Remark 15.95.5. Let $A$ be a ring and let $f \in A$ be a nonzerodivisor. Let $M^\bullet$ be a complex of $f$-torsion free $A$-modules. For every $i$ set $\overline{M}^ i = M^ i/fM^ i$. Denote $B^ i \subset Z^ i \subset \overline{M}^ i$ the boundaries and cocycles for the differentials on the complex $\overline{M}^\bullet = M^\bullet \otimes _ A A/fA$. We claim that there exists a commutative diagram

$\xymatrix{ 0 \ar[r] & B^{i + 1} \ar[r] \ar@{=}[d] & B^{i + 1} \oplus B^ i \ar[r] \ar[d]^{s, s'} & B^ i \ar[r] \ar[d] & 0 \\ 0 \ar[r] & B^{i + 1} \ar[r]^-s & (\eta _ fM)^ i /f(\eta _ fM)^ i \ar[r]^-t & Z^ i \ar[r] & 0 }$

with exact rows. Here are the constructions of the maps

1. If $x \in (\eta _ fM)^ i$ then $x = f^ ix'$ with $d^ i(x') = 0$ in $\overline{M}^{i + 1}$. Hence we can define the map $t$ by sending $x$ to the class of $x'$.

2. If $y \in M^{i + 1}$ has class $\overline{y}$ in $B^{i + 1} \subset \overline{M}^{i + 1}$ then we can write $y = fy' + d^ i(x)$ for $y' \in M^{i + 1}$ and $x \in M^ i$. Hence we can define the map $s$ sending $\overline{y}$ to the class of $f^{i + 1}x$ in $(\eta _ fM)^ i /f(\eta _ fM)^ i$; we omit the verification that this is well defined.

3. If $x \in M^ i$ has class $\overline{x}$ in $B^ i \subset \overline{M}^ i$ then we can write $x = fx' + d^{i - 1}(z)$ for $x' \in M^ i$ and $z \in M^{i - 1}$. We define the map $s'$ by sending $\overline{x}$ to the class of $f^ i d^{i - 1}(z)$ in $(\eta _ fM)^ i/f(\eta _ fM)^ i$. This is well defined because if $fx' + d^{i - 1}(z) = 0$, then $f^ ix'$ is in $(\eta _ fM)^ i$ and consequently $f^ id^{i - 1}(z)$ is in $f(\eta _ fM)^ i$.

We omit the verification that the lower row in the displayed diagram is a short exact sequence of modules. It is immediately clear from these constructions that we have commutative diagrams

$\xymatrix{ B^{i + 1} \oplus B^ i \ar[d]^{s, s'} \ar[r] & B^{i + 2} \oplus B^{i + 1} \ar[d]^{s, s'} \\ (\eta _ fM)^ i /f(\eta _ fM)^ i \ar[r] & (\eta _ fM)^{i + 1} /f(\eta _ fM)^{i + 1} }$

where the upper horizontal arrow is given by the identification of the summands $B^{i + 1}$ in source and target. In other words, we have found an acyclic subcomplex of $\eta _ fM^\bullet / f(\eta _ fM^\bullet ) = \eta _ fM^\bullet \otimes _ A A/fA$ and the quotient by this subcomplex is a complex whose terms $Z^ i/B^ i$ are the cohomology modules of the complex $\overline{M}^\bullet = M^\bullet \otimes _ A A/fA$.

There are also:

• 2 comment(s) on Section 15.95: An operator introduced by Berthelot and Ogus

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
{"ft_lang_label":"__label__en","ft_lang_prob":0.7294374,"math_prob":0.9999883,"size":2036,"snap":"2022-27-2022-33","text_gpt3_token_len":722,"char_repetition_ratio":0.14419292,"word_repetition_ratio":0.036082473,"special_character_ratio":0.3664047,"punctuation_ratio":0.050113894,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000042,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-09T17:33:38Z\",\"WARC-Record-ID\":\"<urn:uuid:e1f2797a-1b29-44c2-a641-508b307e9a92>\",\"Content-Length\":\"16476\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:583d9943-66c8-4b49-8fec-c667c49e3330>\",\"WARC-Concurrent-To\":\"<urn:uuid:2bcf8d00-3afc-4f67-8bfc-5747481752ab>\",\"WARC-IP-Address\":\"128.59.222.85\",\"WARC-Target-URI\":\"https://stacks.math.columbia.edu/tag/0F7S\",\"WARC-Payload-Digest\":\"sha1:CERKPS2X44TPWE64JZ7UDVT6UC66W7IZ\",\"WARC-Block-Digest\":\"sha1:LZOCVSNKPDGQQJ7ASIJYFMY5I3K3J73B\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882571056.58_warc_CC-MAIN-20220809155137-20220809185137-00281.warc.gz\"}"}