forked from stacks/stacks-project
-
Notifications
You must be signed in to change notification settings - Fork 0
/
stacks-perfect.tex
992 lines (900 loc) · 35.1 KB
/
stacks-perfect.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
\input{preamble}
% OK, start here.
%
\begin{document}
\title{Derived Categories of Stacks}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
In this chapter we write about derived categories associated to
algebraic stacks. This mean in particular derived categories
of quasi-coherent sheaves, i.e., we prove analogues of the results
in the chapters entitled ``Derived Categories of Schemes'' and
``Derived Categories of Spaces''. The results in this chapter
are different from those in \cite{LM-B} mainly because we consistently
use the ``big sites''. Before reading this chapter please take a quick
look at the chapters ``Sheaves on Algebraic Stacks'' and
``Cohomology of Algebraic Stacks'' where the terminology we use here is
introduced.
\section{Conventions, notation, and abuse of language}
\label{section-conventions}
\noindent
We continue to use the conventions and the abuse of language
introduced in
Properties of Stacks, Section \ref{stacks-properties-section-conventions}.
We use notation as explained in
Cohomology of Stacks, Section \ref{stacks-cohomology-section-notation}.
\section{The lisse-\'etale and the flat-fppf sites}
\label{section-lisse-etale}
\noindent
The section is the analogue of
Cohomology of Stacks, Section \ref{stacks-cohomology-section-lisse-etale}
for derived categories.
\begin{lemma}
\label{lemma-shriek-derived}
Let $\mathcal{X}$ be an algebraic stack.
Notation as in
Cohomology of Stacks,
Lemmas \ref{stacks-cohomology-lemma-lisse-etale} and
\ref{stacks-cohomology-lemma-lisse-etale-modules}.
\begin{enumerate}
\item The functor
$g_! : \textit{Ab}(\mathcal{X}_{lisse,\etale}) \to
\textit{Ab}(\mathcal{X}_\etale)$
has a left derived functor
$$
Lg_! :
D(\mathcal{X}_{lisse,\etale})
\longrightarrow
D(\mathcal{X}_\etale)
$$
which is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \text{id}$.
\item The functor $g_! :
\textit{Mod}(\mathcal{X}_{lisse,\etale},
\mathcal{O}_{\mathcal{X}_{lisse,\etale}}) \to
\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_{\mathcal{X}})$
has a left derived functor
$$
Lg_! :
D(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})
\longrightarrow
D(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})
$$
which is left adjoint to $g^*$ and such that $g^*Lg_! = \text{id}$.
\item The functor $g_! : \textit{Ab}(\mathcal{X}_{flat,fppf}) \to
\textit{Ab}(\mathcal{X}_{fppf})$
has a left derived functor
$$
Lg_! :
D(\mathcal{X}_{flat, fppf})
\longrightarrow
D(\mathcal{X}_{fppf})
$$
which is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \text{id}$.
\item The functor $g_! :
\textit{Mod}(\mathcal{X}_{flat,fppf},
\mathcal{O}_{\mathcal{X}_{flat,fppf}}) \to
\textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_{\mathcal{X}})$
has a left derived functor
$$
Lg_! :
D(\mathcal{O}_{\mathcal{X}_{flat, fppf}})
\longrightarrow
D(\mathcal{O}_\mathcal{X})
$$
which is left adjoint to $g^*$ and such that $g^*Lg_! = \text{id}$.
\end{enumerate}
Warning: It is not clear (a priori) that $Lg_!$ on modules agrees
with $Lg_!$ on abelian sheaves, see
Cohomology on Sites, Remark
\ref{sites-cohomology-remark-when-derived-shriek-equal}.
\end{lemma}
\begin{proof}
The existence of the functor $Lg_!$ and adjointness to $g^*$ is
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-existence-derived-lower-shriek}.
(For the case of abelian sheaves use the constant sheaf $\mathbf{Z}$
as the structure sheaves.)
Moreover, it is computed on a complex $\mathcal{H}^\bullet$
by taking a suitable left resolution
$\mathcal{K}^\bullet \to \mathcal{H}^\bullet$
and applying the functor $g_!$ to $\mathcal{K}^\bullet$.
Since $g^{-1}g_!\mathcal{K}^\bullet = \mathcal{K}^\bullet$ by
Cohomology of Stacks,
Lemmas \ref{stacks-cohomology-lemma-lisse-etale-modules} and
\ref{stacks-cohomology-lemma-lisse-etale}
we see that the final assertion holds in each case.
\end{proof}
\begin{lemma}
\label{lemma-lisse-etale-functorial-derived}
With assumptions and notation as in
Cohomology of Stacks,
Lemma \ref{stacks-cohomology-lemma-lisse-etale-functorial}.
We have
$$
g^{-1} \circ Rf_* = Rf'_* \circ (g')^{-1}
\quad\text{and}\quad
L(g')_! \circ (f')^{-1} = f^{-1} \circ Lg_!
$$
on unbounded derived categories
(both for the case of modules and for the case of abelian sheaves).
\end{lemma}
\begin{proof}
Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{X}_\etale$
(resp.\ $\mathcal{X}_{fppf}$). We first show that the canonical
(base change) map
$$
g^{-1} Rf_*\mathcal{F} \longrightarrow Rf'_* (g')^{-1}\mathcal{F}
$$
is an isomorphism. To do this let $y$ be an object of
$\mathcal{Y}_{lisse,\etale}$ (resp.\ $\mathcal{Y}_{flat,fppf}$).
