Filtrations induced by tilting torsion pairs alberto tonolo - - PowerPoint PPT Presentation

filtrations induced by tilting torsion pairs
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Filtrations induced by tilting torsion pairs alberto tonolo - - PowerPoint PPT Presentation

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slide-1
SLIDE 1

Filtrations induced by tilting torsion pairs

alberto tonolo Università di Padova

P¨ arnu, July 2019

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slide-2
SLIDE 2
  • L. Fiorot, F. Mattiello, -: A classification theorem for t-structures,
  • J. Algebra 2016
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  • F. Mattiello, S. Pavon, -: Tilting modules and tilting torsion

pairs, (2019)

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slide-3
SLIDE 3

Let A be an associative ring with 1 6= 0.

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progenerator

<latexit sha1_base64="fK0TfvNJNwDsbq8OzlnfeMf0FOs=">ADwXicpVJNb9NAEN3GfJTw1cKRywqrEoco8jqhKYdKFb1wLBJpK8VWtV6PnVX62U9gURW/gbX8rf4N6wbg5KI9sJI9o7fzJu3fprEKFlhEPza6XgPHj56vPuk+/TZ8xcv9/ZfnVflzAoYi1KV9jLhFSipYwSFVwaC7xIFwk16dN/eIb2EqW+gsuDMQFz7XMpODoMjYMgcNlmNpr/b8oP/h6DAcHtKgHwQjFrImCUfDwZAyhzThkzbOrvY7N1FailkBGoXiVTVhocG45halULDsRrMKDBfXPIeJSzUvoIr20sv6YFDUpqV1j0a6S26zqh5UVWLInGdBcdptV1rwH/WGsRgMd/Sx+worqU2MwQtVvLZTFEsaeMLTaUFgWrhEi6sdH9AxZRbLtC5tzE/h7IAtAs3H2GO32WK02Mavg8Mdg/WJY2zpGe4AbvqYaHUq+8pyHyKx4xJHa/P+x+/JnGtuLuQ86zXWtdrbWrOhlT1qMtwWmxKcaXiet7op5BFhc/qGElWe2zZYs2clrz1Gd+WEcJ5FLXieuycr6kPosiP4xAp3+xTRo2NH9wFzGi/uAe9tcV2x/ey3ev4daQrlvpP3tL707Owz4b9MPoX/ysV3uXfKGvCXvCMjckI+kTMyJoIY8oPckJ/eqSc949lVa2en5bwmG+HVvwHlFDuT</latexit>

if

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/ P

n

/ ··· / P / T / 0

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A left A-module AG is a

<latexit sha1_base64="T41Cn0kXeE/sahXRDYeJowHRkU=">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</latexit>

G ≤⊕ A(α)

<latexit sha1_base64="IgUnBrEsCH0jY67lmDjs8bo8fhk=">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</latexit>

A ≤⊕ G(α)

<latexit sha1_base64="xLPgfzSXRUHlLrJPpJW8T5qpLSk=">AD03icpVLPb9MwFPYWfozyq4MjHCy8SUOqiYdgsukAQc4Doluk+qschynseY4nu1AK6uXiSt/A1eu8N/w3+C0EWo72IUnJf7yvfe9F396iRLc2F7v18ZmcOPmrdtbd1p3791/8LC9/ejYlJWmbEBLUerThBgmuGQDy61gp0ozUiSCnSTnb+v8ySemDS/lRztVLC7IWPKMU2I9NWo/3XmNBbuAZ7hUojLw3Znbw0SonDyf7YzaqNftzQNeBWEDEGjiaLS9+Q2nJa0KJi0VxJhGCkbO6Itp4LNWrgyTBF6TsZs6KEkBTOxm19jBnc9k8Ks1P6RFs7ZYUjhTHTIvGVBbG5Wc/V5F9zNaNsMVmb7NXseNSVZJuhifVQLaEtZOwZRrRq2YekCo5v4GkOZE2q9nyv9x6wsmNVT39+yif3MU5sfwOhFT9nW7vJI5S3pKYXtSEZeL75zxcW4PwpDLeLnf/g1jJ0g/oe8Z53Guk5jU3WItOBHtm8WB1FhIjdpJ6fsgwXKHS4ViWZQ+GsYetxUpIUhShyOGFjLl3iqzSfzCAKMUYRZjL9w63KbC1D/X8JMUT9a9QXCzXav1bvX/trTVp+pcP1Bb4KjqNu2O9GHyJ0+KZ7i3wBDwDeyAEL8EheA+OwABQcAm+gx/gZzAIXHAZfFmUbm40msdgJYKvwGrw0Cu</latexit>

P

i ≤⊕ A(α)

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/ A / T0 / ··· / Tn / 0

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Ti ≤⊕ T (α)

<latexit sha1_base64="TVBqIRCT2TpeqXPq7z7zjoDWRg=">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</latexit>

α ∈ N

<latexit sha1_base64="djxm2DKY3rGvlQM0WmT0x5yUbfs=">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</latexit>

A left A-module AT is a n-tilting module, n ∈ N, if

<latexit sha1_base64="9Es96n9xN2JQ6uIko/zUwfoR7+w=">AEMHicpVLbtQwFPV0KC0D9AFLNhZJRbpKM70uajUwoYVKlJf0iQaOY6Tseo4wfbAjKx8C/A18CqYstX4DQR6kyhGywlOTm+57r6xuXnCnt+z86S91Hy49XVp/0nj57vra+sfniQhUTSeg5KXghr2KsKGeCnmumOb0qJcV5zOlfP2u3r/8TKVihTjTs5JGOc4ESxnB2lKjVkoCiYSKjQ8gZymGron7nZeJBNOoWuq0cmZC5mCGJrw1m0oszgyft/f2z0c+J7f3/XR/uGhBb6/dzAIKle425pxzURWwSaRB10RMhHmWI/jGH5wPcjS0YZTa+oF7wPUAge063S0ufQ1TAoyW2xhGOlhigodWSw1IxwWvXCiaIlJtc4o0MLBc6pisxt1RXcskwC0Lax72lr2rMDhXapbHNrIuUy3u1eRf92qm1Pl0wV+nB5FhopxoKkhjn041AWsrwEmTFKi+cwCTCSzJ4BkjCUm2l7WXP6MFjnVcmbzazrVX1ix0cw2PVL3du6a1nalnglLqlsYlDARPM/piwb6yOEmIju5vufg0jw7EtyPbMa1vntW2qv7VIedAiPc7nrTDnkZnW/glNw9xBphmM1DioatnaTgicOMgJTBjTjAkT2yjJphV0UBg6QUhF8oebl+la5gz+JQyhM3hA/alROzsP6u1rZyFJz40Whzg+Ai6KNBP/gYOMdv2+FeBa/Aa/AGILAPjsF7cArOAQE3neXOWme9+637vXvT/dmELnVazUswt7q/fgMx91o</latexit>

AT is a classical n-tilting module, if

<latexit sha1_base64="Se5GX531CFC1+uERY48/N0zwYIk=">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</latexit>

α ∈ N

<latexit sha1_base64="djxm2DKY3rGvlQM0WmT0x5yUbfs=">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</latexit>

A progenerator is a classical 0-tilting module.

<latexit sha1_base64="L+vVjpgNykEvi04sV+Dfat5Yuec=">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</latexit>
slide-4
SLIDE 4

Let AT be a classical n-tilting module, and B := End(AT).

<latexit sha1_base64="LxPk6yzK8sCNa05gdwEsY7l3sHA=">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</latexit>

Morita equivalence

<latexit sha1_base64="Ez6vwpqXV0/Mq/o8byNYdxdLolI=">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</latexit>

classical n-tilting

<latexit sha1_base64="vyQbLPdj+bFNHiu/cLMSwsCMTHQ=">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</latexit>

HomA(T,?) : A-Mod

<latexit sha1_base64="vheblielSh2036bSywom3R9Mx1Q=">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</latexit>

B-Mod : T⊗B?

<latexit sha1_base64="t0shaMwOvbKo6TQVoQADrpEzgk=">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</latexit>

<latexit sha1_base64="quJFplPBqZOIDMoplN74qhOCTBw=">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</latexit>

<latexit sha1_base64="quJFplPBqZOIDMoplN74qhOCTBw=">ADuHicpVJdb9MwFPUaPkb52AaPvFhYk3ioqjgdAiFNmuCFxyHoNqmJhuM4iTXbCfYtIr6E3iFR34X/wanjVDbwV64UuKTc+51zm6a2kgzD8tdMLbt2+c3f3Xv/+g4eP9vYPHp+5amq5GPNKVfYiZU4oacQYJChxUVvBdKrEeXr1ts2fxHWycp8hHktEs0KI3PJGXjqQwzV5T4Jh+Ey8HVAO0BQF6eXB72fcVbxqRYGuGLOTWhUQ9IwC5IrsejHUydqxq9YISYeGqaFS5rlXRf40DMZzivrHwN4ya4rGqadm+vUV2oGpdvOteRfcy1Tg5tzYf8VdJIU09BGL4an08Vhgq3duBMWsFBzT1g3Er/B5iXzDIO3rSN/oWotA79/1BzOCrzKA8xtGLsIb+4frI2lsyqFkt7KqGRtKsvkshixKOKZUmWe/3P35NkYxfyHv2aCzbtDZ1J6tyA2wR1DqzVFMqaSZtfMzkcea0CZuVWneELro2HacMSwjlERNnIpCmib1VbOFpjQOCZRLEz2h9uUQSsjo38JY0xGN6g/r9Tk6Ea9fx1tNen7labC3wdnEVDOhpG70Ny8qZb7l30FD1DzxFL9EJeodO0RhxVKBv6Dv6EbwOPgVFIFelvZ1O8wRtRGB/A85vNxw=</latexit>

Exti

A(T,?) :

KEi

<latexit sha1_base64="HOrTRtxXrRdfKS6SrZK8bAL4WsM=">AD3HicpVJNb9NAEN3GfJTw0RSOXFasIhUpimKnCIRUCVkLgUqWkrxW60Xq/jVe21uzuGRJZv3CquXPkNXOGX8G9YxZKUuiFkex9fjNvxvs0fhYLDYPBr42WdePmrdubd9p3791/sNXZfnis01wxPmJpnKpTn2oeC8lHICDmp5niNPFjfuKfv63yJx+50iKVRzDPuJfQqRShYBQMNel0XeAzKA5mUJ6Jyeudo96rpy/di5wGuM68PygnYtIhg/5gEfgqsBtAUBOHk+3WNzdIWZ5wCSymWo9tJwOvoAoEi3nZdnPNM8rO6ZSPDZQ04dorFvcpcdcwAQ5TZR4JeMEuKwqaD1PfFOZUIj0eq4i/5qrmAyS2dp8CF94hZBZDlyenyYxhSXFmGA6E4g3huAGVKmBtgFlFGRhjV/pPeZpwUHPTvzLvkwg2sPOs0EG7e7yMxY0stoxlVdYztC1t8RF9MI9mxbSG+53/4NfaKmJofMp71Gut6jU3VWYl0DxsEUbI6isaxV8yq+QEP3YTYhVup/LAgdtmw1TgpaUBs4hSuz6dCFr6pUmJWYmK7LnFcLoM/3KoMKhkZ/kvoYjK8Rn1Rq8nutXrz2l1r0jYrba8v8FVw7PTtYd/5MCD7b5rl3kSP0RO0g2z0HO2jd+gQjRBDl+g7+oF+WmfWZ+vS+lKXtjYazSO0EtbX35CERPU=</latexit>

KTi : TorB

i (T,?)

