# Tuesday, February 08 :::{.exercise title="?"} Show that chain-homotopic maps induce the same map in homology. ::: :::{.corollary title="?"} $\complex{\Tor}^{\rmod}(M, N)$ for $M, N\in \rmod$ is well-defined, since the identity $M \mapsvia{\id_M} M$ induces an isomorphism on $\complex{P}\tensor N \to \complex{P}'\tensor N$ for any two projective resolution $\complex{P}, \complex{P}' \covers M$. ::: :::{.proposition title="?"} If $f\in \rmod(M, N)$, then there is an induced morphism $\tilde f\in \Ch\rmod(\complex{P}^M, \complex{P}^N)$ between resolutions $\complex{P}^M\covers M, \complex{P}^N\covers N$, where $\tilde f$ is unique up to homotopy. ::: :::{.proof title="Hint"} For existence, use projectivity to lift through surjections onto kernels. For uniqueness, create $s_0$ such that $d_1 s_0 = d_0-g_0$ and check that $\im f_0 - g_0 \subseteq \ker d_0$ after lifting through $P_1^N\to \ker d_0$: \begin{tikzcd} \vdots && \vdots \\ {P_1^M} && {P_1^N} \\ \\ {P_0^M} && {P_0^N} \\ \\ M && N && {\ker d_0} \\ &&&&& 0 \\ 0 && 0 &&&& {} \arrow[from=6-1, to=8-1] \arrow[from=6-3, to=8-3] \arrow["f", from=6-1, to=6-3] \arrow["{d_0}", from=4-1, to=6-1] \arrow["{d_0'}", from=4-3, to=6-3] \arrow["{f_0, g_0}", from=4-1, to=4-3] \arrow[two heads, from=4-3, to=6-5] \arrow[from=6-5, to=7-6] \arrow["\exists"{pos=0.3}, shift right=2, curve={height=12pt}, dashed, from=4-1, to=6-5] \arrow[from=2-1, to=4-1] \arrow[from=2-3, to=4-3] \arrow["{\exists s}", dashed, from=4-1, to=2-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) ::: :::{.exercise title="?"} Show that a SES of modules induces a SES of chain complexes between their projective resolutions. Hint: use the following diagram. \begin{tikzcd} && \vdots && \vdots && \vdots \\ {} && {P_{21}} && \vdots && {P_{23}} \\ \\ 0 && {P_{11}} && {P_{11} \oplus P_{13}} && {P_{13}} && 0 \\ \\ 0 && A && B && C && 0 \arrow[from=6-1, to=6-3] \arrow[from=6-3, to=6-5] \arrow[from=6-5, to=6-7] \arrow[from=6-7, to=6-9] \arrow[from=4-7, to=4-9] \arrow[from=4-1, to=4-3] \arrow[from=2-3, to=4-3] \arrow[from=4-3, to=6-3] \arrow[from=4-7, to=6-7] \arrow[from=2-7, to=4-7] \arrow["\exists", dashed, hook, from=4-3, to=4-5] \arrow["\exists", dashed, two heads, from=4-5, to=4-7] \arrow["\exists", dashed, from=4-7, to=6-5] \arrow["{\therefore \exist}"', dotted, from=4-5, to=6-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) ::: :::{.exercise title="?"} Show that a SES of chain complexes induces a LES in their homology. There are about eight conditions one needs to check here (as in the snake lemma, of which this is a special case for two-term complexes). ::: :::{.exercise title="?"} Show that given $0\to M_1 \to M_2\to M_3\to 0$, taking projective resolutions and applying $(\wait )\tensor N$ yields a SES \[ 0\to \complex{P}^1\tensor N \to \complex{P}^2\tensor N \to \complex{P}^3\tensor N\to 0 ,\] so there is an induced LES in $\complex{\Tor}^{\rmod}$. ::: :::{.exercise title="?"} Show - $\Tor_0^{\rmod}(M, N) = M\tensor_R N$ - $\tau_{\geq 1} \complex{\Tor}^{\rmod}(M, N) = 0$ if either $M$ or $N$ is projective. - $\complex{\Tor}^R(M, N)$ is uniquely determined by these properties. ::: :::{.remark} Hint: \[ \Tor_0^{\rmod}(M, N) &= H_0(\complex{P}^M \tensor_R N) \\ &= \coker\qty{P_1\tensor N \to P_0\tensor_R N}\\ &= \coker(P_1\to P_0) \tensor_R N \\ &= M\tensor_R N .\] For vanishing, use that projective implies flat and exact complexes have zero higher homology. Note that if $M$ is projective, it is its own projective resolution. For uniqueness, induct on $i$: write $0\to K \to R\sumpower{J}\to M\to 0$, use that free implies projective, and consider the LES: \begin{tikzcd} \cdots && 0 && {\Tor_1^R(M, N) = \ker f} \\ \\ {\Tor_1(K, N)} && {\Tor_1(R\sumpower{I}, N) = 0} && {\Tor_1^R(M, N) = \ker f} \\ \\ {K\tensor_R N} && {R\sumpower{I}\tensor_R N} && {M\tensor_R N} \\ &&&& 0 \arrow[from=5-5, to=6-5] \arrow[from=5-3, to=5-5] \arrow["f", from=5-1, to=5-3] \arrow[from=3-1, to=3-3] \arrow[from=3-3, to=3-5] \arrow[from=3-5, to=5-1] \arrow[from=1-5, to=3-1] \arrow[from=1-3, to=1-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMTAsWzQsNSwiMCJdLFs0LDQsIk1cXHRlbnNvcl9SIE4iXSxbMiw0LCJSXFxzdW1wb3dlcntJfVxcdGVuc29yX1IgTiJdLFswLDQsIktcXHRlbnNvcl9SIE4iXSxbNCwyLCJcXFRvcl8xXlIoTSwgTikgPSBcXGtlciBmIl0sWzIsMiwiXFxUb3JfMShSXFxzdW1wb3dlcntJfSwgTikgPSAwIl0sWzAsMiwiXFxUb3JfMShLLCBOKSJdLFs0LDAsIlxcVG9yXzFeUihNLCBOKSA9IFxca2VyIGYiXSxbMiwwLCIwIl0sWzAsMCwiXFxjZG90cyJdLFsxLDBdLFsyLDFdLFszLDIsImYiXSxbNiw1XSxbNSw0XSxbNCwzXSxbNyw2XSxbOCw3XV0=) Now induct up using the isomorphisms in the LES. ::: :::{.exercise title="?"} Show that \[ \complex{\Tor}^{\rmod}(M, N) \cong \cxH(\complex{P}^M \tensor_R N) \cong \cxH(M\tensor_R \complex{P}^N )\cong \complex{ \Tor}^{\rmod}(N, M) .\] :::