Say $y$ lies over the scheme $V$ such that $y : V \to \mathcal{Y}$ is
smooth (resp.\ flat). Since $g^{-1}$ is the restriction we find that
$$
\left(g^{-1}R^pf_*\mathcal{F}\right)(y) =
H^p_\tau(V \times_{y, \mathcal{Y}} \mathcal{X},\ \text{pr}^{-1}\mathcal{F})
$$
where $\tau = \etale$ (resp.\ $\tau = fppf$), see
Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-pushforward-restriction}.
By
Cohomology of Stacks, Equation
(\ref{stacks-cohomology-equation-pushforward-lisse-etale})
for any sheaf $\mathcal{H}$ on
$\mathcal{X}_{lisse,\etale}$ (resp.\ $\mathcal{X}_{flat,fppf}$)
$$
f'_*\mathcal{H}(y) =
\Gamma((V \times_{y, \mathcal{Y}} \mathcal{X})',
\ (\text{pr}')^{-1}\mathcal{H})
$$
An object of $(V \times_{y, \mathcal{Y}} \mathcal{X})'$ can be seen
as a pair $(x, \varphi)$ where $x$ is an object of
$\mathcal{X}_{lisse,\etale}$ (resp.\ $\mathcal{X}_{flat,fppf}$)
and $\varphi : f(x) \to y$ is a morphism in $\mathcal{Y}$.
We can also think of $\varphi$ as a section of $(f')^{-1}h_y$ over $x$.
Thus $(V \times_\mathcal{Y} \mathcal{X})'$ is the localization
of the site $\mathcal{X}_{lisse,\etale}$
(resp. $\mathcal{X}_{flat,fppf}$) at the sheaf of sets $(f')^{-1}h_y$, see
Sites, Lemma \ref{sites-lemma-localize-topos-site}. The morphism
$$
\text{pr}' : (V \times_{y, \mathcal{Y}} \mathcal{X})'
\to \mathcal{X}_{lisse,\etale}
\ (\text{resp. }
\text{pr}' : (V \times_{y, \mathcal{Y}} \mathcal{X})'
\to \mathcal{X}_{flat,fppf})
$$
is the localization morphism.
In particular, the pullback $(\text{pr}')^{-1}$ preserves
injective abelian sheaves, see
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-cohomology-on-sheaf-sets}.
At this point exactly the same argument as in
Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-pushforward-restriction}
shows that
\begin{equation}
\label{equation-higher-direct-image-lisse-etale}
R^pf'_*\mathcal{H}(y) =
H^p_\tau((V \times_{y, \mathcal{Y}} \mathcal{X})',
\ (\text{pr}')^{-1}\mathcal{H})
\end{equation}
where $\tau = \etale$ (resp.\ $\tau = fppf$). Since $(g')^{-1}$
is given by restriction we conclude that
$$
\left(R^pf'_*(g')^*\mathcal{F}\right)(y) =
H^p_\tau((V \times_{y, \mathcal{Y}} \mathcal{X})',
\ \text{pr}^{-1}\mathcal{F}|_{(V \times_{y, \mathcal{Y}} \mathcal{X})'})
$$
Finally, we can apply
Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-cohomology-on-subcategory}
to see that
$$
H^p_\tau((V \times_{y, \mathcal{Y}} \mathcal{X})',
\ \text{pr}^{-1}\mathcal{F}|_{(V \times_{y, \mathcal{Y}} \mathcal{X})'})
=
H^p_\tau(V \times_{y, \mathcal{Y}} \mathcal{X},\ \text{pr}^{-1}\mathcal{F})
$$
are equal as desired; although we omit the verification of the assumptions
of the lemma we note that the fact that $V \to \mathcal{Y}$ is smooth
(resp.\ flat) is used to verify the second condition.
\medskip\noindent
The rest of the proof is formal. Since cohomology of abelian groups and
sheaves of modules agree we also conclude that
$g^{-1} Rf_*\mathcal{F} = Rf'_* (g')^{-1}\mathcal{F}$ when $\mathcal{F}$
is a sheaf of modules on $\mathcal{X}_\etale$
(resp.\ $\mathcal{X}_{fppf}$).
\medskip\noindent
Next we show that for $\mathcal{G}$ (either sheaf of modules
or abelian groups) on
$\mathcal{Y}_{lisse,\etale}$ (resp.\ $\mathcal{Y}_{flat,fppf}$)
the canonical map
$$
L(g')_!(f')^{-1}\mathcal{G} \to f^{-1}Lg_!\mathcal{G}
$$
is an isomorphism. To see this it is enough to prove for any
injective sheaf $\mathcal{I}$ on $\mathcal{X}_\etale$
(resp.\ $\mathcal{X}_{fppf}$) that the induced map
$$
\Hom(L(g')_!(f')^{-1}\mathcal{G}, \mathcal{I}[n])
\leftarrow
\Hom(f^{-1}Lg_!\mathcal{G}, \mathcal{I}[n])
$$
is an isomorphism for all $n \in \mathbf{Z}$. (Hom's taken
in suitable derived categories.) By the adjointness of
$f^{-1}$ and $Rf_*$, the adjointness of $Lg_!$ and $g^{-1}$, and
their ``primed'' versions this follows from the isomorphism
$g^{-1} Rf_*\mathcal{I} \to Rf'_* (g')^{-1}\mathcal{I}$ proved above.