<latexit sha1_base64="sv5OayzB/4gjrUqUN6ERYQGRno=">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</latexit>

<latexit sha1_base64="quJFplPBqZOIDMoplN74qhOCTBw=">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</latexit>

<latexit sha1_base64="quJFplPBqZOIDMoplN74qhOCTBw=">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</latexit>

Miyashita equivalences

<latexit sha1_base64="8EM67RxDRMOfuk+oJexYpq6Z9w=">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</latexit>

KEi := {M 2 A-Mod : Ext j

A(T,M) = 0 8 j 6= i}

<latexit sha1_base64="luoIDePHBe6b2JDZ213JIeQV2g=">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</latexit>

KTi := {N 2 B-Mod : TorB

j (T,N) = 0 8 j 6= i}

<latexit sha1_base64="uKbLTzQ9DHwPGIjdGNxrf4ZAsQ=">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</latexit>

i = 0,1,...,n

<latexit sha1_base64="hWIKd1D0ExDVDHJeF9XEzPESnzI=">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</latexit>

classical 0-tilting

<latexit sha1_base64="s2lq/XQNJLrT58bTGmlKq5KPAmM=">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</latexit>

=progenerator

<latexit sha1_base64="V8Bnwykz1UWybQs+fU4VDQPLEFg=">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</latexit>
slide-5
SLIDE 5

...

<latexit sha1_base64="GPO/nDTOChP4jh6nlZRQeucDVY=">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</latexit>

A-Mod

<latexit sha1_base64="hHOlSzglkU1w3ATwvVJpEYE/bYM=">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</latexit>

KE0

<latexit sha1_base64="PHeouNkoAjVBQDx6YAx2cjNDbc=">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</latexit>

KE1

<latexit sha1_base64="1iYBlnKa5S3pj9GZjQ05vUVFLCo=">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</latexit>

KEn

<latexit sha1_base64="Lx59bl2XVnqIB3LGCl/AL8c81xs=">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</latexit>

B-Mod

<latexit sha1_base64="2/RfzWTDXw1Y7PV0M+DFMzTyDg=">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</latexit>

KT0

<latexit sha1_base64="lRtCAQJhPBPqLxhgHjerRidWI=">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</latexit>

KT1

<latexit sha1_base64="jLrMEKQ+Iq+dgfLF2RVDh8wOrY=">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</latexit>

KTn

<latexit sha1_base64="mrcME8YNwoTo7LAfB/P3nliNeM=">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</latexit>

Extn

A(T,?)

<latexit sha1_base64="peFK5hTaEjcVr/3r/oDJVE+IvXQ=">ADynicpVJNb9NAEN3GfJTw0ZQeuaywKhUpirxOWpJDRFC4sChSE1bKTbRer2OV12vze4EYlm58T96hZ/Ev2HdBJREtBdGsnf8Zt689dNEhRQGPO/XVsO5d/Bw+1HzcdPnj7bae0+Pzf5VDM+ZLnM9WVEDZdC8SEIkPy0JxmkeQX0dW7un7xlWsjcnUGZcHDjE6USASjYKFxay8APoPq/Qzmn9X47cFZ+82rcv1OoceGRwdYa/jeaTXJzYZDPoWxMQidbhoGafj3cZ1EOdsmnEFTFJjRsQvIKyoBsEknzeDqeEFZVd0wkc2VTjJqxubj/H+xaJcZJr+yjAN+gqo6KZMWUW2c6MQmo2azX4z1qNFJDNvQh6YeVUMUuGIL+WQqMeS4NgjHQnMGsrQJZVrYP8AspZoysDauzZ/wPOgSzu/NvGbiCE9xv6hV0Bzf1WysJa0C1pwveghvlCL75SLSQrHhAgVrs7H79GYSWpvZD1rL20r20qT5rkmljm0GarUtRKcNqVuvHPAkyl1RBzYqSyiXzJVrLKUVjl7h+FUR8IlQV2S4tZnPskiBw/YCr+C+2ToOa5nZvIwbY7d7B/rJgu707+fbV2xjStCv9Z2/x7cm53yHdjv/Jc0/6y+XeRi/QS3SACHqNTtAHdIqGiKESXaMf6Kfz0dFO6VSL1sbWkrOH1sL5/ht0lj2z</latexit>

TorB

n(T,?)

<latexit sha1_base64="3LFQTSktujn/mgLhsIic6v9/jZ0=">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</latexit>

Ext1

A(T,?)

<latexit sha1_base64="PeysUxf34EOdaNLk9sbcYMCXziY=">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</latexit>

TorB

1(T,?)

<latexit sha1_base64="EDcQZnj7hWdazyrKx3/OFfsa8=">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</latexit>

Hom(T,?)

<latexit sha1_base64="Ev36/+2fifOD7faPLyemWod8zf4=">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</latexit>

T⊗B?

<latexit sha1_base64="dpTsJbLV5yjf1Q0KHcowdG+y5Ss=">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</latexit>

...

<latexit sha1_base64="CTV7QCSr3IrLJ0vIBD/oS41BEi8=">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</latexit>

AT

<latexit sha1_base64="bpEoNLR/ORBv3jz4nhmdI/DRA+4=">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</latexit>

classical n-tilting

<latexit sha1_base64="vyQbLPdj+bFNHiu/cLMSwsCMTHQ=">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</latexit>

n ≥ 1

<latexit sha1_base64="lU12UvV6hM+8rjX62Ckmqag1pzw=">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</latexit>
slide-6
SLIDE 6

A torsion pair in an abelian category A

<latexit sha1_base64="2jmOm4tLVcaxT2r6IdEnm/eY8ic=">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</latexit>

is a pair

<latexit sha1_base64="1BpQApYfXtYcKyqTw2Ua9xzZGNI=">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</latexit>

(X ,Y )

<latexit sha1_base64="GJcsTs5B9sbrxsE/wsV4ghwbJAU=">AD0HicpVJNb9NAEN3WfJTw0ROiMuKbaUiRZHXqQSXShVcOBZE2qDYitbrdbzqem12J5DIiqu/AlOXOH38G9YJ1aVpNALI9l+fjNvZvdp4lJC7/e2vbu3X7zt2de637Dx4+2m3vPT6zxcRw0eFKswgZlYoqUfJCgxKI1geazEeXzxps6fxbGykJ/gFkpopyNtUwlZ+CoUfvp/mGYM8g4U3jQuYIfX+yP2sTv+ovA1wFtAEFNnI72tr+HScEnudDAFbN2SIMSoZkFyJeSucWFEyfsHGYuigZrmwUbW4wxwfOCbBaWHcowEv2FVFxXJrZ3nsKutD2s1cTf41VzMl5NON+ZC+iqpywkIzZfj04nCUODaJpxIziomQOMG+lugHnGDOPgzFzrPxZFLsDMXH8QU/giE8iOc/3S2gdrI4snSWdkpXCLGtoIPXyPxNynMExpVJHq/3+x69hVCnmDuQ86zTWdRqb6m8tsh3sEGT5+imVFRN6/mJSMOc0GqxFnFaETpv2Hqc1iwhlARVGIux1FXsqoyczjGhYUiCUOjkiluXQS0jvX8JQ0x6N6g/LdXk6Ea9ex1tNGm5labC3wdnAVd2usG7wJy8rpZ7h30D1Hh4il+gEvUWnqI84ukQ/0E/0y3vTb1L7+uydHur0TxBa+F9+wMPDj8K</latexit>
  • f additive subcategories
<latexit sha1_base64="1mJP7KVdwj+S5QGwxOE6RCJAU=">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</latexit>

such that

<latexit sha1_base64="T0GEfsj3As/fc1hBvpqOCsUxHp4=">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</latexit>

with XM in X and YM in Y

<latexit sha1_base64="WLc5aknBUyl8TxV2wWmgwyMC0Fs=">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</latexit>

HomA (X,Y) = 0 for each X ∈ X and Y ∈ Y

<latexit sha1_base64="aH/HWZYF8C8WiUNMCO9oFJQc3eI=">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</latexit>

for all M in A , there exists a short exact sequence

<latexit sha1_base64="q2XtBZFJbpRditwmI6k1RQ9M8vY=">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</latexit>

0 → XM → M → YM → 0

<latexit sha1_base64="CnpNzdPGlU2/aLV8Bo8IOHiew14=">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</latexit>

Equivalently

<latexit sha1_base64="7XdLQ3hyAQ5vwMiSChN+Pci3u3c=">ADwXicpVJda9swFXjfXTZR9vtcS9iorCHECynsL0Uyspgjx0sbSE2RZavY1FZViW5SzD5G3vt/tb+zeTEjCTd+rILto7Pvede+XBTLYV1Yfhrpxc8evzk6e6z/vMXL1/t7R+8PrdVbTiMeSUrc5kyC1IoGDvhJFxqA6xMJVyk16dt/uIWjBWV+ubmGpKSTZXIBWfOU/Hnm1rcMgnKyfnVPgmH4TLwfUA7QFAXZ1cHvbs4q3hdejmXzNoJjbRLGmac4BIW/bi2oBm/ZlOYeKhYCTZplpde4EPZDivjH+Uw0t2XdGw0tp5mfrKkrnCbuda8q+5ltGunG3Nd/nHpBFK1w4UX43Pa4ldhVtfcCYMcO+CB4wb4f8A84IZxp13b6P/FKoSnJn7/g5m7rvIXHGMR2GoXf9wfaT2lgw02BWNTQSavVdgJgW7phSoZL1fv/j1yRpJPMX8p4NOusGnU3t2YrsAHvkinJzFJMyaWbt/AzyuCS0iVtVmjeELjq2HacUywglUROnMBWqSX2VEbMFJjSOSRSDyv5wmzLXysjoX8IYk9ED6puVmhw9qPevo60mfb/SdHuB74PzaEhHw+hrRE4+dcu9i96id+g9ougDOkFf0BkaI40+oHu0M/gNBCBDsyqtLfTad6gjQia3754Ozg=</latexit>

for all M in A , there exists a filtration

<latexit sha1_base64="rtN2s4dOUrFi8Ib17KvUsyDqMK0=">AD7XicpVJNb9NAEN3GfJTw0RaOXFZsKnGIotipgEulAhcuSEUibaXYqtbrcbzqem12J5DIys/gVsGR/8EV/gD/hnVioSFXpiD/fbtvJnZp4lLJS32+7+2Wt6Nm7dub9p3713/8HO7t7DE1tMjIChKFRhzmJuQUkNQ5So4Kw0wPNYwWl8bq+P/0IxspCv8dZCVHOx1qmUnB01Pnus7QwlCtFO287VGraCXOmeCKvux0KWZgMLUzWEp6lUaBoh6/f6i6BXgd8ARpo4Pt9rfQ2TQkxy0CgUt3bkByVGFTcohYJ5O5xYKLm4GMYOah5DjaqFg+c03HJLQeNS0gW7qh4bu0sj1mPb7dvKvJv97VTIn5dKM/pi+iSupygqDFsn06URQLWntIE2lAoJo5wIWR7gVUZNxwgc7ptfpjKHJAM3P1Eab4SaYHdJBv19ie3+1Zeks6Za8BLPM8QOpl+cM5DjDQ9+XOlqt9z9+jaJKcTeQ86zbWNdtbKr/tch2qUOY5eut3KpE1bTun0Aa5syvFgsTpxXz5w1bt9OaJ8xnQRXGMJa6il2WkdM5ZX4YsiAEnfzh1mVYy9jgX8KQsE16g9LNTu4Vu8+BxtF2m6l/c0FvgpOgp4/6AXvAnb0qlnubfKYPCFPiU+ekyPyhyTIRHknwnP8hPr/A+e5fel2Vqa6vRPCJr4X37DaPhSlY=</latexit>

M ≥ XM ≥ 0

<latexit sha1_base64="br6rhAZlq+D/ckbNF+fOjQH9AKg=">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</latexit>

with M/XM in Y and XM = XM/0 in X .

