id	sid	tid	token	lemma	pos
pc289g57c3s	1	1	in	in	ADP
pc289g57c3s	1	2	this	this	DET
pc289g57c3s	1	3	dissertation	dissertation	NOUN
pc289g57c3s	1	4	,	,	PUNCT
pc289g57c3s	1	5	we	we	PRON
pc289g57c3s	1	6	study	study	VERB
pc289g57c3s	1	7	several	several	ADJ
pc289g57c3s	1	8	problems	problem	NOUN
pc289g57c3s	1	9	in	in	ADP
pc289g57c3s	1	10	commutative	commutative	ADJ
pc289g57c3s	1	11	algebra	algebra	NOUN
pc289g57c3s	1	12	.	.	PUNCT
pc289g57c3s	2	1	the	the	DET
pc289g57c3s	2	2	first	first	ADJ
pc289g57c3s	2	3	problem	problem	NOUN
pc289g57c3s	2	4	we	we	PRON
pc289g57c3s	2	5	study	study	VERB
pc289g57c3s	2	6	is	be	AUX
pc289g57c3s	2	7	the	the	DET
pc289g57c3s	2	8	regularity	regularity	NOUN
pc289g57c3s	2	9	of	of	ADP
pc289g57c3s	2	10	tor	tor	X
pc289g57c3s	2	11	for	for	ADP
pc289g57c3s	2	12	weakly	weakly	ADJ
pc289g57c3s	2	13	stable	stable	ADJ
pc289g57c3s	2	14	ideals	ideal	NOUN
pc289g57c3s	2	15	.	.	PUNCT
pc289g57c3s	3	1	we	we	PRON
pc289g57c3s	3	2	prove	prove	VERB
pc289g57c3s	3	3	that	that	SCONJ
pc289g57c3s	3	4	if	if	SCONJ
pc289g57c3s	3	5	i	i	PRON
pc289g57c3s	3	6	and	and	CCONJ
pc289g57c3s	3	7	j	j	NOUN
pc289g57c3s	3	8	are	be	AUX
pc289g57c3s	3	9	weakly	weakly	ADJ
pc289g57c3s	3	10	stable	stable	ADJ
pc289g57c3s	3	11	ideals	ideal	NOUN
pc289g57c3s	3	12	in	in	ADP
pc289g57c3s	3	13	a	a	DET
pc289g57c3s	3	14	polynomial	polynomial	ADJ
pc289g57c3s	3	15	ring	ring	NOUN
pc289g57c3s	3	16	r	r	NOUN
pc289g57c3s	3	17	=	=	NOUN
pc289g57c3s	3	18	k[x_1,	k[x_1,	SPACE
pc289g57c3s	3	19	...	...	PUNCT
pc289g57c3s	3	20	,x_n	,x_n	PUNCT
pc289g57c3s	3	21	]	]	PUNCT
pc289g57c3s	3	22	over	over	ADP
pc289g57c3s	3	23	a	a	DET
pc289g57c3s	3	24	field	field	NOUN
pc289g57c3s	3	25	k	k	PROPN
pc289g57c3s	3	26	,	,	PUNCT
pc289g57c3s	3	27	then	then	ADV
pc289g57c3s	3	28	the	the	DET
pc289g57c3s	3	29	regularity	regularity	NOUN
pc289g57c3s	3	30	of	of	ADP
pc289g57c3s	3	31	tor^r_i(r	tor^r_i(r	SPACE
pc289g57c3s	3	32	/	/	SYM
pc289g57c3s	3	33	i	i	PROPN
pc289g57c3s	3	34	,	,	PUNCT
pc289g57c3s	3	35	r	r	NOUN
pc289g57c3s	3	36	/	/	SYM
pc289g57c3s	3	37	j	j	NOUN
pc289g57c3s	3	38	)	)	PUNCT
pc289g57c3s	3	39	is	be	AUX
pc289g57c3s	3	40	bounded	bound	VERB
pc289g57c3s	3	41	by	by	ADP
pc289g57c3s	3	42	reg^r	reg^r	PROPN
pc289g57c3s	3	43	r	r	PROPN
pc289g57c3s	3	44	/	/	SYM
pc289g57c3s	3	45	i	i	NOUN
pc289g57c3s	3	46	+	+	CCONJ
pc289g57c3s	3	47	reg_r	reg_r	X
pc289g57c3s	3	48	r	r	NOUN
pc289g57c3s	3	49	/	/	SYM
pc289g57c3s	3	50	j	j	X
pc289g57c3s	3	51	+	+	X
pc289g57c3s	3	52	i.	i.	NOUN
pc289g57c3s	3	53	we	we	PRON
pc289g57c3s	3	54	also	also	ADV
pc289g57c3s	3	55	give	give	VERB
pc289g57c3s	3	56	a	a	DET
pc289g57c3s	3	57	bound	bind	VERB
pc289g57c3s	3	58	for	for	ADP
