The Stacks project

Some grammatical errors in Categories

There are some sentences of the form, for example, “there exists a P and a Q”.
I have changed “exists” into “exist”.
There are other small changes too.

Slight improvement definition essentially constant systems

We need to avoid turning the chapter on categories into some gigantic
monstrosity but I was a bit unhappy with the definition as well.

Thanks to Ingo Blechschmidt and Weixi Lu
https://stacks.math.columbia.edu/tag/05PT#comment-5544
https://stacks.math.columbia.edu/tag/05PT#comment-4831

LaTeX: fix colim

	Introduced the macro

	\def\colim{\mathop{\rm colim}\nolimits}

	and changed all the occurences of \text{colim} into \colim.

LaTeX: fix lim

	Replaced all the occurences of \text{lim} by \lim or
	\lim\nolimits depending on whether the invocation occured in
	display math or not.

Producing K-injective resolutions

	Suppose you admit the existence of K-injective resolutions in
	the category of O-modules on a site. Next, suppose you have a
	morphism f of ringed sites and a complex K^* on the source of f
	whose cohomology sheaves are each acyclic for f_*. Then it
	doesn't seem clear to me that R^if_*K^* is just the pushforward
	of the ith cohomology sheaf of K^*. In fact I would bet this is
	wrong in general. (Any example or counter argument welcome.)

	To see what happens we add a lemma that tells you explicitly how
	to compute a K-injective resolution of a complex where now we
	assume that each of the cohomology sheaves has bounded
	cohomological dimension on sufficiently many objects of the
	site.

Tags: added new tags

Derived functors

	Just the start of the general picture