<latexit sha1_base64="F2OfR3buWw+eZeHyJ/Dl+xelv48=">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</latexit>
slide-7
SLIDE 7

A-Mod

<latexit sha1_base64="hHOlSzglkU1w3ATwvVJpEYE/bYM=">ADwXicpVJdb9MwFPUaPkb52uCRFwtrEg+lqtMheJk02AsvSEOi26QmhzHaw5jrFvoZXVv8Hr+Fv8G5w2Qm0He+FKiU/Ovede5+hmRkHg8GvnU505+69+7sPug8fPX7ydG/2Zmrp5aLEa9VbS8y5oSWoxAghIXxgpWZUqcZ1cnTf78m7BO1voLzI1IKzbRspCcQaCS9wmIGfjXn+p8cblHBv3BMvBNQFtAUBunl/ud6ySv+bQSGrhizo1pbCD1zILkSiy6ydQJw/gVm4hxgJpVwqV+ekFPghMjovahkcDXrLrCs8q5+ZVFiorBqXbzjXkX3MNY6Cabc2H4l3qpTZTEJqvxhdThaHGjS84l1ZwUPMAGLcy/AHmJbOMQ3Bvo/9E1JUAOw/9G/O+yxzKIxy/GRjoHqyPNMGSnmFG2FUNjaVefZdCTko4olTqdL3f/g1Tr1i4ULBs15rXa+1qTkbkevhgKCsNkcxpVI/a+bnokgqQn3SqLCE7po2Wac1iwnlMQ+ycREap+FKitnC0xokpA4ETr/w23KoJGR4b+ECSbDW9RfV2pyeKs+vA63mnTDStPtBb4JzuI+HfbjzwNy/KFd7l30Ar1ErxBFb9Ex+ohO0QhxZNAPdI1+RieRjExkV6WdnVbzHG1E5H8Dkuc63g=</latexit>

KE0

<latexit sha1_base64="PHeouNkoAjVBQDx6YAx2cjNDbc=">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</latexit>

KE1

<latexit sha1_base64="1iYBlnKa5S3pj9GZjQ05vUVFLCo=">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</latexit>

B-Mod

<latexit sha1_base64="2/RfzWTDXw1Y7PV0M+DFMzTyDg=">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</latexit>

KT0

<latexit sha1_base64="lRtCAQJhPBPqLxhgHjerRidWI=">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</latexit>

KT1

<latexit sha1_base64="jLrMEKQ+Iq+dgfLF2RVDh8wOrY=">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</latexit>

Ext1

A(T,?)

<latexit sha1_base64="PeysUxf34EOdaNLk9sbcYMCXziY=">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</latexit>

TorB

1(T,?)

<latexit sha1_base64="EDcQZnj7hWdazyrKx3/OFfsa8=">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</latexit>

Hom(T,?)

<latexit sha1_base64="Ev36/+2fifOD7faPLyemWod8zf4=">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</latexit>

T⊗B?

<latexit sha1_base64="dpTsJbLV5yjf1Q0KHcowdG+y5Ss=">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</latexit>

n = 1

<latexit sha1_base64="DrTF7WKtZSmCw/UgNmCByjTW0=">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</latexit>

(KE0,KE1)

<latexit sha1_base64="QSETx3OuRKG5UDVKCtpNrXf+H6M=">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</latexit>

(KT1,KT0)

<latexit sha1_base64="UAyg2ZYgyABve9s2MNYijmiGbes=">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</latexit>

are torsion pairs in

<latexit sha1_base64="c6Mkx7ZyXViBIrVSzcMz+ZlHBYA=">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</latexit>

A-Mod

<latexit sha1_base64="hHOlSzglkU1w3ATwvVJpEYE/bYM=">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</latexit>

B-Mod

<latexit sha1_base64="2/RfzWTDXw1Y7PV0M+DFMzTyDg=">ADwXicpVJdb9MwFPUWPkb52uCRFwtrEg+lqtMieJk0bS+8IA2JbpOaHIcp7HmOMa+hVZW/wav42/xb3DaCLUd7IUrJT45957naObGSUd9Pu/dnaje/cfPNx71Hn85Omz5/sHL85dPbVcjHitanuZMSeU1GIEpS4NFawKlPiIrs+bfIX34R1stZfYG5EWrGJloXkDAKVnCQgZuDfqrzxdU+6f6y8C3AW0BQW2cXR3s3iR5zaeV0MAVc25MYwOpZxYkV2LRSaZOGMav2USMA9SsEi71y0sv8GFgclzUNjwa8JdV3hWOTevslBZMSjdq4h/5prGAPVbGs+FB9SL7WZgtB8Nb6YKgw1bnzBubSCg5oHwLiV4Q8wL5lHIJ7G/0noq4E2Hno35j3XeZQHuH4Xd9A53B9pAmWdA0zwq5qaCz16rsUclLCEaVSp+v9/sevceoVCxcKnVb67qtTc3ZiFwXBwRltTmKZX6WTM/F0VSEeqTRpUVntBFyzbjtGY5oST2SYmUvsVFk5W2BCk4TEidD5H25TBo2MDP4lTDAZ3KH+ulKT4Z368BpuNemElabC3wbnMc9OujFn4fk+KRd7j30Cr1GbxBF79Ex+ojO0AhxZNAPdIN+RqeRjExkV6W7O63mJdqIyP8Gl4Q64w=</latexit>

M

<latexit sha1_base64="0bUy2vBoGgThcYIBxlUQqMJj9M=">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</latexit>

t(M)

<latexit sha1_base64="YKJAI5+d4RknTqmqUs3ohc20TA=">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</latexit>

M/t(M)

<latexit sha1_base64="C9qWED0M6Wjbd7AgTiw6XRCd0us=">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</latexit>

N/t(N)

<latexit sha1_base64="nElMgYj9HyCNT597Ypn1s6OQ8=">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</latexit>

t(N)

<latexit sha1_base64="+Eat9W5/CwBcteZ+g9SID7oG6gU=">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</latexit>

N

<latexit sha1_base64="hzn5Emo2j/w5qHDbp+cS/vh8vA=">ADtnicpVJNb9NAEN3GfJTw0RaOXFasKnGIq8Ti+RQqYILJ9RKpK0Um2q9Htur2utlvYFEVn4BVzjzu/g3rBODkoj2wkj2jt/Mm7d+mkjlojKu+2uv49y7/+Dh/qPu4ydPnx0cHj2/qMqZ5jDhZV7q4hVkAsJEyNMDldKAyuiHC6jm3dN/fIL6EqU8qNZKAgLlkqRCM6Mhc4/XB8St+/6Y5+62O37Lh0PmQ8Hg19H9O+uwqC2ji7Pur8DOKSzwqQhuesqbUyasmTaC57DsBrMKFOM3LIWpTSUroAr1U2X+NgiMU5KbR9p8ArdZNSsqKpFEdnOgpms2q014D9rDaJMd/RN8korIVUMwOSr+WTWY5NiRszcCw0cJMvbMK4FvYPM+YZtxYy7bmp1AWYPTCzjcwN19FbLIT7PmuMt3jTUlLekpkCve6gn5Po7A5Fm5oRSIcPNef/j1zSsc2YvZD3rtdb1WpuasyFVPWwzkxXbUizPw3re6MeQBAWhdCwoqQmdNmijZyULCaUeHUQSpkHdkuLeZLTGgQEC8AGf/FtmoZHBbcQAk8Ed7M9rNhneybev4c6Qrl3pP3uLb08uvD4d9L3zITl92y73PnqJXqHXiKI36BS9R2dogjgC9A19Rz+ckfPJASdt3b2Ws4LtBWO+g1d3jZp</latexit>

AT

<latexit sha1_base64="bpEoNLR/ORBv3jz4nhmdI/DRA+4=">ADvHicpVJNj9MwEPU2fCzlaxeOXCy8K3GoqjpdBJdFC1w4LtJ2u1ITVY7jNFZtJ9jTpVXUH8EVDvwu/g1OG6G2C3thpMQvb+bNOE+TlEo6PV+7bWCO3fv3d9/0H746PGTpweHzy5dMbNcDHihCnuVMCeUNGIAEpS4Kq1gOlFimEw/1vnhtbBOFuYCFqWINZsYmUnOwFPDo2o5fn9xND4gvW5vFfgmoA0gqInz8WHrZ5QWfKaFAa6YcyMalhBXzILkSizb0cyJkvEpm4iRh4Zp4eJqd8lPvZMirPC+scAXrGbiop5xY68ZWaQe52czX51zNlKDnO/MhextX0pQzEIavx2czhaHAtSU4lVZwUAsPGLfS/wHmObOMgzduq/9EFqAXfj+IObwVaQn+Lwda+E9vHmyNJb0ilZKey6hobSrL9zISc5nFIqTbzZ73/8GsWVYv5C3rNOY12nsak+a5HrYI8g19ujmFJxNa/npyKLNKFVKuSrCJ02bD1OGNYSigJqygRE2mqxFdZOV9iQqOIhJEw6R9uWwa1jPT/JYw6d+i/rJWk5Nb9f51stOk7Vea7i7wTXAZdm/G34OydmHZrn30Qv0Er1CFL1BZ+gTOkcDxNEUfUPf0Y/gXZAG0CvS1t7jeY52org+jedozg7</latexit>

classical 1-tilting

<latexit sha1_base64="1yvA4P9xzYVeQPSWnK1VFGagq8=">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</latexit>
slide-8
SLIDE 8

...

<latexit sha1_base64="GPO/nDTOChP4jh6nlZRQeucDVY=">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</latexit>

A-Mod

<latexit sha1_base64="hHOlSzglkU1w3ATwvVJpEYE/bYM=">ADwXicpVJdb9MwFPUaPkb52uCRFwtrEg+lqtMheJk02AsvSEOi26QmhzHaw5jrFvoZXVv8Hr+Fv8G5w2Qm0He+FKiU/Ovede5+hmRkHg8GvnU505+69+7sPug8fPX7ydG/2Zmrp5aLEa9VbS8y5oSWoxAghIXxgpWZUqcZ1cnTf78m7BO1voLzI1IKzbRspCcQaCS9wmIGfjXn+p8cblHBv3BMvBNQFtAUBunl/ud6ySv+bQSGrhizo1pbCD1zILkSiy6ydQJw/gVm4hxgJpVwqV+ekFPghMjovahkcDXrLrCs8q5+ZVFiorBqXbzjXkX3MNY6Cabc2H4l3qpTZTEJqvxhdThaHGjS84l1ZwUPMAGLcy/AHmJbOMQ3Bvo/9E1JUAOw/9G/O+yxzKIxy/GRjoHqyPNMGSnmFG2FUNjaVefZdCTko4olTqdL3f/g1Tr1i4ULBs15rXa+1qTkbkevhgKCsNkcxpVI/a+bnokgqQn3SqLCE7po2Wac1iwnlMQ+ycREap+FKitnC0xokpA4ETr/w23KoJGR4b+ECSbDW9RfV2pyeKs+vA63mnTDStPtBb4JzuI+HfbjzwNy/KFd7l30Ar1ErxBFb9Ex+ohO0QhxZNAPdI1+RieRjExkV6WdnVbzHG1E5H8Dkuc63g=</latexit>

KE0

<latexit sha1_base64="PHeouNkoAjVBQDx6YAx2cjNDbc=">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</latexit>

KE1

<latexit sha1_base64="1iYBlnKa5S3pj9GZjQ05vUVFLCo=">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</latexit>

KEn

<latexit sha1_base64="Lx59bl2XVnqIB3LGCl/AL8c81xs=">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</latexit>

B-Mod

<latexit sha1_base64="2/RfzWTDXw1Y7PV0M+DFMzTyDg=">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</latexit>

KT0

<latexit sha1_base64="lRtCAQJhPBPqLxhgHjerRidWI=">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</latexit>

KT1

<latexit sha1_base64="jLrMEKQ+Iq+dgfLF2RVDh8wOrY=">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</latexit>

KTn

<latexit sha1_base64="mrcME8YNwoTo7LAfB/P3nliNeM=">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</latexit>

Extn

A(T,?)

