that the sum of all the Betti numbers of such a module must be at least 2^d. We prove the latter holds in a large number of cases.
Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $P$ is called $S$-projective provided that the induced sequence $0rightarrow {rm Hom}_R(P,A)rightarrow {rm Hom}_R(P,B)rightarrow {rm Hom}_R(P,C)rightarrow 0$ is $S$-exact for any $S$-short exact sequence $0rightarrow Arightarrow Brightarrow Crightarrow 0$. Some characterizations and properties of $S$-projective modules are obtained. The notion of $S$-semisimple modules is also introduced. A ring $R$ is called an $S$-semisimple ring provided that every free $R$-module is $S$-semisimple. Several characterizations of $S$-semisimple rings are provided by using $S$-semisimple modules, $S$-projective modules, $S$-injective modules and $S$-split $S$-exact sequences.
We study the topology of a random cubical complex associated to Bernoulli site percolation on a cubical grid. We begin by establishing a limit law for homotopy types. More precisely, looking within an expanding window, we define a sequence of normalized counting measures (counting connected components according to homotopy type), and we show that this sequence of random probability measures converges in probability to a deterministic probability measure. We then investigate the dependence of the limiting homotopy measure on the coloring probability $p$, and our results show a qualitative change in the homotopy measure as $p$ crosses the percolation threshold $p=p_c$. Specializing to the case of $d=2$ dimensions, we also present empirical results that raise further questions on the $p$-dependence of the limiting homotopy measure.
The notion of $times$-homotopy from cite{DocHom} is investigated in the context of the category of pointed graphs. The main result is a long exact sequence that relates the higher homotopy groups of the space $Hom_*(G,H)$ with the homotopy groups of $Hom_*(G,H^I)$. Here $Hom_*(G,H)$ is a space which parametrizes pointed graph maps from $G$ to $H$ (a pointed version of the usual $Hom$ complex), and $H^I$ is the graph of based paths in $H$. As a corollary it is shown that $pi_i big(Hom_*(G,H) big) cong [G,Omega^i H]_{times}$, where $Omega H$ is the graph of based closed paths in $H$ and $[G,K]_{times}$ is the set of $times$-homotopy classes of pointed graph maps from $G$ to $K$. This is similar in spirit to the results of cite{BBLL}, where the authors seek a space whose homotopy groups encode a similarly defined homotopy theory for graphs. The categorical connections to those constructions are discussed.
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