# Wednesday, February 10 :::{.remark} Setup: Let \( \mathcal{A}, \mathcal{B} \) and \( F: \mca \to \mcb \) where \( \mca \) has enough projectives. Let \( P \mapsvia{\eps} A\in \mca \) be a projective resolution, we want to define the left derived functors \( L_i F(A) \da H_i(FP) \). ::: :::{.lemma title="?"} \( L_i F(A) \) is well-defined up to natural isomorphism, i.e. if \( Q\to A \) is a projective resolution, then there are canonical isomorphism \( H_i (FP) \mapsvia{\sim} H_i(FQ) \). In particular, changing projective resolutions yields a new functor \( \hat{L}_iF \) which are naturally isomorphic to $F$. ::: :::{.proof title="?"} We can set up the following situation \begin{tikzcd} P && A && 0 \\ \\ Q && A && 0 \arrow["{\one_A}", from=1-3, to=3-3] \arrow["{\exists f}", from=1-1, to=3-1] \arrow["{\eps_P}"{description}, from=1-1, to=1-3] \arrow["{\eps_Q}"{description}, from=3-1, to=3-3] \arrow[from=3-3, to=3-5] \arrow[from=1-3, to=1-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCJQIl0sWzIsMCwiQSJdLFs0LDAsIjAiXSxbMiwyLCJBIl0sWzAsMiwiUSJdLFs0LDIsIjAiXSxbMSwzLCJcXG9uZV9BIl0sWzAsNCwiXFxleGlzdHMgZiJdLFswLDEsIlxcZXBzX1AiLDFdLFs0LDMsIlxcZXBzX1EiLDFdLFszLDVdLFsxLDJdXQ==) Here \( f \) exists by the comparison theorem, and thus there are induced maps $f_*: H_*(FP) \to H_*(FQ)$ by abuse of notation -- really, this is more like \( (f_*)_i = H_u (Ff) \). We're using that both $F$ and $H_i$ are both additive functors. Note that the lift $f$ of $\one_A$ is not unique, but any other lift is chain homotopic to $f$, i.e. $f-f' = ds + sd$ where $s: P \to Q[1]$. So they induce the same maps on homology, i.e. $f_*' = f_*$. Thus the isomorphism is canonical in the sense that it doesn't depend on the choice of lift. Similarly there exists a $g:Q\to P$ lifting $\one_A$, and so $gf$ and $\one_P$ are both chain maps lifting $\one_A$, since it's the composition of two maps lifting $\one_A$. So they induce the same map on homology by the same reasoning above. We can write \[ g_* f_* = (gf)_* = (\one_{FP})_* = \one_{H_*(FP)} ,\] and similarly $f_* g_* = \one_{H_*(FQ)}$, making $f_*$ an isomorphism. ::: :::{.corollary title="?"} If $A$ is projective, then $L^{>0} FA = 0$. ::: :::{.proof title="?"} Use the projective resolution $\cdots \to 0 \to A \mapsvia{\one_A} A \to 0 \to \cdots$. In this case $H_{>0}(FP) = 0$. ::: :::{.remark} This is an interesting result, since it doesn't depend on the functor! Short aside on $F\dash$acyclic objects -- we don't need something as strong as a *projective* resolution. ::: :::{.definition title="$F\dash$acyclic objects"} An object $Q\in \mca$ is **$F\dash$acyclic** if $L_{>0}FQ = 0$. ::: :::{.remark} Note that projective implies $F\dash$acyclic for every $F$, but not conversely. For example, flat \(R\dash\)modules are acyclic for $\wait \tensor_R \wait$. In general, flat does not imply projective, although projective implies flat. ::: :::{.definition title="$F\dash$acyclic resolutions"} An **$F\dash$acyclic resolution** of $A$ is a left resolution $Q\to A$ for which every $Q_i$ is $F\dash$acyclic. ::: :::{.remark} One can compute $L_iF(A) \cong H_i(FQ)$ for any $F\dash$acyclic resolution. For the $L_i F$ to be functors, we need to define them on maps! ::: :::{.lemma title="?"} If \( f: A\to A' \), there is a natural associated morphism $L_i F(f): L_iF(A) \to L_iF(A')$. ::: :::{.proof title="?"} Again use the comparison theorem: \begin{tikzcd} P && A && 0 \\ \\ {P'} && {A'} && 0 \arrow["f", from=1-3, to=3-3] \arrow[from=3-1, to=3-3] \arrow[from=1-1, to=1-3] \arrow[from=1-3, to=1-5] \arrow[from=3-3, to=3-5] \arrow["{\exists \tilde f}"', from=1-1, to=3-1] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCJQIl0sWzIsMCwiQSJdLFsyLDIsIkEnIl0sWzAsMiwiUCciXSxbNCwwLCIwIl0sWzQsMiwiMCJdLFsxLDIsImYiXSxbMywyXSxbMCwxXSxbMSw0XSxbMiw1XSxbMCwzLCJcXGV4aXN0cyBcXHRpbGRlIGYiLDJdXQ==) Then there is an induced map $\tilde f_*: H_*(FP) \to H_*(FP')$, noting that one first needs to apply $F$ to the above diagram. As before, this is independent of the lift using the same argument as before, using the additivity of $F$ and $H_*$ and the chain homotopy is pushed through $F$ appropriately. So set $(L_i F)(f) \da (\tilde f_*)_i$. ::: :::{.remark} We can now pick up the theorem from the end of last time: ::: :::{.theorem title="Left-derived functors are additive"} $L_iF: \mathcal{A}\to \mathcal{B}$ are additive functors. ::: :::{.proof title="?"} Done last time! ::: :::{.theorem title="Existence of connecting maps for left-derived functors"} Using the same assumptions as before, given a SES \[ 0 \to A' \to A \to A'' \to 0 \] there are natural connecting maps \( \delta \) yielding a LES \begin{tikzcd} &&&& \cdots \\ \\ {L_iF(A')} && {L_iF(A)} && {L_iF(A'')} \\ \\ {L_{i-1}F(A')} && \cdots && \cdots \\ \\ {FA'} && FA && {FA''} && 0 \arrow["{\delta_i}"', from=3-5, to=5-1, out=0, in=180] \arrow[from=3-1, to=3-3] \arrow[from=3-3, to=3-5] \arrow["{\delta_{i+1}}"', from=1-5, to=3-1, out=0, in=180] \arrow[dashed, from=5-1, to=5-3] \arrow[from=7-1, to=7-3] \arrow[from=7-3, to=7-5] \arrow[from=7-5, to=7-7] \arrow[dashed, from=5-3, to=5-5] \arrow["{\delta_1}"', from=5-5, to=7-1, out=0, in=180] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMTEsWzAsMiwiTF9pRihBJykiXSxbMiwyLCJMX2lGKEEpIl0sWzQsMiwiTF9pRihBJycpIl0sWzAsNCwiTF97aS0xfUYoQScpIl0sWzQsMCwiXFxjZG90cyJdLFsyLDQsIlxcY2RvdHMiXSxbMCw2LCJGQSciXSxbMiw2LCJGQSJdLFs0LDYsIkZBJyciXSxbNiw2LCIwIl0sWzQsNCwiXFxjZG90cyJdLFsyLDMsIlxcZGVsdGFfaSIsMl0sWzAsMV0sWzEsMl0sWzQsMCwiXFxkZWx0YV97aSsxfSIsMl0sWzMsNSwiIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzYsN10sWzcsOF0sWzgsOV0sWzUsMTAsIiIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFsxMCw2LCJcXGRlbHRhXzEiLDJdXQ==) ::: :::{.proof title="?"