<latexit sha1_base64="peFK5hTaEjcVr/3r/oDJVE+IvXQ=">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</latexit>

TorB

n(T,?)

<latexit sha1_base64="3LFQTSktujn/mgLhsIic6v9/jZ0=">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</latexit>

Ext1

A(T,?)

<latexit sha1_base64="PeysUxf34EOdaNLk9sbcYMCXziY=">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</latexit>

TorB

1(T,?)

<latexit sha1_base64="EDcQZnj7hWdazyrKx3/OFfsa8=">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</latexit>

Hom(T,?)

<latexit sha1_base64="Ev36/+2fifOD7faPLyemWod8zf4=">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</latexit>

T⊗B?

<latexit sha1_base64="dpTsJbLV5yjf1Q0KHcowdG+y5Ss=">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</latexit>

...

<latexit sha1_base64="CTV7QCSr3IrLJ0vIBD/oS41BEi8=">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</latexit>

AT

<latexit sha1_base64="bpEoNLR/ORBv3jz4nhmdI/DRA+4=">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</latexit>

classical n-tilting

<latexit sha1_base64="vyQbLPdj+bFNHiu/cLMSwsCMTHQ=">ADzHicpVJNb9NAEN3GfJTw0RTEicsKtxKHEHmdBHKpVMGFEyoSaSvFVrRer+1V12uznkCiVa78DcQVfhH/hnViUBLRXhjJ9uybefPWTxOVUlTgeb/2Ws6t23fu7t9r3/w8NFB5/DxeVXMNONjVshCX0a04lIoPgYBkl+WmtM8kvwiunpb1y8+c12JQn2ERcnDnKZKJIJRsNC085RJWlX2KPGROnoJQoJQ6bTjej3v1XBARtjrDT0yIkOb+EPieT4mPW8VLmribHrY+hbEBZvlXMFq4oT4JYSGahBM8mU7mFW8pOyKpnxiU0VzXoVmdf8lPrZIjJNC20cBXqGbDEPzqlrke3MKWTVbq0G/1mrkRLy+Y4+JKPQCFXOgCu2lk9mEkOBa4twLDRnIBc2oUwL+weYZVRTBtbIrfkpL3IOemHnA5/DFxFDdoL9oVdC+3hTsrSWdEtacr3uIb5Q63PGRZrBCSFChZvz/sevSWgktReynUb67qNTfW3JlVdbDPI8m0pKmVo5rV+zJMgd4kJalaUGJcsG7SWU4rGLnF9E0Q8FcpEtkuL+RK7JAhcP+Aq/ot06Cmuf3riAF2+zewP63Z7uBGvn0Ndoa07Ur/2Vt8fXLu90i/53/w3dM3zXLvo2foOXqBCHqNTtE7dIbGiCGDvqMf6Kfz3gHOMt1a2uv4TxBW+F8/Q2QGT6n</latexit>

n > 1

<latexit sha1_base64="u8qaj6xgWDajUmYOwfNgiZGO964=">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</latexit>
slide-9
SLIDE 9

Let k be an algebraically closed field.

<latexit sha1_base64="fPhb3se7J/289J+XQgiP4eyv1QY=">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</latexit>

Let A be the k-algebra associated to the quiver

<latexit sha1_base64="6+lfHEufGkG2sWk/bXWwAQw97nk=">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</latexit>

1

a

/2

b

/3 with the relation ba = 0.

<latexit sha1_base64="GTmeH6iT9e45WTgcNK0gIfCmJg=">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</latexit>

The module T = 2

3 ⊕ 1 2 ⊕ 1 is a classical 2-tilting module.

<latexit sha1_base64="Y6RKsWI7fX5USl5yVFLftpJcN1c=">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</latexit>

HomA(T, 2) ' Ext1

A(T, 2) 6= 0

<latexit sha1_base64="epM6j6wHoJKv6HKaD4A6Kzf3wzY=">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</latexit>

Consider the simple left A-module 2.

<latexit sha1_base64="cskxkH3p6fe5tInDeDcrLYTvauM=">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</latexit>

= ⇒

<latexit sha1_base64="L03kQiWy4Ikui+pftq9Zaw40=">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</latexit>

2 /

∈ KE`,` = 0,1,2.

<latexit sha1_base64="DROVUwa2f/OEvTd2KMywqwUBw=">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</latexit>

There is no way to filtrate the simple left A-module 2 with composition factors in the Miyashita classes.

<latexit sha1_base64="yJCg45MplC0h/9rlByZuzMa5r60=">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</latexit>

[T. 2002]

<latexit sha1_base64="0R8BcKjxofUBPYhfCZKYthNI2w=">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</latexit>

A left A-module M admits a filtration

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M = Mn ≥ Mn−1 ≥ Mn−2 ≥ ··· ≥ M0 ≥ M−1 = 0

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such that for every i = 0,...,n the quotient Mi/Mi−1 belongs to KEi

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if and only if

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for every i 6= j 0,

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TorB

i (T,Extj A(T,M)) = 0.

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slide-10
SLIDE 10

Let us consider the derived category D(A).

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Objects: chain complexes with terms in Mod-A

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Morphisms: chain homotopic equivalence classes of morphisms localised at the set of quasi-isomorphisms

<latexit sha1_base64="w6frEYCzov7YxIBGlyzuIl01kZA=">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</latexit>

Denote by ?[1] the suspension functor, by ?[−1] its inverse, and by ?[i], i ∈ Z, their powers.

<latexit sha1_base64="DvJDNYJMwKhACG8NK4oMfsN6qc0=">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</latexit>
slide-11
SLIDE 11

is a t-structure

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if

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approximating triangle

<latexit sha1_base64="ZkISH/vxaTVlnNp9ER6y2mijJZY=">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</latexit>

S ≤−1 := S ≤0[1] ⊆ S ≤0 and S ≥0 ⊇ S ≥0[−1] =: S ≥1

<latexit sha1_base64="LTHKniFNIjusxpJWq3D8awLRbzg=">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</latexit>

For any complex X• in D(A) there exists the

<latexit sha1_base64="xlLRJRNd1Jpg3oAXPw6Nmfw+9Q=">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</latexit>

σ≤0X•

/ X• / σ≥1X•

+1 /

<latexit sha1_base64="LJrW7hlw9sxJiOhythsTECQm6k=">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</latexit>

<latexit sha1_base64="LTe0l4UyfW3u+RLwscexViKeU=">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</latexit>

S ≤0

<latexit sha1_base64="h2q2u467qKLiMSwlevzOmY2UaZY=">ADynicpVJNb9NAEN3GfJTw0ZQeuVisKnGIq9TCS6VKrhw4FAEaSvVJlqvx/Gq67W7u4FYK9/4H73CT+LfsE4slKTQCyvZO34zb974aZJKcG2C4NdOz7t3/8HD3Uf9x0+ePtsb7D8/0+VcMZiwUpTqIqEaBJcwMdwIuKgU0CIRcJ5cvWvz519BaV7Kz6auIC7oTPKM2ocNB0cRAU1OaPCfmq+2EjAdBMBzgYBcvj3w5IF2DUndPpfu8mSks2L0AaJqjWlySsTGypMpwJaPrRXENF2RWdwaULJS1Ax3Y5feMfOiT1s1K5Rxp/ia4zLC20rovEVbaz6u1cC/41yKVKRZb+iZ7E1suq7kByVby2Vz4pvRbg/yUK2BG1C6gTH3Bz7LqaLMOBs3+s+gLMCo2vU3sDfeGryY38cBJXpH65LVs6SYUrUKsaEnK5+s6Bz3JzTAiX8Xq/HrMraCuoGcZ8POumFnU3u3JD30XWTyYlOKChHbRaufQhYVmNjldiSZxaTp0FZOSpigkMbJTDj0iauSvF42MSRTiMQKZ/sE2aWl4/C9i5OPxHezrFRsf3cl3r6OtJn230mR7gW8HZ+GIjEfhxwCfvO2Wexe9QC/RK0TQa3SC3qNTNEM1egG/UA/vQ+e8mrPrkp7Ox3nAG0c7/tvZtw+Tw=</latexit>

S ≥1

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<latexit sha1_base64="LTe0l4UyfW3u+RLwscexViKeU=">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</latexit>

The full subcategory HS = S ≤0 ∩S ≥0 is called the heart of S .

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The heart is an abelian category.

<latexit sha1_base64="o6eCp25xViqOWAsLBhzypRvI3BU=">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</latexit>

A pair S = (S ≤0,S ≥0) of full subcategories of D(A)

<latexit sha1_base64="ta8lJN3YKmgG9MgrzryieTVJVog=">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</latexit>

HomD(A)(S ≤0,S ≥1) = 0

<latexit sha1_base64="mdNv21bmuwWP1EBMEzb9OPtqbFo=">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</latexit>
slide-12
SLIDE 12

Natural t-structure

<latexit sha1_base64="0PsX3nLPkIsG4stTSJI+YUl/1w=">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</latexit>

D

<latexit sha1_base64="l/CSbt52AcUdgqRrsiz42H/9Qec=">ADwHicpVJNb9NAEN3GfJTw1cKRy4pVJQ5R5E2K4FKpKhw4Fom0lWKrWq/H8artbs7gQrP4Mr/V38G9aJhZIUemEle8dv5s0bP01SaeUwDH/tdIJ79x83H3Ufzk6bPne/svzlw5tRJGstSlvUiEA60MjFChovKgigSDefJ1Ycmf/4VrFOl+YLzCuJCTIzKlBToXFUCMyl0PXHxeUeC/vh8tDbAW8DRtpzernfuYnSUk4LMCi1cG7MBxXGtbCopIZFN5o6qIS8EhMY+9CIAlxcL2de0AOPpDQrX8M0iW6zqhF4dy8SHxlM6PbzjXgX3MNUmEx29LH7H1cK1NEYxcyWdTbGkjS0VRYk6rkPhLTK/wGVubBCojdvo/8EygLQzn1/hBl+UynmR3T4Nqywe7AuWXlLepWowK5q+ECZ1XcOapLjEefKxOv9/sevcVxr4QfynvVa63qtTc3dkFyP+gjzYlNKaB3Xs0Y/hSwqGK+XW5FkNeOLFm3kjBEp42xQRwlMlKkTX2XVbEZjyI2iMCkf7BNGjY0NvwXMaJseAf7esVmh3fy/etwq0nXrzTfXuDbwdmgz4f9weQHZ+0y71LXpHX5A3h5B05Jp/IKRkRSUryg/wkN8FJkAdlcL0q7ey0nJdk4wTfwO7Sjqg</latexit>

