add first part of introduction

2022-04-06 11:49:53 +02:00
parent 9aab12b331
commit 1b3657879f
3 changed files with 86 additions and 11 deletions
@@ -21,6 +21,32 @@

 \section{Einleitung}

+Aus der kommutativen Algebra ist für einen kommutativen Ring $A$ und $A$-Moduln
+$M, N, P$ die Adjunktion
+\[
+\text{Hom}_A(M \otimes_A N, P) = \text{Hom}_A(M, \text{Hom}_A(N, P))
+\] bekannt. In der klassischen homologischen Algebra definiert man außerdem
+die Funktoren $\text{Ext}_A^{i}(N, -)$ und $\text{Tor}_A^{i}(-, N)$, als
+Ableitungen der Funktoren $\text{Hom}_A(N, -)$ und $- \otimes_A N$. Nun stellt sich
+die Frage, ob zwischen diesen ein analoges Adjunktionsresultat gilt.
+
+Die Antwort ist nein, denn angenommen $\text{Ext}^{i}(N, -)$ wäre rechtsadjungiert, dann
+folgte, dass $\text{Ext}_A^{i}(N, -)$ linksexakt sei. Die exakte Folge
+\[
+\begin{tikzcd}
+    0 \arrow{r} & \Z \arrow{r} & \Z \arrow{r} & \Z / 2 \Z \arrow{r} & 0
+\end{tikzcd}
+\] in $\Z$-Mod lieferte dann die exakte Folge
+\[
+\begin{tikzcd}
+    \text{Ext}^{0}_{\Z}(\Z, \Z / 2 \Z) \arrow{r} & \text{Ext}^{1}_{\Z}(\Z, \Z) \arrow{r}
+                                            & \text{Ext}^{1}_{\Z}(\Z, \Z)
+\end{tikzcd}
+.\] Da $\text{Ext}^{0}_{\Z}(\Z, \Z / 2 \Z) = \text{Hom}_{\Z}(\Z, \Z / 2 \Z) \neq 0$, ist
+jedoch 
+
+\newpage
+
 \section{Derivierte Kategorien und abgeleitete Funktoren}

 Seien $\mathcal{A}, \mathcal{B}$ abelsche Kategorien und sei $F\colon \mathcal{A} \to 
@@ -2187,17 +2213,20 @@ Jetzt können wir alles zusammentragen und erhalten:
    $\com{\text{Hom}}(\com{N}, \com{P})$ K-injektiv ist.
 \end{proof}

-% TODO: zitate richtig machen
-\begin{thebibliography}{9}
-\bibitem{hartshorne}
-Hartshorne, R. \emph{Residues and duality.} Lecture Notes in Math. 20, Springer-Verlag (1966)
-\bibitem{spaltenstein}
-%N. Spaltenstein. Resolutions of unbounded complexes. \emph{Composito Mathematica 65}. (1988)
-Spaltenstein, N. \emph{Resolutions of unbounded complexes.} Compositio Mathematica, Tome 65 (1988) no. 2, pp. 121-154. %http://www.numdam.org/item/CM_1988__65_2_121_0/
-\bibitem{set-theoretic}
-%Charles A. Weibel. An introduction to homological algebra. \emph{Cambridge studies in advanced mathematics}. 38 (1988)
-Weibel, C. \emph{An Introduction to Homological Algebra}. Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press (1994).%  doi:10.1017/CBO9781139644136
+\bibliographystyle{plain}
+\bibliography{refs}

-\end{thebibliography}
+%% TODO: zitate richtig machen
+%\begin{thebibliography}{9}
+%\bibitem{hartshorne}
+%Hartshorne, R. \emph{Residues and duality.} Lecture Notes in Math. 20, Springer-Verlag (1966)
+%\bibitem{spaltenstein}
+%%N. Spaltenstein. Resolutions of unbounded complexes. \emph{Composito Mathematica 65}. (1988)
+%Spaltenstein, N. \emph{Resolutions of unbounded complexes.} Compositio Mathematica, Tome 65 (1988) no. 2, pp. 121-154. %http://www.numdam.org/item/CM_1988__65_2_121_0/
+%\bibitem{set-theoretic}
+%%Charles A. Weibel. An introduction to homological algebra. \emph{Cambridge studies in advanced mathematics}. 38 (1988)
+%Weibel, C. \emph{An Introduction to Homological Algebra}. Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press (1994).%  doi:10.1017/CBO9781139644136
+%
+%\end{thebibliography}

 \end{document}
@@ -0,0 +1,46 @@
+@article {spaltenstein,
+    AUTHOR = {Spaltenstein, N.},
+     TITLE = {Resolutions of unbounded complexes},
+   JOURNAL = {Compositio Math.},
+  FJOURNAL = {Compositio Mathematica},
+    VOLUME = {65},
+      YEAR = {1988},
+    NUMBER = {2},
+     PAGES = {121--154},
+      ISSN = {0010-437X},
+   MRCLASS = {18E25 (32C35)},
+  MRNUMBER = {932640},
+MRREVIEWER = {Michael M. Kapranov},
+       URL = {http://www.numdam.org/item?id=CM_1988__65_2_121_0},
+}
+
+@book {hartshorne,
+    AUTHOR = {Hartshorne, Robin},
+     TITLE = {Residues and duality},
+    SERIES = {Lecture Notes in Mathematics, No. 20},
+      NOTE = {Lecture notes of a seminar on the work of A. Grothendieck,
+              given at Harvard 1963/64,
+              With an appendix by P. Deligne},
+ PUBLISHER = {Springer-Verlag, Berlin-New York},
+      YEAR = {1966},
+     PAGES = {vii+423},
+   MRCLASS = {14.55},
+  MRNUMBER = {0222093},
+MRREVIEWER = {R. L. Knighten},
+}
+
+@book {set-theoretic,
+    AUTHOR = {Weibel, Charles A.},
+     TITLE = {An introduction to homological algebra},
+    SERIES = {Cambridge Studies in Advanced Mathematics},
+    VOLUME = {38},
+ PUBLISHER = {Cambridge University Press, Cambridge},
+      YEAR = {1994},
+     PAGES = {xiv+450},
+      ISBN = {0-521-43500-5; 0-521-55987-1},
+   MRCLASS = {18-01 (16-01 17-01 20-01 55Uxx)},
+  MRNUMBER = {1269324},
+MRREVIEWER = {Kenneth A. Brown},
+       DOI = {10.1017/CBO9781139644136},
+       URL = {https://doi.org/10.1017/CBO9781139644136},
+}