\medskip\noindent
In the case of a bounded complex $\mathcal{G}^\bullet$
(of modules or abelian groups) on
$\mathcal{Y}_{lisse,\etale}$ (resp.\ $\mathcal{Y}_{fppf}$)
the canonical map
\begin{equation}
\label{equation-to-show}
L(g')_!(f')^{-1}\mathcal{G}^\bullet \to f^{-1}Lg_!\mathcal{G}^\bullet
\end{equation}
is an isomorphism as follows from the case of a sheaf by the usual arguments
involving truncations and the fact that the functors
$L(g')_!(f')^{-1}$ and $f^{-1}Lg_!$ are exact functors of
triangulated categories.
\medskip\noindent
Suppose that $\mathcal{G}^\bullet$ is a bounded above complex
(of modules or abelian groups) on
$\mathcal{Y}_{lisse,\etale}$ (resp.\ $\mathcal{Y}_{fppf}$).
The canonical map (\ref{equation-to-show})
is an isomorphism because we can use the stupid truncations
$\sigma_{\geq -n}$ (see
Homology, Section \ref{homology-section-truncations}) to write
$\mathcal{G}^\bullet$ as a colimit
$\mathcal{G}^\bullet = \colim \mathcal{G}_n^\bullet$
of bounded complexes. This gives a distinguished triangle
$$
\bigoplus\nolimits_{n \geq 1} \mathcal{G}_n^\bullet \to
\bigoplus\nolimits_{n \geq 1} \mathcal{G}_n^\bullet \to
\mathcal{G}^\bullet \to \ldots
$$
and each of the functors $L(g')_!$, $(f')^{-1}$, $f^{-1}$, $Lg_!$
commutes with direct sums (of complexes).
\medskip\noindent
If $\mathcal{G}^\bullet$ is an arbitrary complex
(of modules or abelian groups) on
$\mathcal{Y}_{lisse,\etale}$ (resp.\ $\mathcal{Y}_{fppf}$)
then we use the canonical truncations $\tau_{\leq n}$ (see
Homology, Section \ref{homology-section-truncations})
to write $\mathcal{G}^\bullet$ as a colimit of bounded above complexes
and we repeat the argument of the paragraph above.
\medskip\noindent
Finally, by the adjointness of
$f^{-1}$ and $Rf_*$, the adjointness of $Lg_!$ and $g^{-1}$, and
their ``primed'' versions we conclude that the first
identity of the lemma follows from the second in full generality.
\end{proof}
\begin{lemma}
\label{lemma-higher-shriek-quasi-coherent}
Let $\mathcal{X}$ be an algebraic stack. Notation as in
Cohomology of Stacks,
Lemma \ref{stacks-cohomology-lemma-lisse-etale}.
\begin{enumerate}
\item Let $\mathcal{H}$ be a quasi-coherent
$\mathcal{O}_{\mathcal{X}_{lisse,\etale}}$-module
on the lisse-\'etale site of $\mathcal{X}$. For all $p \in \mathbf{Z}$
the sheaf $H^p(Lg_!\mathcal{H})$ is a locally quasi-coherent module with
the flat base change property on $\mathcal{X}$.
\item Let $\mathcal{H}$ be a quasi-coherent
$\mathcal{O}_{\mathcal{X}_{flat,fppf}}$-module
on the flat-fppf site of $\mathcal{X}$. For all $p \in \mathbf{Z}$
the sheaf $H^p(Lg_!\mathcal{H})$ is a locally quasi-coherent module with the
flat base change property on $\mathcal{X}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Pick a scheme $U$ and a surjective smooth morphism $x : U \to \mathcal{X}$. By
Modules on Sites, Definition \ref{sites-modules-definition-site-local}
there exists an \'etale (resp.\ fppf) covering
$\{U_i \to U\}_{i \in I}$ such that each pullback $f_i^{-1}\mathcal{H}$
has a global presentation (see
Modules on Sites, Definition \ref{sites-modules-definition-global}).
Here $f_i : U_i \to \mathcal{X}$ is the composition
$U_i \to U \to \mathcal{X}$ which is a morphism of algebraic stacks.
(Recall that the pullback ``is'' the restriction to $\mathcal{X}/f_i$, see
Sheaves on Stacks, Definition \ref{stacks-sheaves-definition-pullback}
and the discussion following.)
After refining the covering we may assume each $U_i$ is an affine scheme.
Since each $f_i$ is smooth (resp.\ flat) by
Lemma \ref{lemma-lisse-etale-functorial-derived}
we see that $f_i^{-1}Lg_!\mathcal{H} = Lg_{i, !}(f'_i)^{-1}\mathcal{H}$.
Using
Cohomology of Stacks,
Lemma \ref{stacks-cohomology-lemma-check-lqc-fbc-on-covering}
we reduce the statement of the lemma to the case where $\mathcal{H}$
has a global presentation and where $\mathcal{X} = (\Sch/X)_{fppf}$
for some affine scheme $X = \Spec(A)$.
