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zvz3i-!|u(oc>9wPfLq+{&hU{CfPJiZXrFSg8`C=WRWkk+T6f^UAZ4`$o&=SXAIQPU Ni$qH+ttx}`e*hW?_V@q* diff --git a/sose2022/bachelorarbeit/arbeit.tex b/sose2022/bachelorarbeit/arbeit.tex index ccff3df..0ac19dd 100644 --- a/sose2022/bachelorarbeit/arbeit.tex +++ b/sose2022/bachelorarbeit/arbeit.tex @@ -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 - -\end{thebibliography} +\bibliographystyle{plain} +\bibliography{refs} + +%% 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} diff --git a/sose2022/bachelorarbeit/refs.bib b/sose2022/bachelorarbeit/refs.bib new file mode 100644 index 0000000..3fa5257 --- /dev/null +++ b/sose2022/bachelorarbeit/refs.bib @@ -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}, +}