--- title: "The annihilator of the Lefschetz motive" status: read-2x tags: projects/talbot-talk --- # The annihilator of the Lefschetz motive ## Meta - CiteKey: "Zak17b" - Type: journalArticle - Author: "Zakharevich, Inna;" - Year: 2017 - DOI: 10.1215/00127094-0000016X - Collections: "Syllabus; Talbot 2022," - **Original URL**: - Open local files directly: [arXiv.org Snapshot](file:///home/zack/Zotero/storage/RG2V2BWK/1506.html); [Zakharevich - 2017 - The annihilator of the Lefschetz motive.pdf](file:///home/zack/Zotero/storage/ERN8H528/Zakharevich%20-%202017%20-%20The%20annihilator%20of%20the%20Lefschetz%20motive.pdf) - **Open in Zotero**: [Zotero](zotero://select/library/items/AKCWP7FT) ## Abstract In this paper we study a spectrum $K(\mathcal{V}_k)$ such that $\pi_0 K(\mathcal{V}_k)$ is the Grothendieck ring of varieties and such that the higher homotopy groups contain more geometric information about the geometry of varieties. We use the topology of this spectrum to analyze the structure of $K_0[\mathcal{V}_k]$ and show that classes in the kernel of multiplication by $[\mathbb{A}^1]$ can always be represented as $[X]-[Y]$ where $X$ and $Y$ are varieties such that $[X] \neq [Y]$, $X\times \mathbb{A}^1$ and $Y\times \mathbb{A}^1$ are not piecewise isomorphic, but $[X\times \mathbb{A}^1] =[Y\times \mathbb{A}^1]$ in $K_0[\mathcal{V}_k]$. Along the way we present new proofs of the result of Larsen--Lunts on the structure on $K_0[\mathcal{V}_k]/([\mathbb{A}^1])$. ---- ## Extracted Annotations Annotations(6/8/2022, 2:45:10 PM) ![[Zak17b_G7JFII3U.png]] [(Zakharevich, 2017, p. 1)](zotero://open-pdf/library/items/ERN8H528?page=1&annotation=BM8M8KM6) - This ring is quite complicated; for example, it is not an integral domain [(Zakharevich, 2017, p. 1)](zotero://open-pdf/library/items/ERN8H528?page=1&annotation=4MF3WCEI) ![[Zak17b_5379ESYH.png]] [(Zakharevich, 2017, p. 1)](zotero://open-pdf/library/items/ERN8H528?page=1&annotation=HIKB3R95) ![[Zak17b_KW5523SD.png]] [(Zakharevich, 2017, p. 1)](zotero://open-pdf/library/items/ERN8H528?page=1&annotation=C9QY2WE9) - In fact, Borisov’s main result was to construct an element in the kernel of multiplication by L, and, seemingly coincidentally, his method also constructed an element in the kernel of ψn. [(Zakharevich, 2017, p. 1)](zotero://open-pdf/library/items/ERN8H528?page=1&annotation=XIFNYXJ6)- Borisov's coincidence. ![[Zak17b_3NJAHFIX.png]] [(Zakharevich, 2017, p. 2)](zotero://open-pdf/library/items/ERN8H528?page=2&annotation=4VDWMS8N) - Theorem A ![[Zak17b_RBTASIL7.png]] [(Zakharevich, 2017, p. 2)](zotero://open-pdf/library/items/ERN8H528?page=2&annotation=KGSEQA7N) - Theorem B ![[Zak17b_E53CJSLX.png]] [(Zakharevich, 2017, p. 