\medskip\noindent
Say our presentation looks like
$$
\bigoplus\nolimits_{j \in J} \mathcal{O} \longrightarrow
\bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow
\mathcal{H} \longrightarrow 0
$$
where $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{lisse,\etale}}$
(resp.\ $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}$).
Note that the site $\mathcal{X}_{lisse,\etale}$
(resp.\ $\mathcal{X}_{flat,fppf}$) has a final object, namely
$X/X$ which is quasi-compact (see
Cohomology on Sites, Section \ref{sites-cohomology-section-limits}).
Hence we have
$$
\Gamma(\bigoplus\nolimits_{i \in I} \mathcal{O}) =
\bigoplus\nolimits_{i \in I} A
$$
by Sites, Lemma \ref{sites-lemma-directed-colimits-sections}. Hence the map
in the presentation corresponds to a similar presentation
$$
\bigoplus\nolimits_{j \in J} A \longrightarrow
\bigoplus\nolimits_{i \in I} A \longrightarrow
M \longrightarrow 0
$$
of an $A$-module $M$. Moreover, $\mathcal{H}$ is equal to the restriction
to the lisse-\'etale (resp.\ flat-fppf) site of the quasi-coherent sheaf
$M^a$ associated to $M$. Choose a resolution
$$
\ldots \to F_2 \to F_1 \to F_0 \to M \to 0
$$
by free $A$-modules. The complex
$$
\ldots \mathcal{O} \otimes_A F_2 \to \mathcal{O} \otimes_A F_1 \to
\mathcal{O} \otimes_A F_0 \to \mathcal{H} \to 0
$$
is a resolution of $\mathcal{H}$ by free $\mathcal{O}$-modules because
for each object $U/X$ of $\mathcal{X}_{lisse,\etale}$
(resp.\ $\mathcal{X}_{flat,fppf}$) the structure morphism $U \to X$
is flat. Hence by construction the value of $Lg_!\mathcal{H}$ is
$$
\ldots \to
\mathcal{O}_\mathcal{X} \otimes_A F_2 \to
\mathcal{O}_\mathcal{X} \otimes_A F_1 \to
\mathcal{O}_\mathcal{X} \otimes_A F_0 \to 0 \to \ldots
$$
Since this is a complex of quasi-coherent modules on
$\mathcal{X}_\etale$ (resp.\ $\mathcal{X}_{fppf}$)
it follows from
Cohomology of Stacks,
Proposition \ref{stacks-cohomology-proposition-lcq-flat-base-change}
that $H^p(Lg_!\mathcal{H})$ is quasi-coherent.
\end{proof}
\section{Derived categories of quasi-coherent modules}
\label{section-derived}
\noindent
Let $\mathcal{X}$ be an algebraic stack. As the inclusion functor
$\textit{QCoh}(\mathcal{O}_\mathcal{X}) \to
\textit{Mod}(\mathcal{O}_\mathcal{X})$ isn't exact, we cannot define
$D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X})$ as the full subcategory
of $D(\mathcal{O}_\mathcal{X})$ consisting of complexes with quasi-coherent
cohomology sheaves. In stead we define the category as follows.
\begin{definition}
\label{definition-derived}
Let $\mathcal{X}$ be an algebraic stack. Let
$\mathcal{M}_\mathcal{X} \subset \textit{Mod}(\mathcal{O}_\mathcal{X})$
denote the category of locally quasi-coherent
$\mathcal{O}_\mathcal{X}$-modules with the flat base change property.
Let $\mathcal{P}_\mathcal{X} \subset \mathcal{M}_\mathcal{X}$
be the full subcategory consisting of parasitic objects.
We define the {\it derived category of $\mathcal{O}_\mathcal{X}$-modules with
quasi-coherent cohomology sheaves} as the Verdier quotient\footnote{This
definition is different from the one in the literature, see
\cite[6.3]{olsson_sheaves}, but it agrees with that definition
by Lemma \ref{lemma-derived-quasi-coherent}.}
$$
D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X}) =
D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})/
D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})
$$
\end{definition}
\noindent
This definition makes sense: By
Cohomology of Stacks,
Proposition \ref{stacks-cohomology-proposition-lcq-flat-base-change}
we see that $\mathcal{M}_\mathcal{X}$ is a weak Serre subcategory
of $\textit{Mod}(\mathcal{O}_\mathcal{X})$
hence $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$
is a strictly full, saturated triangulated subcategory of
$D(\mathcal{O}_\mathcal{X})$, see
Derived Categories, Lemma \ref{derived-lemma-cohomology-in-serre-subcategory}.
Since parasitic modules form a Serre subcategory of
$\textit{Mod}(\mathcal{O}_\mathcal{X})$ (by
Cohomology of Stacks,
Lemma \ref{stacks-cohomology-lemma-parasitic}) we see that
$\mathcal{P}_\mathcal{X} = \text{Parasitic} \cap \mathcal{M}_\mathcal{X}$
is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_\mathcal{X})$ and
hence $D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$
is a strictly full, saturated triangulated subcategory of
$D(\mathcal{O}_\mathcal{X})$. Since clearly
$$
D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})
\subset
D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})
$$
we conclude that the first is a strictly full, saturated triangulated
subcategory of the second. Hence the Verdier quotient exists. A morphism
$a : E \to E'$ of
$D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ becomes an
isomorphism in $D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X})$ if and
only if the cone $C(a)$ has parasitic cohomology sheaves, see
Derived Categories, Section \ref{derived-section-quotients} and especially
Lemma \ref{derived-lemma-operations}.