D≤0 =

  • X• ∈ D(A) : Hi(X•) = 0 for every i > 0
<latexit sha1_base64="EdNc3fOL0vPyvoPAs9fQBh0sZG4=">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</latexit>

D≥0 =

  • X• ∈ D(A) : Hi(X•) = 0 for every i < 0
<latexit sha1_base64="1TETnWcvMtbmBeOsjTi+P/ypraA=">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</latexit>

Let T be a n-tilting left A-module

<latexit sha1_base64="ChTOusJP3Bzl15/CZJP5di4CcAw=">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</latexit>

Tilting t-structure T

<latexit sha1_base64="td7zVWdwK4JMWKEH8m6HfFdt3xo=">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</latexit>

T ≤0 =

  • X• ∈ D(A) : HomD(A)(T,X•[i]) = 0 for every i > 0
<latexit sha1_base64="Hj05Is4EVn+jvTbaJZnqj9ezNqg=">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</latexit>

T ≥0 =

  • X• ∈ D(A) : HomD(A)(T,X•[i]) = 0 for every i < 0
<latexit sha1_base64="UzZcp4Kky/+IJKqJsT/9x/AzBk=">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</latexit>

(not necessarily classical)

<latexit sha1_base64="V3It/0pXS4Kucp06VPT7qlv1gag=">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</latexit>

D≤−n ⊆ T ≤0 ⊆ D≤0 and D≥0 ⊆ T ≥0 ⊆ D≥−n

<latexit sha1_base64="v2tbhE6WxBSBciVtf1RX50cMJuw=">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</latexit>
slide-13
SLIDE 13

Tilting t-structure T

<latexit sha1_base64="td7zVWdwK4JMWKEH8m6HfFdt3xo=">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</latexit>

T ≤0 =

  • X• ∈ D(A) : HomD(A)(T,X•[i]) = 0 for every i > 0
<latexit sha1_base64="Hj05Is4EVn+jvTbaJZnqj9ezNqg=">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</latexit>

T ≥0 =

  • X• ∈ D(A) : HomD(A)(T,X•[i]) = 0 for every i < 0
<latexit sha1_base64="UzZcp4Kky/+IJKqJsT/9x/AzBk=">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</latexit>

then

<latexit sha1_base64="9lGCQanhVRmFXDNklkUBKGZ/2J4=">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</latexit>

Let M be a left A-module;

<latexit sha1_base64="VMekfucNHWBzsEmkyBXkBiSbVrg=">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</latexit>

since

<latexit sha1_base64="YPpm4oEQKANKcgOa4J0ZWxbLv8w=">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</latexit>

Exti

A(T,M) '

<latexit sha1_base64="eVOgAn8Vq2XhJ8KlAUbPHJxOQUI=">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</latexit>

M belongs to KEe if and only if

<latexit sha1_base64="V7J0zwf/tle7D26+OkZ/VOktjA=">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</latexit>

HomD(A)(T,M[i]) = 0 8i 6= e

<latexit sha1_base64="e0JX/wPDOKaMy70/Ix7kodBI1uQ=">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</latexit>

HomD(A)(T,M[i])

<latexit sha1_base64="Nk16YdEd4UZb8c/Ub7r4XlFdOI=">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</latexit>

M ∈ A-Mod∩

<latexit sha1_base64="OHxrp0NYJ5Un/w5+ZmNd1/XnN0s=">ADzHicpVJNb9NAEN3GfJTw0RTEicsKqxKHENlOEVwqFbhwKSoSaStlrWi9Xser2utldwKJVr7yNxBX+EX8G9aJhZIUemEke5/fzJtZP02iCmEgCH7tdLwbN2/d3r3TvXv/oO93v7DM1PNOMjVhWVvkio4YWQfAQCn6hNKdlUvDz5PJtkz/zLURlfwIC8Xjk6lyASj4KhJ7/EJERK/JsDnYJ+fVGlNGFWTnh8MgmXgqyBsgY/aOJ3sd76RtGKzktgBTVmHEYKYks1CFbwuktmhivKLumUjx2UtOQmtsv71/jAMSnOKu0eCXjJrisLY1ZlImrLCnkZjvXkH/NYyCcr41H7JXsRVSzYBLthqfzQoMFW4swqnQnEGxcIAyLdwfYJZTRk4Izf6T3lVctAL178x8ItIT/CwxeBgu7B+kjlLOkrqrhe1YSRkKvnItpDkdhKGS83u9/BrHtqDuQs6zfmtdv7WpORuR6WOHIC83R9GiO28mZ/yjJR+aEmjSjLrh3XLNuOkpKkf+pElCZ8KaRNXpcW8xn5IiB8RLtM/3KYMGpk/JeQYH94jfrTSu0fXqt3r8OtJl230uH2Al8FZ9EgHA6iD4F/KZd7l30BD1Fz1CIXqJj9A6dohFiyKLv6Af6b3wLNevSrt7LSaR2gjvK+/AWstPpo=</latexit>

T ≤e∩

<latexit sha1_base64="DcD+s0+6Bb/l6bsSH7dX3xJO7M=">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</latexit>

T ≥e =

<latexit sha1_base64="nzEwWMo90n07B7mdIEzNun+mwvU=">ADy3icpVJNb9NAEN3GfJTw0RS4cVmxqsQhirxOEVwiVXDhglSkpq0Um2i9Hjurtfu7gaSmhz5HXCFf8S/YZ1YKEmhF1ayd/xm3rzx08SlFMb6/q+dlnfr9p27u/fa9x8fLTX2X98aoqp5jDkhSz0ecwMSKFgaIWVcF5qYHks4Sy+eFvnz6BNqJQJ3ZeQpSzTIlUcGYdNO48DXNmJ5xJfPKxCjO4xLAYjDvE7/nLg68HtAkIas7xeL/1LUwKPs1BWS6ZMSMalDaqmLaCS1i0w6mBkvELlsHIhYrlYKJqOf4CHzgkwWmh3aMsXqLrjIrlxsz2FXWw5rtXA3+NVcjpc1nW/o2fR1VQpVTC4qv5NOpxLbAtUM4ERq4lXMXMK6F+wPMJ0wzbp2PG/0zKHKweu76W5jZzyKxkwHuv/RL2z5YlydJd2SlaBXNTQavU9AZFN7IBSoaL1fv/j1yiqJHMDOc+6jXdxqb6rkmi1kJ/mFJMyqma1fgJpmBNaLdcjTitCFw1aynFEkJUIUxZEJVsavSYrbAhIYhCUJQyR9sk2ZrGun/ixhi0r+Bfblik8Mb+e51uNWk7Vabi/w9eA06NF+L/jgk6M3zXLvomfoOXqBKHqFjtA7dIyGiKMr9B39QD+957xrwvq9LWTsN5gjaO9/U3pOw+FA=</latexit>

if and only if

<latexit sha1_base64="5pVb6aoMbYRcm3p9ICZPVKTGC4=">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</latexit>

A-Mod∩ ⇣ HT [−e] ⌘

<latexit sha1_base64="VjLmQohfIYe34iBiSERKcnYO8aE=">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</latexit>
slide-14
SLIDE 14

D(A)

<latexit sha1_base64="vLXUYSGhNH8HP+i69T50Cebj6vQ=">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</latexit>

HT

<latexit sha1_base64="CphsjDLEypWUrnSfjC8Dg5eY3g=">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</latexit>

HT[−1]

<latexit sha1_base64="g/s046twNuyE2anMJMUQfQp8tao=">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</latexit>

...

<latexit sha1_base64="GPO/nDTOChP4jh6nlZRQeucDVY=">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</latexit>

KE0

<latexit sha1_base64="PHeouNkoAjVBQDx6YAx2cjNDbc=">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</latexit>

KE1

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KEn

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...

<latexit sha1_base64="q/Bv8JOfn3ueWB2w8NGIZ8Mrhdo=">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</latexit>

A-Mod

<latexit sha1_base64="hHOlSzglkU1w3ATwvVJpEYE/bYM=">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</latexit>

HT[−n]

<latexit sha1_base64="YbI1soH8dQIQjWeAMiqA+ePUtik=">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</latexit>
slide-15
SLIDE 15

n = 1

<latexit sha1_base64="DrTF7WKtZSmCw/UgNmCByjTW0=">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</latexit>

D≤−1 ⊆ T ≤0 ⊆ D≤0 and D≥0 ⊆ T ≥0 ⊆ D≥−1

<latexit sha1_base64="CTSZIdSNGwKodL/fqwG4q0SwqI=">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</latexit>

[Polishchuk, 2007]

<latexit sha1_base64="yvlxb5TDl5+FV2g1vJj6YyubFPA=">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</latexit>

The classes:

<latexit sha1_base64="m7fzQGfGAyr7pz/Litgta5ptmSA=">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</latexit>

= A-Mod

<latexit sha1_base64="us5U7NJT2Agm+3faHxzqFhjUTjM=">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</latexit>

form a torsion pair (X ,Y ) in HD

<latexit sha1_base64="XprPStTucVF2r6J910+e9YOlDA=">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</latexit>

X = HD ∩T ≤0

<latexit sha1_base64="nyDNB08A1kvHCa+b2WQ1QJ61WIs=">AD8XicpVJda9swFXjfXTZV9o+7kXMFMYIwXYKG4NA2frQxw6aNhB7QZblWFSWVUnZEoR/yJ5W9jb2B/YL9rq+7t9MjkNI0q0vE9g6Ove6XDjQWjSnve762Gc+fuvfvbD5oPHz1+8rS1s3umionEpI8LVshBjBRhlJO+pqRgZAE5TEj5/HFuyp+/pFIRQt+qmeCRDkac5pSjLSlRq03Y50hEzg7K3xMflaImPyhAjsTyelh9MyMilV45artfx5gveBP4CuIe7P46u9769PBntNL6ESYEnOeEaM6TU0A+EjgySmJGymY4UQgfIHGZGghRzlRkZk/soT7lklgWkj7cQ3n7KrCoFypWR7bzOquajNWkX+NVYzQ+XSjv05fR4ZyMdGE47p9OmFQF7DyESZUEqzZzAKEJbUvgDhDEmFt3V6rPyZFTrSc2fqaTPUnmuisB7ueJ3Rzf7WlsJa0BRJE1jl+QHl9zgdZ7rn+5RHq/X+x69hZBiyF7KetRfWtRc2VXslUm1okc7y9VaIschMq/4JScPc9c18OuLUuH65YKt2nKPE9d3AhDEZU25imyXptISuH4ZuEBKeLl1ma5kbvdfwhC63VvUl7XaPbhVb38HG0WadqT9zQG+Cc6Cjt/tBO/tbL8F9doGz8Bz8AL4BU4BMfgBPQBlfgJ/gFrh3lfHaunK91amNrodkDa8v5/gcA+FJ0</latexit>

= HD ∩HT

<latexit sha1_base64="C5KxAg/hzoWlTKYp0nPmq4ck+v0=">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</latexit>

Y = HD ∩T ≥1

<latexit sha1_base64="p5yhiRvQl751+Jh+2Kfj0pMUylE=">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</latexit>

= HD ∩HT [−1]

<latexit sha1_base64="6j2YlBIwLl7+N2TBk3Xeg79EFPk=">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</latexit>