2)](zotero://open-pdf/library/items/ERN8H528?page=2&annotation=G4I9H9CL) - Theorem C ![[Zak17b_3CXY8EAF.png]] [(Zakharevich, 2017, p. 2)](zotero://open-pdf/library/items/ERN8H528?page=2&annotation=7TAV5JLB) - Theorem D ![[Zak17b_GU6HQVW9.png]] [(Zakharevich, 2017, p. 2)](zotero://open-pdf/library/items/ERN8H528?page=2&annotation=EM3Q6B3E) - Theorem E. This morphism only takes *smooth* varieties to their single birational isomorphism class. ![[Zak17b_9IS68UVN.png]] [(Zakharevich, 2017, p. 3)](zotero://open-pdf/library/items/ERN8H528?page=3&annotation=JKTX8FAQ) - Liu-Sebag's result ![[Zak17b_4M78YSVI.png]] [(Zakharevich, 2017, p. 3)](zotero://open-pdf/library/items/ERN8H528?page=3&annotation=I7M6PB3S) ![[Zak17b_EFAU7I7Q.png]] [(Zakharevich, 2017, p. 3)](zotero://open-pdf/library/items/ERN8H528?page=3&annotation=IIPVAVXR) - Notation. ![[Zak17b_YG2GUSQS.png]] [(Zakharevich, 2017, p. 3)](zotero://open-pdf/library/items/ERN8H528?page=3&annotation=7JPXZBBM) - Definition of assemblers. ![[Zak17b_4GC6KMLE.png]] [(Zakharevich, 2017, p. 4)](zotero://open-pdf/library/items/ERN8H528?page=4&annotation=NBWYVGIE) - Fundamental theorem of $\Asm$. Relations between generators left imprecise. ![[Zak17b_3665F2KA.png]] [(Zakharevich, 2017, p. 4)](zotero://open-pdf/library/items/ERN8H528?page=4&annotation=G48E64UE) ![[Zak17b_772FZWHX.png]] [(Zakharevich, 2017, p. 4)](zotero://open-pdf/library/items/ERN8H528?page=4&annotation=RFSSD26K) ![[Zak17b_WI8P7SA7.png]] [(Zakharevich, 2017, p. 4)](zotero://open-pdf/library/items/ERN8H528?page=4&annotation=HQRW648R) ![[Zak17b_QWMRAMYC.png]] [(Zakharevich, 2017, p. 5)](zotero://open-pdf/library/items/ERN8H528?page=5&annotation=8FQC5VCY) ![[Zak17b_2F9S2LDJ.png]] [(Zakharevich, 2017, p. 5)](zotero://open-pdf/library/items/ERN8H528?page=5&annotation=9PR7V7RC) ![[Zak17b_835TG39A.png]] [(Zakharevich, 2017, p. 5)](zotero://open-pdf/library/items/ERN8H528?page=5&annotation=RKV5DWBA) ![[Zak17b_NKGEIK6X.png]] [(Zakharevich, 2017, p. 5)](zotero://open-pdf/library/items/ERN8H528?page=5&annotation=5A4SPDDH) ![[Zak17b_J9ULDSL8.png]] [(Zakharevich, 2017, p. 5)](zotero://open-pdf/library/items/ERN8H528?page=5&annotation=X7GXYKUI) - #todo write as a tower with cofibers on the side. ![[Zak17b_GQETJSTK.png]] [(Zakharevich, 2017, p. 5)](zotero://open-pdf/library/items/ERN8H528?page=5&annotation=7FN438RF) - devissage for assemblers ![[Zak17b_QKJF73F3.png]] [(Zakharevich, 2017, p. 5)](zotero://open-pdf/library/items/ERN8H528?page=5&annotation=DIU2BHJE) ![[Zak17b_IBSSKKTW.png]] [(Zakharevich, 2017, p. 6)](zotero://open-pdf/library/items/ERN8H528?page=6&annotation=PZ8BNPZ4) - Theorem A follows directly from several results in [ZakA]. Here, we give an outline of the proof by reducing of the theorem to those results. [(Zakharevich, 2017, p. 6)](zotero://open-pdf/library/items/ERN8H528?page=6&annotation=G66WS396)- Proof of theorem A ![[Zak17b_CJXFFUDH.png]] [(Zakharevich, 2017, p. 6)](zotero://open-pdf/library/items/ERN8H528?page=6&annotation=VXXT7CZC) ![[Zak17b_VWVIEEQA.png]] [(Zakharevich, 2017, p. 6)](zotero://open-pdf/library/items/ERN8H528?page=6&annotation=WTNZF77B) - $\Sigma \cat{C}.