} Using the Horseshoe lemma, we can obtain the following map: \begin{tikzcd} &&& 0 \\ \\ {P'} &&& {A'} \\ \\ P &&& A && 0 \\ \\ {P''} &&& {A''} \\ \\ &&& 0 \arrow["\exists", dashed, from=5-1, to=5-4] \arrow[from=7-1, to=7-4] \arrow[from=7-4, to=9-4] \arrow[from=5-4, to=7-4] \arrow[from=5-4, to=5-6] \arrow[from=3-4, to=5-4] \arrow[from=3-1, to=3-4] \arrow[from=3-1, to=5-1] \arrow[from=5-1, to=7-1] \arrow[from=1-4, to=3-4] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsOSxbMCwyLCJQJyJdLFswLDQsIlAiXSxbMCw2LCJQJyciXSxbMyw2LCJBJyciXSxbMyw0LCJBIl0sWzMsMiwiQSciXSxbMywwLCIwIl0sWzMsOCwiMCJdLFs1LDQsIjAiXSxbMSw0LCJcXGV4aXN0cyIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFsyLDNdLFszLDddLFs0LDNdLFs0LDhdLFs1LDRdLFswLDVdLFswLDFdLFsxLDJdLFs2LDVdXQ==) So we get a SES of complexes over \( \mathcal{A} \), $0 \to P' \to P \to P'' \to 0$. One can use that $P = P' \oplus P''$, or alternatively that each $P_n''$ is a projective \(R\dash\)module, to see that there are splittings \begin{tikzcd} 0 && {P'} && P && {P''} && 0 \arrow[from=1-1, to=1-3] \arrow["f", from=1-3, to=1-5] \arrow["g", from=1-5, to=1-7] \arrow[from=1-7, to=1-9] \arrow["{g'}"{description}, curve={height=18pt}, dashed, from=1-7, to=1-5] \arrow["{f'}"{description}, curve={height=18pt}, dashed, from=1-5, to=1-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCIwIl0sWzIsMCwiUCciXSxbNCwwLCJQIl0sWzYsMCwiUCcnIl0sWzgsMCwiMCJdLFswLDFdLFsxLDIsImYiXSxbMiwzLCJnIl0sWzMsNF0sWzMsMiwiZyciLDEseyJjdXJ2ZSI6Mywic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzIsMSwiZiciLDEseyJjdXJ2ZSI6Mywic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d) Note that this can be phrased in terms of $g'g = \one, f'f = \one$, or $g'g + f'f = \one$. Since $F$ is additive, it preserves all of these relations, particularly the ones that define being split exact. So additive functors preserve split exact sequences. Thus $0 \to FP' \to FP \to FP'' \to 0$ is still split exact, even though $F$ is only right exact. Now take homology and use the LES in homology to get the desired LES above, and $\delta$ is the connecting morphism that comes from the snake lemma. Proving naturality: we start with the following setup. % https://q.uiver.app/?q=WzAsMTAsWzAsMCwiMCJdLFsyLDAsIkEnIl0sWzQsMCwiQSJdLFs2LDAsIkEnJyJdLFs4LDAsIjAiXSxbMCwyLCIwIl0sWzIsMiwiQiciXSxbNCwyLCJCIl0sWzYsMiwiQicnIl0sWzgsMiwiMCJdLFsxLDYsImcnIl0sWzIsNywiZyJdLFszLDgsImcnJyJdLFswLDFdLFsxLDJdLFsyLDNdLFszLDRdLFs1LDZdLFs2LDddLFs3LDhdLFs4LDldXQ== \begin{tikzcd} 0 && {A'} && A && {A''} && 0 \\ \\ 0 && {B'} && B && {B''} && 0 \arrow["{g'}", from=1-3, to=3-3] \arrow["g", from=1-5, to=3-5] \arrow["{g''}", from=1-7, to=3-7] \arrow[from=1-1, to=1-3] \arrow[from=1-3, to=1-5] \arrow[from=1-5, to=1-7] \arrow[from=1-7, to=1-9] \arrow[from=3-1, to=3-3] \arrow[from=3-3, to=3-5] \arrow[from=3-5, to=3-7] \arrow[from=3-7, to=3-9] \end{tikzcd} Naturality