Sheaves of abelian \(l\)-groups
Let \(G\) denote an abelian \(l\)-group (lattice ordered group) and let \(R\) denote an integral domain with quotient field \(K.\) A GCD domain \(R\) (every pair of nonzero elements has a GCD) has the property that its group of divisibility \(G(R)=\{uR:u\in K\backslash \{0\}\}\) ordered by \(uR\geq vR\) if and only if \(uR\subseteq vR\) is an \(l\)-group, with \(uR\vee vR=uR\cap vR\) and \(uR\wedge vR=((uR+vR)^{-1})^{-1}\) where, for a fractional ideal \(A,\) we have \(A^{-1}=\{x\in K:\) \(xA\subseteq R\}.\) A Bézout domain (every two generated ideal is principal) is a GCD domain for which \(uR\wedge vR\) gets a more direct description \(uR\wedge vR=uR+vR\) because if \(A=uR\) then \(A^{-1}=\frac{1 }{u}R\) and \((A^{-1})^{-1}=uR.\) So the link between Bézout domains and \(l\) -groups is more direct. Perhaps that is why \textit{P. Jaffard} [J. Math. Pures Appl. (9) 32, 203--280 (1953; Zbl 0051.01303)] and \textit{J. Ohm} [Trans. Am. Math. Soc. 122, 321--333 (1966; Zbl 0142.28502)] were able to show that every \(l\)-group can be realized as the group of divisibility \(G(R)\) of a Bézout domain \(R\), using an easy group algebra construction. \textit{W. Rump} and \textit{Y. C. Yang} [Bull. Lond. Math. Soc. 40, No. 2, 263--273 (2008; Zbl 1151.13001)] studied the correspondence between abelian \(l\)-groups and Bézout domains. They verified Anderson's conjecture (every \(l\)-embedding \(G(R)\rightarrow H\) comes from an extension \(R\rightarrow S\) of Bézout domains) and used the refinement provided by it to extend the Jaffard Ohm correspondence to the correspondence between sheaves and spectral spaces, using \textit{M. Hochster}'s work [Trans. Am. Math. Soc. 142, 43--60 (1969; Zbl 0184.29401)]. They also show that the spectrum of a Bézout domain and the spectrum of its corresponding abelian \(l\)-group present a perfect example for Hochster's duality of spectral spaces. The notion of the spectrum of an abelian \(l\)-group and the corresponding sheaf was introduced by \textit{K. Keimel} in [Lect. Notes Math. 248, 1--98 (1971; Zbl 0231.06023)]. The points of Keimel's spectrum are the proper prime \(l\) -ideals, i.e., the convex subgroups \(p\) \(\subsetneq \) \(G\) such that \(G/p\) is totally ordered. As \(I\) varies in the set of \(l\)-ideals of \(G\), the sets \( U(I)=\{p\in \mathrm{Spec}(G)|I\nsubseteq p\}\) are the open sets of a topology of \( \mathrm{Spec}(G)\). Now \(\mathrm{Spec}(G)\) is not quasi-compact, generally. One can correct this problem by adding the improper prime \(l\)-ideal \(G\) to \(\mathrm{Spec}(G)\) with open sets defined as above. The resulting topological space is a spectral space in Hochster's terminology. Now every spectral space \(X\) can be retopologized by using the quasi-compact open sets as a basis of closed sets of a new topology. This new topology is called the inverse topology and the topological space the inverse space and is denoted by \(X_{\mathrm{inv}}.\) According to Hochster \(X_{\mathrm{inv}}\) is spectral too. So \(\mathrm{Spec}_{\mathrm{inv}}(G),\) with the modified spectrum, is a spectral space. \(\mathrm{Spec}_{\mathrm{inv}}(G)\) for an \(l\)-group showed up in the above mentioned work of Rump and Yang. The author of the paper under review presents a direct and elementary approach without appealing to Jaffard-Ohm, studying \(\mathrm{Spec}_{\mathrm{inv}}(G)\) and the associated sheaf. He analyzes the relationship between an \(l\)-group \(H\) and an \(l\)-subgroup \(G\) via the structure sheaves on the inverse spectra. The structure sheaves are to give a sheaf-theoretic presentation of the valuation theory of a field. He identifies the group of Cartier divisors of an integral affine scheme canonically with an \(l\)-group and pays special attention to Prüfer domains, where the ties between rings and \(l\)-groups are particularly close. Reviewer's remarks: (1) It appears that in multiplicative ideal theory there are several types of integral domains to which \(l\)-groups can be associated. For example a, so-called, Prüfer \(v\)- Multiplication domain \(D\) is characterized by the fact that the group of \(t\)-invertible \(t\)-ideals of \(D,\) under \(t\)-multiplication, is an \(l\)-group (see Proposition 2.4 [\textit{M. Zafrullah}, in: Non-Noetherian commutative ring theory. Dordrecht: Kluwer Academic Publishers. Math. Appl., Dordr. 520, 429--457 (2000; Zbl 0988.13003)]). It may be noted that ``\(t\)-invertible'' generalizes ``invertible''. Similar instances of association of \(l\)-groups with specific domains can be found in [\textit{D. D. Anderson} et al., Commun. Algebra 37, No. 9, 3018--3043 (2009; Zbl 1185.13003)]. (2) A portion of Proposition 4.11 is missing. Philip Scowcroft, in his review, in Mathematical Reviews, of the article reports that the complete Proposition is: ``Let \(\phi :G\rightarrow H\)and \(\psi :H\rightarrow K\) be \(l\)-subgroups. If \(G=\mathcal{E}_{K,G}\)(\(\mathrm{Spec}_{\mathrm{inv}}(K)\)) then also\(G=\mathcal{E} _{H,G}(\mathrm{Spec}_{\mathrm{inv}}(H))\)''.
scientific article