\medskip\noindent
Consider the functors
$$
D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})
\xrightarrow{H^i}
\mathcal{M}_\mathcal{X}
\xrightarrow{Q}
\textit{QCoh}(\mathcal{O}_\mathcal{X})
$$
Note that $Q$ annihilates the subcategory $\mathcal{P}_\mathcal{X}$, see
Cohomology of Stacks,
Lemma \ref{stacks-cohomology-lemma-adjoint-kernel-parasitic}.
By
Derived Categories, Lemma \ref{derived-lemma-universal-property-quotient}
we obtain a cohomological functor
\begin{equation}
\label{equation-Hi-quasi-coherent}
H^i :
D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X})
\longrightarrow
\textit{QCoh}(\mathcal{O}_\mathcal{X})
\end{equation}
Moreover, note that $E \in D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X})$
is zero if and only if $H^i(E) = 0$ for all $i \in \mathbf{Z}$.
\medskip\noindent
Note that the categories $\mathcal{P}_\mathcal{X}$ and
$\mathcal{M}_\mathcal{X}$ are also weak Serre subcategories of the
abelian category
$\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$
of modules in the \'etale topology, see
Cohomology of Stacks,
Proposition \ref{stacks-cohomology-proposition-lcq-flat-base-change} and
Lemma \ref{stacks-cohomology-lemma-parasitic}.
Hence the statement of the following lemma makes sense.
\begin{lemma}
\label{lemma-compare-etale-fppf-QCoh}
Let $\mathcal{X}$ be an algebraic stack. The comparison morphism
$\epsilon : \mathcal{X}_{fppf} \to \mathcal{X}_\etale$
induces a commutative diagram
$$
\xymatrix{
D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \ar[r] &
D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \ar[r] &
D(\mathcal{O}_\mathcal{X}) \\
D_{\mathcal{P}_\mathcal{X}}(
\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})
\ar[r] \ar[u]^{\epsilon^*} &
D_{\mathcal{M}_\mathcal{X}}(
\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})
\ar[r] \ar[u]^{\epsilon^*} &
D(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})
\ar[u]^{\epsilon^*}
}
$$
Moreover, the left two vertical arrows are equivalences of triangulated
categories, hence we also obtain an equivalence
$$
D_{\mathcal{M}_\mathcal{X}}(
\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})
/
D_{\mathcal{P}_\mathcal{X}}(
\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})
\longrightarrow
D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X})
$$
\end{lemma}
\begin{proof}
Since $\epsilon^*$ is exact it is clear that we obtain a diagram as
in the statement of the lemma. We will show the middle vertical
arrow is an equivalence by applying
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-compare-topologies-derived-adequate-modules}
to the following situation:
$\mathcal{C} = \mathcal{X}$,
$\tau = fppf$,
$\tau' = \etale$,
$\mathcal{O} = \mathcal{O}_\mathcal{X}$,
$\mathcal{A} = \mathcal{M}_\mathcal{X}$, and
$\mathcal{B}$ is the set of objects of $\mathcal{X}$ lying over
affine schemes. To see the lemma applies we have to check conditions
(1), (2), (3), (4). Conditions (1) and (2) are clear from the discussion
above (explicitly this follows from
Cohomology of Stacks,
Proposition \ref{stacks-cohomology-proposition-lcq-flat-base-change}).
Condition (3) holds because every scheme has a Zariski
open covering by affines. Condition (4) follows from
Descent, Lemma \ref{descent-lemma-quasi-coherent-and-flat-base-change}.
\medskip\noindent
We omit the verification that the equivalence of
categories $\epsilon^* :
D_{\mathcal{M}_\mathcal{X}}(
\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})
\to
D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$
induces an equivalence of the subcategories of complexes
with parasitic cohomology sheaves.
\end{proof}
\noindent
It turns out that $D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X})$
is the same as the derived category of complexes of modules
with quasi-coherent cohomology sheaves on the lisse-\'etale or
flat-fppf site.
\begin{lemma}
\label{lemma-derived-quasi-coherent}
Let $\mathcal{X}$ be an algebraic stack.
Let $\mathcal{F}^\bullet$ be an object of
$D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$.