$ ![[Zak17b_D3JULKGH.png]] [(Zakharevich, 2017, p. 7)](zotero://open-pdf/library/items/ERN8H528?page=7&annotation=NR77JIKI) - Cofiber of maps between assemblers ![[Zak17b_C8HIUGSS.png]] [(Zakharevich, 2017, p. 7)](zotero://open-pdf/library/items/ERN8H528?page=7&annotation=32KQQTKN) ![[Zak17b_NK4APXN2.png]] [(Zakharevich, 2017, p. 7)](zotero://open-pdf/library/items/ERN8H528?page=7&annotation=BQTVNPPP) - Embedded exact sequences ![[Zak17b_Z7QRHLWY.png]] [(Zakharevich, 2017, p. 7)](zotero://open-pdf/library/items/ERN8H528?page=7&annotation=4Z7LBT2N) ![[Zak17b_7QGR2ITT.png]] [(Zakharevich, 2017, p. 8)](zotero://open-pdf/library/items/ERN8H528?page=8&annotation=8MCPX6G8) ![[Zak17b_TJQG3MAW.png]] [(Zakharevich, 2017, p. 8)](zotero://open-pdf/library/items/ERN8H528?page=8&annotation=AGBVYB8F) ![[Zak17b_HEL2Q2FV.png]] [(Zakharevich, 2017, p. 8)](zotero://open-pdf/library/items/ERN8H528?page=8&annotation=2KBVNJVK) ![[Zak17b_GI5GZT8R.png]] [(Zakharevich, 2017, p. 9)](zotero://open-pdf/library/items/ERN8H528?page=9&annotation=FBM367RQ) - Proof. [(Zakharevich, 2017, p. 9)](zotero://open-pdf/library/items/ERN8H528?page=9&annotation=QH7BQF79) - Thus the boundary map in the long exact sequence associated to the inclusion of one filtration degree into the next measures the error of a birational automorphism of the variety extending to a piecewise automorphism [(Zakharevich, 2017, p. 9)](zotero://open-pdf/library/items/ERN8H528?page=9&annotation=GHG4495C) ![[Zak17b_JSV89AMC.png]] [(Zakharevich, 2017, p. 9)](zotero://open-pdf/library/items/ERN8H528?page=9&annotation=C9XPT8P5) - Proof. [(Zakharevich, 2017, p. 9)](zotero://open-pdf/library/items/ERN8H528?page=9&annotation=VKRHMIWD) ![[Zak17b_VG29FLCM.png]] [(Zakharevich, 2017, p. 10)](zotero://open-pdf/library/items/ERN8H528?page=10&annotation=HMJQYKCM) - Proof. [(Zakharevich, 2017, p. 10)](zotero://open-pdf/library/items/ERN8H528?page=10&annotation=UWC2HRXR) ![[Zak17b_7MQX3AWT.png]] [(Zakharevich, 2017, p. 10)](zotero://open-pdf/library/items/ERN8H528?page=10&annotation=YJNVQAFY) ![[Zak17b_UAV7EWUC.png]] [(Zakharevich, 2017, p. 11)](zotero://open-pdf/library/items/ERN8H528?page=11&annotation=S2QKWBE7) - Proof. [(Zakharevich, 2017, p. 11)](zotero://open-pdf/library/items/ERN8H528?page=11&annotation=AVPUJ9MZ) ![[Zak17b_CANN4BGG.png]] [(Zakharevich, 2017, p. 14)](zotero://open-pdf/library/items/ERN8H528?page=14&annotation=Q3VDPT9F) - Proof. [(Zakharevich, 