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In this paper, we prove a result similar to results of Itoh and Hong-Ulrich, proving that integral closure of an ideal is compatible with specialization by a general element of that ideal for ideals of height at least two in a large class of rings. Moreover, we show integral closure of sufficiently large powers of the ideal is compatible with specialization by a general element of the original ideal. In a polynomial ring over an infinite field, we give a class of squarefree monomial ideals for which the integral closure is compatible with specialization by a general linear form.

This dissertation is based on joint work with Lindsey Hill. There are two main parts, which are linked by the common theme of the integral closure of the Rees algebra. In the first part of this dissertation, comprised of Chapter 3 and Chapter 4, we study the integral closure of the Rees algebra directly. In Chapter 3 we identify a bound for the multiplicity of the Rees algebra R[It] of a homogeneous ideal I generated in the same degree, and combine this result with theorems of Ulrich and Vasconcelos to obtain upper bounds on the number of generators of the integral closure of the Rees algebra as a module over R[It]. We also find various other upper bounds for this number, and compare them in the case of a monomial ideal generated in the same degree. In Chapter 4, inspired by the large depth assumption on the integral closure of R[It] in the results of Chapter 3, we obtain a lower bound for the depth of the associated graded ring and the Rees algebra of the integral closure filtration in terms of the dimension of the Cohen-Macaulay local ring R and the equimultiple ideal I. We finish the first part of this dissertation with a characterization of when the integral closure of R[It] is Cohen-Macaulay for height 2 ideals. In the second part of this dissertation, Chapter 5, we use the integral closure of the Rees algebra as a tool to discuss specialization of the integral closure of an ideal I. We prove that for ideals of height at least two in a large class of rings, the integral closure of I is compatible with specialization modulo general elements of I. This result is analogous to a result of Itoh and an extension by Hong and Ulrich which show that for ideals of height at least two in a large class of rings, the integral closure of I is compatible with specialization modulo generic elements of I. We then discuss specialization modulo a general element of the maximal ideal, rather than modulo a general element of the ideal I itself. In general it is not the case that the operations of integral closure and specialization modulo a general element of the maximal ideal are compatible, even under the assumptions of our main theorem. We prove that the two operations are compatible for local excellent algebras over fields of characteristic zero whenever R/I is reduced with depth at least 2, and conclude with a class of ideals for which the two operations appear to be compatible based on computations in Macaulay2.

Background: We previously reported that, in response to chemotherapy, gastro-oesophageal cancer cells (GEC) activate a human epidermal receptor (HER)-mediated survival response abrogated by HER inhibitors. In order to elucidate potential biomarkers for response to chemotherapy treatment and potential novel targets, we investigated the pathways which mediate chemo-induced HER activity in GEC cells. Methods: Apoptosis was measured by Flow Cytometry or PARP, Caspase 8/3 cleavage. HER1/HER2/HER3, Akt expression/activity and Heregulin (HRG) expression levels were determined by Western blotting. ADAM17 activity was measured using a fluorometric (MCA) assay kit while sTGF-α levels were analysed by ELISA. Results: Following treatment with chemotherapy, we found that increased HER1/HER2/HER3 activity was associated with increased Src-family kinase (SFK) activity. Using the SFK inhibitor AZD0530, we found that chemotherapy-induced HER1 activation was abolished, indicating that HER1 activation was mediated via SFKs. Furthermore, we found that the dual HER1/HER2 inhibitor lapatinib had no effect on chemo-induced SFK activity, indicating that SFKs act upstream of HER. Furthermore, we found that chemotherapy treatment resulted in increased ADAM-17 activity, TGF-α and heregulin levels in culture medium of GEC cells. ADAM17 inhibition attenuated chemotherapy-induced TGF-α and heregulin levels and HER1/3 activity. When cells were co-treated with heregulin-neutralizing antibodies, we found strongest inhibition of chemotherapy-induced HER1/HER3 and Akt activity. In addition, following treatment with AZD0530, we found complete inhibition of chemotherapy-induced ADAM17 activity and ligand shedding. Finally, when HER1/HER2/HER3, ADAM17 or SFK inhibitors were combined with chemotherapy, we found a potent increase in apoptosis in GEC cells. Conclusions: We provide strong evidence that chemotherapy induces HER1/HER2/HER3 activation via a ligand dependent mechanism in GEC cells and have demonstrated that SFK's, ADAM17 and heregulin are critical mediators of HER1/HER2/HER3 activation following chemotherapy. We propose a model in which chemotherapy results in SFK activation that in turn activates ADAM17-mediated shedding of heregulin. Heregulin then activates HER1/HER2/HER3 pro-survival signalling in an autocrine and paracrine fashion. Thus, inhibiting HER1/HER2/HER3, SFKs or ADAM17 may have therapeutic potential for sensitizing GEC tumors to chemotherapy. Citation Information: Mol Cancer Ther 2009;8(12 Suppl):C177.