\begin{enumerate}
\item With $g$ as in
Cohomology of Stacks,
Lemma \ref{stacks-cohomology-lemma-lisse-etale}
for the lisse-\'etale site we have
\begin{enumerate}
\item $g^{-1}\mathcal{F}^\bullet$ is in
$D_{\textit{QCoh}}(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$,
\item $g^{-1}\mathcal{F}^\bullet = 0$ if and only if
$\mathcal{F}^\bullet$ is in
$D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$,
\item $Lg_!\mathcal{H}^\bullet$ is in
$D_{\mathcal{M}_\mathcal{X}}(
\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$
for $\mathcal{H}^\bullet$ in
$D_{\textit{QCoh}}(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$, and
\item the functors $g^{-1}$ and $Lg_!$ define mutually inverse functors
$$
\xymatrix{
D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X}) \ar@<1ex>[r]^-{g^{-1}} &
D_{\textit{QCoh}}(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})
\ar@<1ex>[l]^-{Lg_!}
}
$$
\end{enumerate}
\item With $g$ as in
Cohomology of Stacks,
Lemma \ref{stacks-cohomology-lemma-lisse-etale}
for the flat-fppf site we have
\begin{enumerate}
\item $g^{-1}\mathcal{F}^\bullet$ is in
$D_{\textit{QCoh}}(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$,
\item $g^{-1}\mathcal{F}^\bullet = 0$ if and only if
$\mathcal{F}^\bullet$ is in
$D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$,
\item $Lg_!\mathcal{H}^\bullet$ is in
$D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$
for $\mathcal{H}^\bullet$ in
$D_{\textit{QCoh}}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$, and
\item the functors $g^{-1}$ and $Lg_!$ define mutually inverse functors
$$
\xymatrix{
D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X}) \ar@<1ex>[r]^-{g^{-1}} &
D_{\textit{QCoh}}(\mathcal{O}_{\mathcal{X}_{flat,fppf}}) \ar@<1ex>[l]^-{Lg_!}
}
$$
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof}
The functor $g^{-1}$ is exact, hence (a) and (b) follow from
Cohomology of Stacks,
Lemmas \ref{stacks-cohomology-lemma-quasi-coherent} and
\ref{stacks-cohomology-lemma-parasitic-in-terms-flat-fppf}.
\medskip\noindent
The construction of $Lg_!$ in Lemma \ref{lemma-shriek-derived}
(via Cohomology on Sites,
Lemma \ref{sites-cohomology-lemma-existence-derived-lower-shriek}
which in turn uses
Derived Categories, Proposition \ref{derived-proposition-left-derived-exists})
shows that $Lg_!$ on any object $\mathcal{H}^\bullet$ of
$D(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$ is computed
as
$$
Lg_!\mathcal{H}^\bullet = \colim g_!\mathcal{K}_n^\bullet =
g_! \colim \mathcal{K}_n^\bullet
$$
(termwise colimits) where the quasi-isomorphism
$\colim \mathcal{K}_n^\bullet \to \mathcal{H}^\bullet$
induces quasi-isomorphisms
$\mathcal{K}_n^\bullet \to \tau_{\leq n} \mathcal{H}^\bullet$.
Since
$\mathcal{M}_\mathcal{X} \subset
\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$
(resp.\
$\mathcal{M}_\mathcal{X} \subset \textit{Mod}(\mathcal{O}_\mathcal{X})$)
is preserved under colimits we see that it suffices to prove (c)
on bounded above complexes $\mathcal{H}^\bullet$ in
$D_{\textit{QCoh}}(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$
(resp.\ $D_{\textit{QCoh}}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$).
In this case to show that $H^n(Lg_!\mathcal{H}^\bullet)$ is
in $\mathcal{M}_\mathcal{X}$ we can argue by induction on the integer
$m$ such that $\mathcal{H}^i = 0$ for $i > m$. If $m < n$, then
$H^n(Lg_!\mathcal{H}^\bullet) = 0$ and the result holds. In general
consider the distinguished triangle
$$
\tau_{\leq m - 1}\mathcal{H}^\bullet \to \mathcal{H}^\bullet \to
H^m(\mathcal{H}^\bullet)[-m] \to \ldots
$$
(Derived Categories, Remark
\ref{derived-remark-truncation-distinguished-triangle})
and apply the functor $Lg_!$. Since $\mathcal{M}_\mathcal{X}$
is a weak Serre subcategory of the module category it suffices to
prove (c) for two out of three. We have the result for
$Lg_!\tau_{\leq m - 1}\mathcal{H}^\bullet$ by induction and we
have the result for $Lg_!H^m(\mathcal{H}^\bullet)[-m]$ by
Lemma \ref{lemma-higher-shriek-quasi-coherent}. Whence (c) holds.
\medskip\noindent
Let us prove (2)(d). By (a) and (b) the functor $g^{-1} = g^*$ induces
a functor
$$
c :
D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X})
\longrightarrow
D_{\textit{QCoh}}(\mathcal{O}_{\mathcal{X}_{flat, fppf}})
$$
see
Derived Categories, Lemma \ref{derived-lemma-universal-property-quotient}.
Thus we have the following diagram of triangulated categories
$$
\xymatrix{
D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})
\ar[rd]^{g^{-1}} \ar[rr]_q & &
D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X}) \ar[ld]^c \\
& D_{\textit{QCoh}}(\mathcal{O}_{\mathcal{X}_{flat, fppf}})
\ar@<1ex>[lu]^{Lg_!}
}
$$
where $q$ is the quotient functor, the inner triangle is commutative, and
$g^{-1}Lg_! = \text{id}$.
For any object of $E$ of $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$
the map $a : Lg_!g^{-1}E \to E$ maps to a quasi-isomorphism in
$D(\mathcal{O}_{\mathcal{X}_{flat, fppf}})$. Hence the cone on
$a$ maps to zero under $g^{-1}$ and by (b) we see that $q(a)$ is
an isomorphism. Thus $q \circ Lg_!$ is a quasi-inverse to $c$.
\medskip\noindent
In the case of the lisse-\'etale site exactly the same argument as above
proves that
$$
D_{\mathcal{M}_\mathcal{X}}(
\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})
/
D_{\mathcal{P}_\mathcal{X}}(
\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})
$$
is equivalent to
$D_{\textit{QCoh}}(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$.