=

<latexit sha1_base64="VIQwZxaunMyjckVIPB+cP5Wtxo=">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</latexit>

KE0

<latexit sha1_base64="1M6oDj1XVtP1BH5IRLCe5RmGYyg=">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</latexit>

A-Mod∩HT =

<latexit sha1_base64="mURrpjeHE8DaCM7N+7Qcz9Vd/OU=">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</latexit>

=

<latexit sha1_base64="VIQwZxaunMyjckVIPB+cP5Wtxo=">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</latexit>

KE1

<latexit sha1_base64="NYJKruNm07197Z5og8DV6S6B40Y=">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</latexit>

A-Mod∩HT [−1] =

<latexit sha1_base64="tGCpW3W2Y8bpE2imvzWewq5sl8g=">AD5HicpVJNb9MwGPZWPkb5WAdHpMkimsShreK0Wseh0oDLkibtG6TmqhyHKex5jgmcaGVlSM3bhNX+BvjCr+C38CfwF461FZsFywlfv287/M+9qM3lJwVynV/ra3X7ty9d3/jQf3ho8dPNhtbT0+KbJITOiAZz/KzEBeUM0EHilOz2ROcRpyehqev7X50w80L1gmjtVM0iDFY8FiRrAy0KjReu0rOlW69S6LSp9g6adYJQRzfVCO9PUBHpfDFgr6o4bjtl/t7XrdXei2XbeHPGQDr9ftdCEyiF3O/vbl0e9P25eHo631b36UkUlKhSIcF8UQeVIFGueKEU7Luj8pqMTkHI/p0IQCp7QI9NW7SrhjkAjGW4+oeAVusjQOC2KWRqaSnvRYjVnwX/mLCJVOl3RV/FeoJmQE0UFqeTjCYcqg9Y6GLGcEsVnJsAkZ+YFkCQ4x0QZg5f6j2mWUpXPTH/r7kcWqaQPO64rVX1nUVIaS5oS5pXNchjojonlI0T1UeIiWCx3/4NQw0x+ZCxrPm3Lrm3Ca7W1LRhCZSboshTkP9NTqRzT2UwdVoxHG2kHlHLVyQuDIQY6n/ZCOmdChqcrZtIQO8n3H86mI/mLNGVpTucmog+dzi3s9xXb6d7KN7/uSpO6GenruYU3BydeG3Xa3pGZ7TegWhvgOXgBXgIEemAfHIBDMAEXIDv4Af4WYtrn2sXtS9V6franPMLK3a1z8cd0ye</latexit>
slide-16
SLIDE 16

n > 1

<latexit sha1_base64="u8qaj6xgWDajUmYOwfNgiZGO964=">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</latexit>

D≤−n ⊆ T ≤0 ⊆ D≤0 and D≥0 ⊆ T ≥0 ⊆ D≥−n

<latexit sha1_base64="v2tbhE6WxBSBciVtf1RX50cMJuw=">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</latexit>

D≥0

i

= D≥−i ∩T ≥0

<latexit sha1_base64="sH2DMEOFgBPDX9ishQ57k3hFZSA=">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</latexit>

are coaisles for i = 0,...n!

<latexit sha1_base64="PkBAK6bh2i3bZhzN8Sk0V9FYO+c=">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</latexit>

It is D≥0

i−1 ⊆ D≥0 i

⊆ D≥0

i−1[1] for i = 1,2,...,n.

<latexit sha1_base64="N0AUsblgUIuarnXId8VUwQ86Vk=">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</latexit>

Denote by Hi the heart of the t-structure Di = (D≤0

i

,D≥0

i

).

<latexit sha1_base64="jLA6EUTj6pZyDfepxp7K9vKI4Xc=">AENnicpVJLb9NAEN40PErKo+Vx4rLCqVSkENlOJbhUqCHotEH1JtovV6HK+6Xru7Y0hkRfwU/gZ/hQu3is/gXVioSFXhjJ3plv5puZnZ2okMKg635vrbVv3b5zd/1eZ+P+g4ePNrcen5i81ByOeS5zfRYxA1IoOEaBEs4KDSyLJxGF+9q/+kn0Ebk6gNOCgzNlIiEZyhYabXwKVCxWDQnoAKkeg0YR2g4xhypmkh0PRpZgCTYFpHkyM7rYfWVQlxLDQvRB0Oxt7NofawCZfUnfZW0NEMfdntDzcdt+/OhF5XvEZxSCNHw621r0Gc8zKzLXPJjDn3/ALDyrYnuIRpJygNFIxfsBGcW1WxDExYzSY1pdsWiWmSa/vZK8/QRUbFMmMmWQj64bNq8G/+qrkQKz8Up9TN6ElVBFiaD4vHxSo5rR+DxkIDRzmxCuNa2BtQnjLNONonW8o/gjwD1BObH2GMn0WM6R4duG6Bne3FkoUdSa9gBeh5jOcLNbdTEKMU9zxPqHAx3/M6zysJLMN2Zn1mtH1mjHVZ0yPWo1TLPlUkzKsBrX9WNIgszxqtmKREnleNMGrcspxWLHc/wqiGAkVBXZKC3GU+p4QeD4Aaj4D7ZMw5rmDP5FDKgzuIF9OWc7uzfy7W93JUnHrS3usDXlRO/7w36/nvf2X/bLPc6eU5ekB3ikdknxySI3JMOLlqbSetp61v7V/tK/aP+eha62G84QsSfvXb4g7Y8g=</latexit>

⇣ Xi := Hi ∩Hi+1,

<latexit sha1_base64="DdDN8m9tmJTNWAmns4FKWIUBKOo=">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</latexit>

Yi := Hi ∩Hi+1[−1] ⌘

<latexit sha1_base64="q+WqsEmwNgSVKlmb4aGjEgEyYA=">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</latexit>

is a torsion pair in Hi

<latexit sha1_base64="Trzsrxb+E1zJ6KhZ9+PzZCY9+0=">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</latexit>

We get n+1 t-structures Di with D0 = D and Dn = T

<latexit sha1_base64="i5HdJK31wib+XH9EykmCiqF8LGw=">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</latexit>
slide-17
SLIDE 17

The t-tree

<latexit sha1_base64="68GMEFLpxfQCucoJcrXwHY1jM0=">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</latexit>

X

<latexit sha1_base64="VI1soU2Ee5YOjSb+NrbRU7RWmyM=">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</latexit>

∈ A-Mod

<latexit sha1_base64="IVlFPJPeYknoK84nBjQCN4ptchA=">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</latexit>

X0

<latexit sha1_base64="fmuD/Dw9yH+pjgl0EF7lVCdDOE=">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</latexit>

X1

<latexit sha1_base64="iW1E6fw7j7HKLWFUv+y9RHXnNWs=">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</latexit>

<latexit sha1_base64="sNIv1nbsZihxJOrIdoTl1JoWn8=">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</latexit>

H1

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<latexit sha1_base64="sNIv1nbsZihxJOrIdoTl1JoWn8=">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</latexit> H1[−1] <latexit sha1_base64="SegqlpulEgyM2VoPgNqflVpx7c=">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</latexit>

(X0,Y0)

<latexit sha1_base64="wjZYpK0BOtTzJZ8Xzf0vGZhE940=">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</latexit>

(X1,Y1)

<latexit sha1_base64="kTh2GI3UhKdj6mPLJuTVXgaKejw=">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</latexit>

X00

<latexit sha1_base64="QcqOe/v2Xtzse7KBY3utpdj5NPk=">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</latexit>

X01

<latexit sha1_base64="gTU0lQVZdYJ0Fu0KQA/pM1ADj0=">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</latexit>

X10

<latexit sha1_base64="18r8ib2Zi/sDAFriNAnkt3pcyek=">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</latexit>

X11

<latexit sha1_base64="+z0aEUJo1l7jgE0lW10jdPAx7UE=">ADyXicpVJNb9NAEN3GfJTwlZYjlxXbShyiyOtUgkulCi5IXIpE2kixFa3X63jV9drdnYCDlRP/A47wl/g3rBMLJSn0wkj2Pr+ZN7N+mrhU0oLv/9reHfu3ru/6D78NHjJ097B4cXtpgbLka8UIUZx8wKJbUYgQlxqURLI+VuIyv3jb5y0/CWFnoj7AoRZSzmZap5AwcNe0dhrqQOhEa8NF4WlO6PJr2iD/wV4FvAtoCgto4nx50vodJwe568IVs3ZCgxKimhmQXIlN5xbUTJ+xWZi4qBmubBRvbr8Eh87JsFpYdzjbrFiNxU1y61d5LGrzBlkdjfXkH/NUwJebUzH9LXUS1OQeh+Xp8OlcYCtz4gxNpBAe1cIBxI90fYJ4xwzg4F7f6z0SRCzAL1x9EBZ9lAtkpHvp+Cd3jzZGls6RfslKYdQ0NpF5/Z0LOMjilVOpos9/+DWJasXchZxn/da6fmtTczYi28cOQZvj2JKRXVzE9EGuaE1mGjitOa0GXLNuO0ZgmhJKjDWMykrmNXZWS1xISGIQlCoZM/3LYMGhkZ/ksYjK8RX29VpOTW/XudbLTpOtWmu4u8E1wEQzocB8CMjZm3a59Fz9AK9RBS9QmfoHTpHI8Rhb6hH+in9679irvy7q0s9dqnqGt8L7+Bms/PNY=</latexit>

< l a t e x i t s h a 1 _ b a s e 6 4 = " s N I v 1 n b s Z i h x J O r I d
  • T
l 1 J
  • W
n 8 = " > A A A D x n i c p V J N b 9 N A E N 3 G f J T w R S O X F Z s K 3 G I I q 9 T
  • B
w q V e L S Y 5 F I W 6 m 2
  • v
V 6 H a + 6 X r u 7 Y 5 r I i s T / 4 M A F f h T / h n V i U B J
  • L
4 x k 7 / j N v H n r p 4 l L J S 3 4 / s + t j n f v /
  • O
H 2 4 + 6 j 5 8 8 f b b T 2 3 1 + Z
  • v
K c D H i h S r M R c y s U F K L E U h Q 4 q I g u W x E u f x 1 Y e m f v 5 Z G C s L / Q l m p Y h y N t E y l Z y B g 8 a 9 n V A X U i d C A 9 4 L p d 4 b 9 4 g / e O P T 9 2 8 p / j u h A 3 8 R B L V x O t 7 t f A u T g l e 5 G 8 E V s / a S B i V E N T M g u R L z b l h Z U T J + x S b i q W a 5 c J G 9 e L m c 7 z v k A S n h X G P u 8 I C X W X U L L d 2 l s e u M 2 e Q 2 c 1 a A / 6 z 1 i A l 5 N M N f U g P
  • 1
r q s g K h + V I + r R S G A j f m 4 E Q a w U H N X M K 4 k e 4 P M M + Y Y R y c h W v z J 6 L I B Z i Z m w 9 i C j c y g e w I D 3 2 / h O 7 + q m T p L O m X r B R m 2 U M D q Z f f m Z C T D I 4
  • l
T p a n f c / f l 1 G t W L u Q s 6 z f m t d v 7 W p O R u S 7 W O X Q Z a v S z G l
  • n
r a 6 C c i D X N C 6 7 B h x W l N 6 L x F G z m t W U I
  • C
e
  • w
F h O p 6 9 h 1 G T m d Y L D k A S h M k f b J G D Y M b y O G m A z v Y F 8 v 2 e T g T r 5 7 H W w M 6 b q V / r 2 3 + P b k L B j Q 4 S D 4 G J D j w 3 a 5 t 9 F L 9 A q 9 R h S 9 Q 8 f
  • B
J 2 i E e K
  • Q
l / R d / T D O / G V 3 k 3 y 9 b O V s t 5 g d b C + / I L 1 8 8 E g = = < / l a t e x i t >

