{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T07:13:16Z","timestamp":1775805196205,"version":"3.50.1"},"reference-count":9,"publisher":"American Mathematical Society (AMS)","issue":"6","license":[{"start":{"date-parts":[[2005,5,10]],"date-time":"2005-05-10T00:00:00Z","timestamp":1115683200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Trans. Amer. Math. Soc."],"abstract":"<p>\n                    In this paper we compute some derived functors\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E x t\">\n                        <mml:semantics>\n                          <mml:mi>Ext<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\operatorname {Ext}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of the internal homomorphism functor in the category of modules over the representation Green functor. This internal homomorphism functor is the left adjoint of the box product. When the group is a cyclic\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2\">\n                        <mml:semantics>\n                          <mml:mn>2<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -group, we construct a projective resolution of the module fixed point functor, and that allows a direct computation of the graded Green functor\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E x t\">\n                        <mml:semantics>\n                          <mml:mi>Ext<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\operatorname {Ext}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . When the group is\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G equals double-struck upper Z slash 2 times double-struck upper Z slash 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>G<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>\n                              \u00d7\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">G=\\mathbb {Z}\/2\\times \\mathbb {Z}\/2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , we can still build a projective resolution, but we do not have explicit formulas for the differentials. The resolution is built from long exact sequences of projective modules over the representation functor for the subgroups of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G\">\n                        <mml:semantics>\n                          <mml:mi>G<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">G<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    by using exact functors between these categories of modules. This induces a filtration which gives a spectral sequence which converges to the desired\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E x t\">\n                        <mml:semantics>\n                          <mml:mi>Ext<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\operatorname {Ext}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    functors.\n                  <\/p>","DOI":"10.1090\/s0002-9947-04-03566-4","type":"journal-article","created":{"date-parts":[[2005,2,28]],"date-time":"2005-02-28T12:43:12Z","timestamp":1109594592000},"page":"2253-2289","source":"Crossref","is-referenced-by-count":1,"special_numbering":"841","title":["Homological algebra for the representation Green functor for abelian groups"],"prefix":"10.1090","volume":"357","author":[{"given":"Joana","family":"Ventura","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2004,5,10]]},"reference":[{"key":"1","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0095821","volume-title":"Green functors and $G$-sets","volume":"1671","author":"Bouc, Serge","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/3540635505"},{"key":"2","first-page":"183","article-title":"Contributions to the theory of induced representations","author":"Dress, Andreas W. M.","year":"1973"},{"key":"3","unstructured":"L. Gaunce Lewis, Jr. The box product of Mackey functors. Unpublished."},{"key":"4","unstructured":"L. Gaunce Lewis, Jr. The theory of Green functors. Unpublished notes, 1981."},{"key":"5","series-title":"Graduate Texts in Mathematics","isbn-type":"print","volume-title":"Categories for the working mathematician","volume":"5","author":"Mac Lane, Saunders","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0387984038","edition":"2"},{"key":"6","series-title":"Cambridge Studies in Advanced Mathematics","isbn-type":"print","volume-title":"A user's guide to spectral sequences","volume":"58","author":"McCleary, John","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0521567599","edition":"2"},{"issue":"6","key":"7","doi-asserted-by":"publisher","first-page":"1865","DOI":"10.2307\/2154915","article-title":"The structure of Mackey functors","volume":"347","author":"Th\u00e9venaz, Jacques","year":"1995","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"issue":"23","key":"8","first-page":"299","article-title":"Simple Mackey functors","author":"Th\u00e9venaz, Jacques","year":"1990","journal-title":"Rend. Circ. Mat. 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