Applying the last equivalence of
Lemma \ref{lemma-compare-etale-fppf-QCoh}
finishes the proof.
\end{proof}
\noindent
The following lemma tells us that the quotient functor
$D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \to
D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X})$ is a Bousfield
colocalization (insert future reference here).
\begin{lemma}
\label{lemma-bousfield-colocalization}
Let $\mathcal{X}$ be an algebraic stack.
Let $E$ be an object of $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$.
There exists a canonical distinguished triangle
$$
E' \to E \to P \to E'[1]
$$
in $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ such that
$P$ is in $D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$
and
$$
\Hom_{D(\mathcal{O}_\mathcal{X})}(E', P') = 0
$$
for all $P'$ in $D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$.
\end{lemma}
\begin{proof}
Consider the morphism of ringed topoi
$g : \Sh(\mathcal{X}_{flat, fppf}) \longrightarrow \Sh(\mathcal{X}_{fppf})$.
Set $E' = Lg_!g^{-1}E$ and let $P$ be the cone on the adjunction
map $E' \to E$. Since $g^{-1}E' \to g^{-1}E$ is an isomorphism we see that
$P$ is an object of $D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ by
Lemma \ref{lemma-derived-quasi-coherent} (2)(b).
Finally, $\Hom(E', P') = \Hom(Lg_!g^{-1}E, P') = \Hom(g^{-1}E, g^{-1}P') = 0$
as $g^{-1}P' = 0$.
\medskip\noindent
Uniqueness. Suppose that $E'' \to E \to P'$ is a second distinguished
triangle as in the statement of the lemma. Since $\Hom(E', P') = 0$
the morphism $E' \to E$ factors as $E' \to E'' \to E$, see
Derived Categories, Lemma \ref{derived-lemma-representable-homological}.
Similarly, the morphism $E'' \to E$ factors as $E'' \to E' \to E$.
Consider the composition $\varphi : E' \to E'$ of the maps $E' \to E''$ and
$E'' \to E'$. Note that $\varphi - 1 : E' \to E'$ fits into the commutative
diagram
$$
\xymatrix{
E' \ar[d]^{\varphi - 1} \ar[r] & E \ar[d]^0 \\
E' \ar[r] & E
}
$$
hence factors through $P[-1] \to E$. Since $\Hom(E', P[-1]) = 0$
we see that $\varphi = 1$. Whence the maps $E' \to E''$ and $E'' \to E'$
are inverse to each other.
\end{proof}
\section{Derived pushforward of quasi-coherent modules}
\label{section-derived-pushforward}
\noindent
As a first application of the material above we construct the derived
pushforward. In
Examples, Section \ref{examples-section-derived-push-quasi-coherent}
the reader can find an example of a quasi-compact and quasi-separated
morphism $f : \mathcal{X} \to \mathcal{Y}$ of algebraic stacks such
that the direct image functor $Rf_*$ does not induce a functor
$D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X}) \to
D_{\textit{QCoh}}(\mathcal{O}_\mathcal{Y})$. Thus restricting to bounded
below complexes is necessary.
\begin{proposition}
\label{proposition-derived-direct-image-quasi-coherent}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and
quasi-separated morphism of algebraic stacks.
The functor $Rf_*$ induces a commutative diagram
$$
\xymatrix{
D^{+}_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})
\ar[r] \ar[d]^{Rf_*} &
D^{+}_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})
\ar[r] \ar[d]^{Rf_*} &
D(\mathcal{O}_\mathcal{X})
\ar[d]^{Rf_*} \\
D^{+}_{\mathcal{P}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y}) \ar[r] &
D^{+}_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y}) \ar[r] &
D(\mathcal{O}_\mathcal{Y})
}
$$
and hence induces a functor
$$
Rf_{\textit{QCoh}, *} :
D^{+}_{\textit{QCoh}}(\mathcal{X})
\longrightarrow
D^{+}_{\textit{QCoh}}(\mathcal{Y})
$$
on quotient categories. Moreover, the functor $R^if_{\textit{QCoh}}$
of
Cohomology of Stacks,
Proposition \ref{stacks-cohomology-proposition-direct-image-quasi-coherent}
are equal to $H^i \circ Rf_{\textit{QCoh}, *}$ with $H^i$ as in
(\ref{equation-Hi-quasi-coherent}).
\end{proposition}
\begin{proof}
We have to show that $Rf_*E$ is an object of
$D^{+}_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$ for
$E$ in $D^{+}_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$.
This follows from
Cohomology of Stacks,
Proposition \ref{stacks-cohomology-proposition-lcq-flat-base-change}
and the spectral sequence $R^if_*H^j(E) \Rightarrow R^{i + j}f_*E$.
The case of parasitic modules works the same way using
Cohomology of Stacks, Lemma
\ref{stacks-cohomology-lemma-pushforward-parasitic}.
The final statement is clear from the definition of
$H^i$ in (\ref{equation-Hi-quasi-coherent}).