H2[−2]

<latexit sha1_base64="vb/vCLYHwpFPwlA7p2nx6fxOJ5M=">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</latexit>

< l a t e x i t s h a 1 _ b a s e 6 4 = " s N I v 1 n b s Z i h x J O r I d
  • T
l 1 J
  • W
n 8 = " > A A A D x n i c p V J N b 9 N A E N 3 G f J T w R S O X F Z s K 3 G I I q 9 T
  • B
w q V e L S Y 5 F I W 6 m 2
  • v
V 6 H a + 6 X r u 7 Y 5 r I i s T / 4 M A F f h T / h n V i U B J
  • L
4 x k 7 / j N v H n r p 4 l L J S 3 4 / s + t j n f v /
  • O
H 2 4 + 6 j 5 8 8 f b b T 2 3 1 + Z
  • v
K c D H i h S r M R c y s U F K L E U h Q 4 q I g u W x E u f x 1 Y e m f v 5 Z G C s L / Q l m p Y h y N t E y l Z y B g 8 a 9 n V A X U i d C A 9 4 L p d 4 b 9 4 g / e O P T 9 2 8 p / j u h A 3 8 R B L V x O t 7 t f A u T g l e 5 G 8 E V s / a S B i V E N T M g u R L z b l h Z U T J + x S b i q W a 5 c J G 9 e L m c 7 z v k A S n h X G P u 8 I C X W X U L L d 2 l s e u M 2 e Q 2 c 1 a A / 6 z 1 i A l 5 N M N f U g P
  • 1
r q s g K h + V I + r R S G A j f m 4 E Q a w U H N X M K 4 k e 4 P M M + Y Y R y c h W v z J 6 L I B Z i Z m w 9 i C j c y g e w I D 3 2 / h O 7 + q m T p L O m X r B R m 2 U M D q Z f f m Z C T D I 4
  • l
T p a n f c / f l 1 G t W L u Q s 6 z f m t d v 7 W p O R u S 7 W O X Q Z a v S z G l
  • n
r a 6 C c i D X N C 6 7 B h x W l N 6 L x F G z m t W U I
  • C
e
  • w
F h O p 6 9 h 1 G T m d Y L D k A S h M k f b J G D Y M b y O G m A z v Y F 8 v 2 e T g T r 5 7 H W w M 6 b q V / r 2 3 + P b k L B j Q 4 S D 4 G J D j w 3 a 5 t 9 F L 9 A q 9 R h S 9 Q 8 f
  • B
J 2 i E e K
  • Q
l / R d / T D O / G V 3 k 3 y 9 b O V s t 5 g d b C + / I L 1 8 8 E g = = < / l a t e x i t >

H2[−1]

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(Xn−1,Yn−1)

<latexit sha1_base64="I8PY4qsNi02Pk4j7SQN4RTYGhCI=">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</latexit>

X0...0

|{z}

n

<latexit sha1_base64="sWKhJL31YDgaubtWFSHzMzBy5wA=">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</latexit>

X0...01

| {z }

n

<latexit sha1_base64="QsyLMZJ4t3cMtKuvTizWhaepAM=">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</latexit>

X1...1

|{z}

n

<latexit sha1_base64="4u1uGsTxiwQOc4gRIsACRCLZf8U=">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</latexit>

X1...10

| {z }

n

<latexit sha1_base64="2AznQdQBW7mZjo9PDws4TgANQ=">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</latexit>

<latexit sha1_base64="sNIv1nbsZihxJOrIdoTl1JoWn8=">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</latexit>

HT

<latexit sha1_base64="wBTCTpQkT3vgPTdSYn7C4vRO7Gs=">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</latexit>

< l a t e x i t s h a 1 _ b a s e 6 4 = " s N I v 1 n b s Z i h x J O r I d
  • T
l 1 J
  • W
n 8 = " > A A A D x n i c p V J N b 9 N A E N 3 G f J T w R S O X F Z s K 3 G I I q 9 T
  • B
w q V e L S Y 5 F I W 6 m 2
  • v
V 6 H a + 6 X r u 7 Y 5 r I i s T / 4 M A F f h T / h n V i U B J
  • L
4 x k 7 / j N v H n r p 4 l L J S 3 4 / s + t j n f v /
  • O
H 2 4 + 6 j 5 8 8 f b b T 2 3 1 + Z
  • v
K c D H i h S r M R c y s U F K L E U h Q 4 q I g u W x E u f x 1 Y e m f v 5 Z G C s L / Q l m p Y h y N t E y l Z y B g 8 a 9 n V A X U i d C A 9 4 L p d 4 b 9 4 g / e O P T 9 2 8 p / j u h A 3 8 R B L V x O t 7 t f A u T g l e 5 G 8 E V s / a S B i V E N T M g u R L z b l h Z U T J + x S b i q W a 5 c J G 9 e L m c 7 z v k A S n h X G P u 8 I C X W X U L L d 2 l s e u M 2 e Q 2 c 1 a A / 6 z 1 i A l 5 N M N f U g P
  • 1
r q s g K h + V I + r R S G A j f m 4 E Q a w U H N X M K 4 k e 4 P M M + Y Y R y c h W v z J 6 L I B Z i Z m w 9 i C j c y g e w I D 3 2 / h O 7 + q m T p L O m X r B R m 2 U M D q Z f f m Z C T D I 4
  • l
T p a n f c / f l 1 G t W L u Q s 6 z f m t d v 7 W p O R u S 7 W O X Q Z a v S z G l
  • n
r a 6 C c i D X N C 6 7 B h x W l N 6 L x F G z m t W U I
  • C
e
  • w
F h O p 6 9 h 1 G T m d Y L D k A S h M k f b J G D Y M b y O G m A z v Y F 8 v 2 e T g T r 5 7 H W w M 6 b q V / r 2 3 + P b k L B j Q 4 S D 4 G J D j w 3 a 5 t 9 F L 9 A q 9 R h S 9 Q 8 f
  • B
J 2 i E e K
  • Q
l / R d / T D O / G V 3 k 3 y 9 b O V s t 5 g d b C + / I L 1 8 8 E g = = < / l a t e x i t >

HT [−n]

<latexit sha1_base64="IfEXfUnzB0Ik7XMSHDc/n6esFTE=">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</latexit>

HT [−1]

<latexit sha1_base64="sLcdSVfKZudSNC1pDHJs5ptfTHg=">AD3nicpVLPb9MwFPYafozyq4MjFwtvgkOp4nTAOEyaxGXHIa3bpCYqjuM01hzH2C60inrlNnHlxh/BFf4Q/hucJkxtYbtgKcmX73vedPL1aCG+v7vzZa3o2bt25v3mnfvXf/wcPO1qMTU0w0ZQNaiEKfxcQwSUbWG4FO1OakTwW7DQ+f1vFTz8ybXghj+1MsSgnY8lTol1KjzLJQFlwmTFm6HObEZJQIejspLfDwfvsDR9qiD/N5LH795heHfAPf8xUGgOUejrda3MCnoJHelqSDGDHGgbFQSbTkVbN4OJ4YpQs/JmA0dlCRnJioXN5rDHckMC20e9xoC3ZUZLcmFkeu8xqULMeq8h/xipG2Xy61t+me1HJpZpYJmndPp0IaAtYmQYTrhm1YuYAoZq7G0CaEU2odau1B+zImdWz1x9y6b2E09stg/7vq9se2e5pXKWdBVRTNc5OCy/s8YH2d2H2Muo+V6/+PXMCoFcQM5z7qNd3GpupbiUwXOmSzfLUVESIqp1X/hKVhjnC9GnFaIjxv2KqdlCRBGAVlGLMxl2XsjSfziHCYiCkMnkluV2UqG+lcJQ4j616g/1Gq0e63evXbXirTdSv/ZW3g1OAl6uN8L3gXoYK9Z7k3wBDwFzwEGr8EBOARHYAouADfwQ/w03vfYuvC91amuj0TwGK8f7+hus8kWE</latexit>

< l a t e x i t s h a 1 _ b a s e 6 4 = " s N I v 1 n b s Z i h x J O r I d
  • T
l 1 J
  • W
n 8 = " > A A A D x n i c p V J N b 9 N A E N 3 G f J T w R S O X F Z s K 3 G I I q 9 T
  • B
w q V e L S Y 5 F I W 6 m 2
  • v
V 6 H a + 6 X r u 7 Y 5 r I i s T / 4 M A F f h T / h n V i U B J
  • L
4 x k 7 / j N v H n r p 4 l L J S 3 4 / s + t j n f v /
  • O
H 2 4 + 6 j 5 8 8 f b b T 2 3 1 + Z
  • v
K c D H i h S r M R c y s U F K L E U h Q 4 q I g u W x E u f x 1 Y e m f v 5 Z G C s L / Q l m p Y h y N t E y l Z y B g 8 a 9 n V A X U i d C A 9 4 L p d 4 b 9 4 g / e O P T 9 2 8 p / j u h A 3 8 R B L V x O t 7 t f A u T g l e 5 G 8 E V s / a S B i V E N T M g u R L z b l h Z U T J + x S b i q W a 5 c J G 9 e L m c 7 z v k A S n h X G P u 8 I C X W X U L L d 2 l s e u M 2 e Q 2 c 1 a A / 6 z 1 i A l 5 N M N f U g P
  • 1
r q s g K h + V I + r R S G A j f m 4 E Q a w U H N X M K 4 k e 4 P M M + Y Y R y c h W v z J 6 L I B Z i Z m w 9 i C j c y g e w I D 3 2 / h O 7 + q m T p L O m X r B R m 2 U M D q Z f f m Z C T D I 4
  • l
T p a n f c / f l 1 G t W L u Q s 6 z f m t d v 7 W p O R u S 7 W O X Q Z a v S z G l
  • n
r a 6 C c i D X N C 6 7 B h x W l N 6 L x F G z m t W U I
  • C
e
  • w
F h O p 6 9 h 1 G T m d Y L D k A S h M k f b J G D Y M b y O G m A z v Y F 8 v 2 e T g T r 5 7 H W w M 6 b q V / r 2 3 + P b k L B j Q 4 S D 4 G J D j w 3 a 5 t 9 F L 9 A q 9 R h S 9 Q 8 f
  • B
J 2 i E e K
  • Q
l / R d / T D O / G V 3 k 3 y 9 b O V s t 5 g d b C + / I L 1 8 8 E g = = < / l a t e x i t >

HT [−n+1]

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H2

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H2[−1]

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·········

<latexit sha1_base64="uKGFnvhRiji6T7UKqGRTjYJwbaY=">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</latexit>

·········

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···

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slide-18
SLIDE 18

A= k-algebra associated to

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1

a

/2

b

/3 with ba = 0.

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T = 2

3 ⊕ 1 2 ⊕ 1

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D0 = D(A)0 = h1, 2, 3, 1

2, 2 3,

  • 2

3 ! 1 2i

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T 0 = h1, 2, 3[2], 1

2, 2 3, 2 3 !

  • 1

2i

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D≥0

2

=

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D0

1

= h1, 2, 3[1], 1

2, 2 3, 2 3 !

