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What facts in commutative algebra fail miserably for simplicial commutative rings, even up to homotopy?
938:
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370:; in particular, it is a graded abelian group. Next, to multiply two elements, writing
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363:
67:
1089:
What is a simplicial commutative ring from the point of view of homotopy theory?
783:
189:
1104:
585:{\displaystyle (S^{1})^{\wedge i}\times (S^{1})^{\wedge j}\to A\times A\to A}
1037:
of the category of simplicial commutative rings; an object corresponding to
16:
Simplicial commutative ring: commutative monoid in simplicial abelian groups
1076:
226:
492:{\displaystyle x:(S^{1})^{\wedge i}\to A,\,\,y:(S^{1})^{\wedge j}\to A}
59:
1088:
1098:
1093:
671:{\displaystyle (S^{1})^{\wedge i}\wedge (S^{1})^{\wedge j}\to A}
325:
the structure of a graded-commutative graded ring as follows.
18:
259:
be a simplicial commutative ring. Then the ring structure of
90:
is a simplicial commutative ring, then it can be shown that
775:{\displaystyle \pi _{i}A\times \pi _{j}A\to \pi _{i+j}A}
1147:
923:{\displaystyle S^{1}\wedge S^{1}\to S^{1}\wedge S^{1}}
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Simplicial presheaves and derived algebraic geometry
318:{\displaystyle \pi _{*}A=\oplus _{i\geq 0}\pi _{i}A}
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714:. We have thus defined the graded multiplication
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8:
1099:Reference request - CDGA vs. sAlg in char. 0
32:It has been suggested that this article be
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987:has the structure of a graded module over
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95:
957:), then the similar argument shows that
1118:Model categories and simplicial methods
1060:{\displaystyle \operatorname {Spec} A}
1029:By definition, the category of affine
596:the second map the multiplication of
7:
1128:
1126:
786:because the smash product is. It is
678:. This in turn gives an element in
857:{\displaystyle xy=(-1)^{|x||y|}yx}
499:be two maps. Then the composition
14:
229:-counterpart of this notion is a
1130:
23:
1116:P. Goerss and K. Schemmerhorn,
1105:Simplicial commutative rings, I
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1:
244:polynomial differential forms
84:category of commutative rings
1146:. You can help Knowledge by
707:{\displaystyle \pi _{i+j}A}
64:simplicial commutative ring
1224:
1125:
1208:Commutative algebra stubs
1010:{\displaystyle \pi _{*}A}
980:{\displaystyle \pi _{*}M}
930:introduces a minus sign.
355:{\displaystyle \pi _{*}A}
231:commutative ring spectrum
215:{\displaystyle \pi _{0}A}
181:{\displaystyle \pi _{*}A}
158:over that ring (in fact,
147:{\displaystyle \pi _{i}A}
113:{\displaystyle \pi _{0}A}
76:simplicial abelian groups
49:Proposed since July 2024.
951:simplicial abelian group
864:) since the involution
362:is the homology of the
330:Dold–Kan correspondence
1142:-related article is a
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390:{\displaystyle S^{1}}
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251:Graded ring structure
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1203:Algebraic structures
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1193:Commutative algebra
1140:commutative algebra
1041:will be denoted by
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953:with an action of
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788:graded-commutative
772:
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68:commutative monoid
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1035:opposite category
939:simplicial module
399:simplicial circle
366:corresponding to
80:simplicial object
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39:Simplicial group
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364:chain complex
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246:on simplexes.
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1148:expanding it
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1038:
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954:
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597:
595:
367:
327:
260:
256:
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242:The ring of
224:
87:
63:
57:
48:
37:
1198:Ring theory
1103:A. Mathew,
784:associative
190:graded ring
1187:Categories
1083:References
945:(that is,
600:, induces
1110:B. Toën,
1052:
1000:∗
996:π
970:∗
966:π
908:∧
895:→
882:∧
810:−
755:π
751:→
739:π
735:×
723:π
687:π
663:→
655:∧
634:∧
626:∧
577:→
571:×
565:→
557:∧
536:×
528:∧
484:→
476:∧
441:→
433:∧
345:∗
341:π
304:π
295:≥
288:⊕
276:∗
272:π
201:π
171:∗
167:π
133:π
99:π
1077:E_n-ring
1071:See also
782:. It is
397:for the
237:Examples
227:topology
72:category
1033:is the
790:(i.e.,
328:By the
156:modules
82:in the
70:in the
60:algebra
44:Discuss
401:, let
263:gives
34:merged
1138:This
1017:(cf.
949:is a
941:over
937:is a
192:over
188:is a
120:is a
86:. If
66:is a
36:into
1144:stub
1049:Spec
1025:Spec
255:Let
154:are
124:and
122:ring
62:, a
1021:).
933:If
222:.)
74:of
58:In
42:. (
1189::
1067:.
332:,
233:.
225:A
1175:e
1168:t
1161:v
1150:.
1107:.
1055:A
1039:A
1005:A
975:M
955:A
947:M
943:A
935:M
916:1
912:S
903:1
899:S
890:1
886:S
877:1
873:S
852:x
849:y
843:|
839:y
835:|
830:|
826:x
822:|
817:)
813:1
807:(
804:=
801:y
798:x
770:A
765:j
762:+
759:i
748:A
743:j
732:A
727:i
702:A
697:j
694:+
691:i
666:A
658:j
651:)
645:1
641:S
637:(
629:i
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616:1
612:S
608:(
598:A
592:,
580:A
574:A
568:A
560:j
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547:1
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531:i
524:)
518:1
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487:A
479:j
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466:1
462:S
458:(
455::
452:y
447:,
444:A
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423:1
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412::
409:x
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368:A
350:A
313:A
308:i
298:0
292:i
284:=
281:A
261:A
257:A
210:A
205:0
176:A
142:A
137:i
108:A
103:0
88:A
46:)
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