\end{proof}
\section{Derived pullback of quasi-coherent modules}
\label{section-derived-pullback}
\noindent
Derived pullback of complexes with quasi-coherent cohomology
sheaves exists in general.
\begin{proposition}
\label{proposition-derived-pullback-quasi-coherent}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
The exact functor $f^*$ induces a commutative diagram
$$
\xymatrix{
D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \ar[r] &
D(\mathcal{O}_\mathcal{X}) \\
D_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})
\ar[r] \ar[u]^{f^*} &
D(\mathcal{O}_\mathcal{Y}) \ar[u]^{f^*}
}
$$
The composition
$$
D_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})
\xrightarrow{f^*}
D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})
\xrightarrow{q_\mathcal{X}}
D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X})
$$
is left deriveable with respect to the localization
$D_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y}) \to
D_{\textit{QCoh}}(\mathcal{O}_\mathcal{Y})$
and we may define $Lf^*_{\textit{QCoh}}$ as its left derived functor
$$
Lf_{\textit{QCoh}}^* :
D_{\textit{QCoh}}(\mathcal{O}_\mathcal{Y})
\longrightarrow
D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X})
$$
(see
Derived Categories,
Definitions \ref{derived-definition-right-derived-functor-defined} and
\ref{derived-definition-everywhere-defined}). If $f$ is quasi-compact
and quasi-separated, then $Lf^*_{\textit{QCoh}}$ and $Rf_{\textit{QCoh}, *}$
satisfy the following adjointness:
$$
\Hom_{D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X})}(Lf^*_{\textit{QCoh}}A, B)
=
\Hom_{D_{\textit{QCoh}}(\mathcal{O}_\mathcal{Y})}(A, Rf_{\textit{QCoh}, *}B)
$$
for $A \in D_{\textit{QCoh}}(\mathcal{O}_\mathcal{Y})$ and
$B \in D^{+}_{\textit{QCoh}}(\mathcal{O}_\mathcal{X})$.
\end{proposition}
\begin{proof}
To prove the first statement, we have to show that $f^*E$ is an object of
$D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ for
$E$ in $D_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$.
Since $f^* = f^{-1}$ is exact this follows immediately from the fact that
$f^*$ maps $\mathcal{M}_\mathcal{Y}$ into $\mathcal{M}_\mathcal{X}$.
\medskip\noindent
Set $\mathcal{D} = D_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$.
Let $S$ be the collection of morphisms in $\mathcal{D}$
whose cone is an object of
$D_{\mathcal{P}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$.
Set $\mathcal{D}' = D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X})$.
Set $F = q_\mathcal{X} \circ f^* : \mathcal{D} \to \mathcal{D}'$.
Then $\mathcal{D}, S, \mathcal{D}', F$ are as in
Derived Categories, Situation \ref{derived-situation-derived-functor} and
Definition \ref{derived-definition-right-derived-functor-defined}.
Let us prove that $LF(E)$ is defined for any object $E$ of $\mathcal{D}$.
Namely, consider the triangle
$$
E' \to E \to P \to E'[1]
$$
constructed in Lemma \ref{lemma-bousfield-colocalization}.
Note that $s : E' \to E$ is an element of $S$. We claim that $E'$ computes
$LF$. Namely, suppose that $s' : E'' \to E$ is another element of $S$, i.e.,
fits into a triangle $E'' \to E \to P' \to E''[1]$ with $P'$ in
$D_{\mathcal{P}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$. By
Lemma \ref{lemma-bousfield-colocalization} (and its proof)
we see that $E' \to E$ factors through $E'' \to E$. Thus we see that
$E' \to E$ is cofinal in the system $S/E$. Hence it is clear that
$E'$ computes $LF$.
\medskip\noindent
To see the final statement, write $B = q_\mathcal{X}(H)$ and
$A = q_\mathcal{Y}(E)$.
Choose $E' \to E$ as above.
We will use on the one hand that
$Rf_{\textit{QCoh}, *}(B) = q_\mathcal{Y}(Rf_*H)$
and on the other that
$Lf^*_{\textit{QCoh}}(A) = q_\mathcal{X}(f^*E')$.
\begin{align*}
\Hom_{D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X})}(Lf^*_{\textit{QCoh}}A, B)
& =
\Hom_{D_{\textit{QCoh}}(\mathcal{O}_\mathcal{X})}(q_\mathcal{X}(f^*E'),
q_\mathcal{X}(H)) \\
& =
\colim_{H \to H'} \Hom_{D(\mathcal{O}_\mathcal{X})}(f^*E', H') \\
& = \colim_{H \to H'} \Hom_{D(\mathcal{O}_\mathcal{Y})}(E', Rf_*H') \\
& = \Hom_{D(\mathcal{O}_\mathcal{Y})}(E', Rf_*H) \\
& =
\Hom_{D_{\textit{QCoh}}(\mathcal{O}_\mathcal{Y})}(A, Rf_{\textit{QCoh}, *}B)
\end{align*}
Here the colimit is over morphisms $s : H \to H'$ in
$D^+_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$
whose cone $P(s)$ is an object of
$D^+_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$.
The first equality we've seen above.
The second equality holds by construction of the Verdier quotient.
The third equality holds by
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-adjoint}.
Since $Rf_*P(s)$ is an object of
$D^+_{\mathcal{P}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$ by
Proposition \ref{proposition-derived-direct-image-quasi-coherent}
we see that $\Hom_{D(\mathcal{O}_\mathcal{Y})}(E', Rf_*P(s)) = 0$.
Thus the fourth equality holds. The final equality
holds by construction of $E'$.
\end{proof}
\input{chapters}
\bibliography{my}
\bibliographystyle{amsalpha}
\end{document}