2017, p. 14)](zotero://open-pdf/library/items/ERN8H528?page=14&annotation=D9WCXRU8) - Proof of (2): [(Zakharevich, 2017, p. 15)](zotero://open-pdf/library/items/ERN8H528?page=15&annotation=AYK2Q87P) - Proof of (3): [(Zakharevich, 2017, p. 15)](zotero://open-pdf/library/items/ERN8H528?page=15&annotation=3E56GHD5) ![[Zak17b_WDBBSQ77.png]] [(Zakharevich, 2017, p. 15)](zotero://open-pdf/library/items/ERN8H528?page=15&annotation=8GQ2DUAP) - Proof. [(Zakharevich, 2017, p. 15)](zotero://open-pdf/library/items/ERN8H528?page=15&annotation=EMLJUNZJ) ![[Zak17b_86PVTSUV.png]] [(Zakharevich, 2017, p. 15)](zotero://open-pdf/library/items/ERN8H528?page=15&annotation=63UIISB6) ![[Zak17b_XG2X2EUX.png]] [(Zakharevich, 2017, p. 16)](zotero://open-pdf/library/items/ERN8H528?page=16&annotation=9RH7TGVA) ![[Zak17b_7YGMIV73.png]] [(Zakharevich, 2017, p. 16)](zotero://open-pdf/library/items/ERN8H528?page=16&annotation=AGPC7MP9) - Proof. [(Zakharevich, 2017, p. 16)](zotero://open-pdf/library/items/ERN8H528?page=16&annotation=DHYB8CT5) ![[Zak17b_M6SYD2AH.png]] [(Zakharevich, 2017, p. 16)](zotero://open-pdf/library/items/ERN8H528?page=16&annotation=UBQHW2M5) ![[Zak17b_FZYQ3ZZE.png]] [(Zakharevich, 2017, p. 16)](zotero://open-pdf/library/items/ERN8H528?page=16&annotation=GLNPQC7B) ![[Zak17b_IURAI8UB.png]] [(Zakharevich, 2017, p. 17)](zotero://open-pdf/library/items/ERN8H528?page=17&annotation=I9FYPB2S) - Proof. [(Zakharevich, 2017, p. 17)](zotero://open-pdf/library/items/ERN8H528?page=17&annotation=EBWECYJH) ![[Zak17b_MVRVE36K.png]] [(Zakharevich, 2017, p. 18)](zotero://open-pdf/library/items/ERN8H528?page=18&annotation=6BJJTZ5T) - Proof. [(Zakharevich, 2017, p. 18)](zotero://open-pdf/library/items/ERN8H528?page=18&annotation=WXFB5I75) ![[Zak17b_NLL8FFEU.png]] [(Zakharevich, 2017, p. 18)](zotero://open-pdf/library/items/ERN8H528?page=18&annotation=26X4HRKI) ![[Zak17b_Z9MT45K5.png]] [(Zakharevich, 2017, p. 19)](zotero://open-pdf/library/items/ERN8H528?page=19&annotation=JXWVRIQU) - Proof of Theorem E. [(Zakharevich, 2017, p. 19)](zotero://open-pdf/library/items/ERN8H528?page=19&annotation=CAL7W6WW) ![[Zak17b_K6C6FTB7.png]] [(Zakharevich, 2017, p. 19)](zotero://open-pdf/library/items/ERN8H528?page=19&annotation=H2WZSWRN) - Proof of Theorem C. [(Zakharevich, 2017, p. 19)](zotero://open-pdf/library/items/ERN8H528?page=19&annotation=ERIQZFY9) - Proof of Theorem D. [(Zakharevich, 2017, p. 20)](zotero://open-pdf/library/items/ERN8H528?page=20&annotation=CW9IE8Y7) - Conjecture ![[Zak17b_8DBW796J.png]] [(Zakharevich, 2017, p. 20)](zotero://open-pdf/library/items/ERN8H528?page=20&annotation=P4438DU9) - Conjecture ![[Zak17b_93H4Q2QM.png]] [(Zakharevich, 2017, p. 21)](zotero://open-pdf/library/items/ERN8H528?page=21&annotation=N5HTWTNK)