  • 1

2i

<latexit sha1_base64="U9RpSkTHe8b7KHkMhWcuao35shI=">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</latexit>

X0 = H0 ∩H1 =

  • 1, 2, 1

2, 2 3

, Y0 = H0 ∩H1[−1] = {3}

<latexit sha1_base64="kXYMTceHm6IrL32SgfdEgY04v8=">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</latexit>

X1 = H1∩H2 = ⇢

1, 1 2, 2 3, 2 3 →

  • 1

2

  • , Y1 = H1∩H2[−1] = {3[1]}
<latexit sha1_base64="Js2lU20tYvReKnEjmwuWTG7aSgM=">AFcXicpVLdatswFHa9Zuyv3a7GoMiagqDZSWyC9tNoGwercU1p8RmSDLciIqy51siUYX+459hrdo+wZdrkXmBSnJUl/KPSA7aNP5ztH+vxFuRQa2u0/K+691cb9B2sPm48eP3n6bH3j+ZHORgXjhyTWXESUc2lUPwQBEh+khecpHkx9HpR7t/J0XWmTqC0xyHqZ0oEQiGAUD9TdWfpOUwpBRWZ5UfYw6GK9b9aE0Xwe8G2B5AmQsolMlCTiA6FKnVIpTV0hxhUmXMULSNW6pD4V9Re29gnJLh7PYFsykCIZFYZDoY7kpJDdftDIlKIwRCIuRlBpt2FRl9vI2LvLQ5vIWTQw+ENo/vrXnunPQ10OcGzxNvbPDv4+3PzrNvfcH+ROGOjlCtgkmrdw34OYUkLEzyqklGmueUndIB75lU0ZTrsJyarELbBolRkhXmUYCm6DyjpKnWkzQylfauenPglfuWSHdLw0H5L3YSlUPgKuWD0+GUkEGbI+RrEoOAM5MQlhTA3QGxIC8rA/NOF/gOepRyKiekPfAw/RAzDgqCdg7N7fmRuZGkldOcF3UN9oWq10NuJe9gLFQ43+8uevXCUlJzIKNZayZdayaT/VqSbiGTwTBdHGXMEJZjOz/mCUk9XE7dFSWlh6sZascpRWMPe/65t6KZhZBnTO3Vrj7HFmlgaV5wHZEgL7iB/a1me7s38s1rd6lJ01gaLxv4cnLk7+Bgxz8w3v7g1LHmvHK2nNcOdt45e86+03UOHeYi95P72e2u/mu8bKDGVl3qrsw4L5yFaLz5DzYQ3jk=</latexit>

A-Mod =

  • 1, 2, 3, 1

2, 2 3

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D(A) =

  • 1, 2, 3, 1

2, 2 3, 2 3 → 1 2

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slide-19
SLIDE 19

<latexit sha1_base64="sNIv1nbsZihxJOrIdoTl1JoWn8=">ADxnicpVJNb9NAEN3GfJTw0RSOXFZsK3GIq9ToBwqVeLSY5FIW6m2ovV6Ha+6Xru7Y5rIisT/4MAFfhT/hnViUBJoL4xk7/jNvHnrp4lLJS34/s+tjnfv/oOH24+6j58fbT231+ZovKcDHihSrMRcysUFKLEUhQ4qI0guWxEufx1Yemfv5ZGCsL/QlmpYhyNtEylZyBg8a9nVAXUidCA94Lpd4b94g/eOPT928p/juhA38RBLVxOt7tfAuTgle5G8EVs/aSBiVENTMguRLzblhZUTJ+xSbi0qWa5cJG9eLmc7zvkASnhXGPu8ICXWXULd2lseuM2eQ2c1aA/6z1iAl5NMNfUgPo1rqsgKh+VI+rRSGAjfm4EQawUHNXMK4ke4PM+YRychWvzJ6LIBZiZmw9iCjcygewID32/hO7+qmTpLOmXrBRm2UMDqZfmZCTDI4olTpanfc/fl1GtWLuQs6zfmtdv7WpORuS7WOXQZavSzGlonra6CciDXNC67BhxWlN6LxFGzmtWUIoCeowFhOp69h1GTmdY0LDkASh0MkfbJ0GDY0MbyOGmAzvYF8v2eTgTr57HWwM6bqV/r23+PbkLBjQ4SD4GJDjw3a5t9FL9Aq9RhS9Q8foBJ2iEeKoQl/Rd/TDO/G0V3k3y9bOVst5gdbC+/IL1808Eg=</latexit>

H1

<latexit sha1_base64="et/13RjTKaTQjinVDhTd860EAdo=">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</latexit>

<latexit sha1_base64="sNIv1nbsZihxJOrIdoTl1JoWn8=">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</latexit> H1[−1] <latexit sha1_base64="SegqlpulEgyM2VoPgNqflVpx7c=">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</latexit>

< l a t e x i t s h a 1 _ b a s e 6 4 = " s N I v 1 n b s Z i h x J O r I d
  • T
l 1 J
  • W
n 8 = " > A A A D x n i c p V J N b 9 N A E N 3 G f J T w R S O X F Z s K 3 G I I q 9 T
  • B
w q V e L S Y 5 F I W 6 m 2
  • v
V 6 H a + 6 X r u 7 Y 5 r I i s T / 4 M A F f h T / h n V i U B J
  • L
4 x k 7 / j N v H n r p 4 l L J S 3 4 / s + t j n f v /
  • O
H 2 4 + 6 j 5 8 8 f b b T 2 3 1 + Z
  • v
K c D H i h S r M R c y s U F K L E U h Q 4 q I g u W x E u f x 1 Y e m f v 5 Z G C s L / Q l m p Y h y N t E y l Z y B g 8 a 9 n V A X U i d C A 9 4 L p d 4 b 9 4 g / e O P T 9 2 8 p / j u h A 3 8 R B L V x O t 7 t f A u T g l e 5 G 8 E V s / a S B i V E N T M g u R L z b l h Z U T J + x S b i q W a 5 c J G 9 e L m c 7 z v k A S n h X G P u 8 I C X W X U L L d 2 l s e u M 2 e Q 2 c 1 a A / 6 z 1 i A l 5 N M N f U g P
  • 1
r q s g K h + V I + r R S G A j f m 4 E Q a w U H N X M K 4 k e 4 P M M + Y Y R y c h W v z J 6 L I B Z i Z m w 9 i C j c y g e w I D 3 2 / h O 7 + q m T p L O m X r B R m 2 U M D q Z f f m Z C T D I 4
  • l
T p a n f c / f l 1 G t W L u Q s 6 z f m t d v 7 W p O R u S 7 W O X Q Z a v S z G l
  • n
r a 6 C c i D X N C 6 7 B h x W l N 6 L x F G z m t W U I
  • C
e
  • w
F h O p 6 9 h 1 G T m d Y L D k A S h M k f b J G D Y M b y O G m A z v Y F 8 v 2 e T g T r 5 7 H W w M 6 b q V / r 2 3 + P b k L B j Q 4 S D 4 G J D j w 3 a 5 t 9 F L 9 A q 9 R h S 9 Q 8 f
  • B
J 2 i E e K
  • Q
l / R d / T D O / G V 3 k 3 y 9 b O V s t 5 g d b C + / I L 1 8 8 E g = = < / l a t e x i t >

H2[−2]

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< l a t e x i t s h a 1 _ b a s e 6 4 = " s N I v 1 n b s Z i h x J O r I d
  • T
l 1 J
  • W
n 8 = " > A A A D x n i c p V J N b 9 N A E N 3 G f J T w R S O X F Z s K 3 G I I q 9 T
  • B
w q V e L S Y 5 F I W 6 m 2
  • v
V 6 H a + 6 X r u 7 Y 5 r I i s T / 4 M A F f h T / h n V i U B J
  • L
4 x k 7 / j N v H n r p 4 l L J S 3 4 / s + t j n f v /
  • O
H 2 4 + 6 j 5 8 8 f b b T 2 3 1 + Z
  • v
K c D H i h S r M R c y s U F K L E U h Q 4 q I g u W x E u f x 1 Y e m f v 5 Z G C s L / Q l m p Y h y N t E y l Z y B g 8 a 9 n V A X U i d C A 9 4 L p d 4 b 9 4 g / e O P T 9 2 8 p / j u h A 3 8 R B L V x O t 7 t f A u T g l e 5 G 8 E V s / a S B i V E N T M g u R L z b l h Z U T J + x S b i q W a 5 c J G 9 e L m c 7 z v k A S n h X G P u 8 I C X W X U L L d 2 l s e u M 2 e Q 2 c 1 a A / 6 z 1 i A l 5 N M N f U g P
  • 1
r q s g K h + V I + r R S G A j f m 4 E Q a w U H N X M K 4 k e 4 P M M + Y Y R y c h W v z J 6 L I B Z i Z m w 9 i C j c y g e w I D 3 2 / h O 7 + q m T p L O m X r B R m 2 U M D q Z f f m Z C T D I 4
  • l
T p a n f c / f l 1 G t W L u Q s 6 z f m t d v 7 W p O R u S 7 W O X Q Z a v S z G l
  • n
r a 6 C c i D X N C 6 7 B h x W l N 6 L x F G z m t W U I
  • C
e
  • w
F h O p 6 9 h 1 G T m d Y L D k A S h M k f b J G D Y M b y O G m A z v Y F 8 v 2 e T g T r 5 7 H W w M 6 b q V / r 2 3 + P b k L B j Q 4 S D 4 G J D j w 3 a 5 t 9 F L 9 A q 9 R h S 9 Q 8 f
  • B
J 2 i E e K
  • Q
l / R d / T D O / G V 3 k 3 y 9 b O V s t 5 g d b C + / I L 1 8 8 E g = = < / l a t e x i t >

H2[−1]

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H2

<latexit sha1_base64="zXuBQsFWuEGZdXeIypmzS6Zlg=">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</latexit>

<latexit sha1_base64="sNIv1nbsZihxJOrIdoTl1JoWn8=">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</latexit>

H2[−1]

<latexit sha1_base64="1s/4J1Y/y30f3g9WtNiE2hwxWY=">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</latexit>

<latexit sha1_base64="sNIv1nbsZihxJOrIdoTl1JoWn8=">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</latexit>

∈ A-Mod

<latexit sha1_base64="IVlFPJPeYknoK84nBjQCN4ptchA=">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</latexit>

2

<latexit sha1_base64="UiKOAgvnYAneHl2Lh/48H/7T5NI=">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</latexit>

2

<latexit sha1_base64="UiKOAgvnYAneHl2Lh/48H/7T5NI=">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</latexit> <latexit sha1_base64="+31ynBYsMd3DFqzZxnRVdEPnsI=">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</latexit>

2 3

<latexit sha1_base64="yL7drEuXohKkGYoSQNp/06mRLuw=">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</latexit>

3[1]

<latexit sha1_base64="x5oR687qR84b2/ClzgOCzkRuQA=">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</latexit> <latexit sha1_base64="+31ynBYsMd3DFqzZxnRVdEPnsI=">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</latexit> <latexit sha1_base64="+31ynBYsMd3DFqzZxnRVdEPnsI=">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</latexit>

(X0,Y0)

<latexit sha1_base64="wjZYpK0BOtTzJZ8Xzf0vGZhE940=">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</latexit>

{1, 1

2, 2 3, 2 3 →

  • 1

2}

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= ⇣ {1, 2, 1

2, 2 3},

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(X1,Y1)

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{3[1]} ⌘

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= ⇣ {1, 1

2, 2 3, 2 3 →

  • 1

2},

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