of $\delta$ will be showing that the following square commutes: \begin{tikzcd} L_{i+1}F(A'') \ar[r, "\delta"] \ar[d] & L_iF(A') \ar[d] \\ L_{i+1}F(B'') \ar[r, "\delta"] & L_iF(B') \end{tikzcd} We now apply the horseshoe lemma several times: % https://q.uiver.app/?q=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 \begin{tikzcd} 0 && {P'} && \textcolor{rgb,255:red,214;green,92;blue,92}{P} && {P''} && 0 \\ \\ \\ & 0 && {A'} && A && {A''} && 0 \\ \\ & 0 && {B'} && B && {B''} && 0 \\ \\ \\ 0 && {Q'} && \textcolor{rgb,255:red,214;green,92;blue,92}{Q} && {Q''} && 0 \arrow["{g'}", from=4-4, to=6-4] \arrow["g", from=4-6, to=6-6] \arrow["{g''}", from=4-8, to=6-8] \arrow[from=4-2, to=4-4] \arrow[from=4-4, to=4-6] \arrow[from=4-6, to=4-8] \arrow[from=4-8, to=4-10] \arrow[from=6-2, to=6-4] \arrow[from=6-4, to=6-6] \arrow[from=6-6, to=6-8] \arrow[from=6-8, to=6-10] \arrow[from=1-7, to=4-8] \arrow[from=1-3, to=4-4] \arrow[color={rgb,255:red,214;green,92;blue,92}, dashed, from=1-5, to=4-6] \arrow[from=9-3, to=6-4] \arrow[from=9-7, to=6-8] \arrow[color={rgb,255:red,214;green,92;blue,92}, dashed, from=9-5, to=6-6] \arrow[from=9-3, to=9-5] \arrow[from=9-5, to=9-7] \arrow["{\exists G'}"{description}, curve={height=-6pt}, dotted, from=1-3, to=9-3] \arrow["{\exists G''}"{description}, curve={height=-12pt}, dotted, from=1-7, to=9-7] \arrow["{\exists G}"{description}, curve={height=6pt}, dashed, from=1-5, to=9-5] \arrow[from=1-3, to=1-5] \arrow[from=1-5, to=1-7] \end{tikzcd} It turns out (details omitted see Weibel p. 46) that $G$ can be chosen such that we get a commutative diagram of chain complexes with exact rows: \begin{tikzcd} 0 && {P'} && P && {P''} && 0 \\ \\ 0 && {Q'} && Q && {Q''} && 0 \arrow[from=3-1, to=3-3] \arrow[from=3-3, to=3-5] \arrow[from=3-5, to=3-7] \arrow[from=3-7, to=3-9] \arrow[from=1-1, to=1-3] \arrow[from=1-3, to=1-5] \arrow[from=1-5, to=1-7] \arrow[from=1-7, to=1-9] \arrow["{G'}"{description}, from=1-3, to=3-3] \arrow["G"{description}, from=1-5, to=3-5] \arrow["{G''}"{description}, from=1-7, to=3-7] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMTAsWzAsMCwiMCJdLFsyLDAsIlAnIl0sWzQsMCwiUCJdLFs2LDAsIlAnJyJdLFsyLDIsIlEnIl0sWzQsMiwiUSJdLFs2LDIsIlEnJyJdLFswLDIsIjAiXSxbOCwwLCIwIl0sWzgsMiwiMCJdLFs3LDRdLFs0LDVdLFs1LDZdLFs2LDldLFswLDFdLFsxLDJdLFsyLDNdLFszLDhdLFsxLDQsIkcnIiwxXSxbMiw1LCJHIiwxXSxbMyw2LCJHJyciLDFdXQ==) We proved naturality of the connecting maps $\bd$ in the corresponding LES in homology in general (see prop. 1.3.4). This translates to naturality of the maps $\delta_i: L_{i} (A'') \to L_{i-1}(A')$. ::: :::{.remark} See exercise 2.4.3 for "dimension shifting". This is a useful tool for inductive arguments. :::