pc289g57c3s	3	59	the	the	DET
pc289g57c3s	3	60	regularity	regularity	NOUN
pc289g57c3s	3	61	of	of	ADP
pc289g57c3s	3	62	ext^i_r(r	ext^i_r(r	NOUN
pc289g57c3s	3	63	/	/	SYM
pc289g57c3s	3	64	i	i	PROPN
pc289g57c3s	3	65	,	,	PUNCT
pc289g57c3s	3	66	r	r	NOUN
pc289g57c3s	3	67	)	)	PUNCT
pc289g57c3s	3	68	for	for	ADP
pc289g57c3s	3	69	i	i	PRON
pc289g57c3s	3	70	a	a	DET
pc289g57c3s	3	71	weakly	weakly	ADJ
pc289g57c3s	3	72	stable	stable	ADJ
pc289g57c3s	3	73	ideal	ideal	NOUN
pc289g57c3s	3	74	.	.	PUNCT
pc289g57c3s	4	1	for	for	ADP
pc289g57c3s	4	2	the	the	DET
pc289g57c3s	4	3	second	second	ADJ
pc289g57c3s	4	4	problem	problem	NOUN
pc289g57c3s	4	5	,	,	PUNCT
pc289g57c3s	4	6	we	we	PRON
pc289g57c3s	4	7	define	define	VERB
pc289g57c3s	4	8	an	an	DET
pc289g57c3s	4	9	operation	operation	NOUN
pc289g57c3s	4	10	on	on	ADP
pc289g57c3s	4	11	monomial	monomial	ADJ
pc289g57c3s	4	12	ideals	ideal	NOUN
pc289g57c3s	4	13	known	know	VERB
pc289g57c3s	4	14	as	as	ADP
pc289g57c3s	4	15	the	the	DET
pc289g57c3s	4	16	lcm	lcm	NOUN
pc289g57c3s	4	17	-	-	PUNCT
pc289g57c3s	4	18	dual	dual	ADJ
pc289g57c3s	4	19	.	.	PUNCT
pc289g57c3s	5	1	we	we	PRON
pc289g57c3s	5	2	study	study	VERB
pc289g57c3s	5	3	the	the	DET
pc289g57c3s	5	4	properties	property	NOUN
pc289g57c3s	5	5	of	of	ADP
pc289g57c3s	5	6	the	the	DET
pc289g57c3s	5	7	lcm	lcm	NOUN
pc289g57c3s	5	8	-	-	NOUN
pc289g57c3s	5	9	dual	dual	ADJ
pc289g57c3s	5	10	for	for	ADP
pc289g57c3s	5	11	several	several	ADJ
pc289g57c3s	5	12	classes	class	NOUN
pc289g57c3s	5	13	of	of	ADP
pc289g57c3s	5	14	monomial	monomial	ADJ
pc289g57c3s	5	15	ideals	ideal	NOUN
pc289g57c3s	5	16	.	.	PUNCT
pc289g57c3s	6	1	the	the	DET
pc289g57c3s	6	2	first	first	ADJ
pc289g57c3s	6	3	class	class	NOUN
pc289g57c3s	6	4	of	of	ADP
pc289g57c3s	6	5	ideals	ideal	NOUN
pc289g57c3s	6	6	that	that	PRON
pc289g57c3s	6	7	arises	arise	VERB
pc289g57c3s	6	8	from	from	ADP
pc289g57c3s	6	9	graph	graph	NOUN
pc289g57c3s	6	10	theory	theory	NOUN
pc289g57c3s	6	11	are	be	AUX
pc289g57c3s	6	12	ferrers	ferrer	NOUN
pc289g57c3s	6	13	ideals	ideal	NOUN
pc289g57c3s	6	14	,	,	PUNCT
pc289g57c3s	6	15	which	which	PRON
pc289g57c3s	6	16	are	be	AUX
pc289g57c3s	6	17	edge	edge	NOUN
pc289g57c3s	6	18	ideals	ideal	NOUN
pc289g57c3s	6	19	of	of	ADP
pc289g57c3s	6	20	a	a	DET
pc289g57c3s	6	21	ferrers	ferrer	NOUN
pc289g57c3s	6	22	graph	graph	NOUN
pc289g57c3s	6	23	.	.	PUNCT
pc289g57c3s	7	1	we	we	PRON
pc289g57c3s	7	2	show	show	VERB
pc289g57c3s	7	3	that	that	SCONJ
pc289g57c3s	7	4	the	the	DET
pc289g57c3s	7	5	lcm	lcm	NOUN
pc289g57c3s	7	6	-	-	NOUN
pc289g57c3s	7	7	dual	dual	NOUN
pc289g57c3s	7	8	of	of	ADP
pc289g57c3s	7	9	a	a	DET
pc289g57c3s	7	10	ferrers	ferrer	NOUN
pc289g57c3s	7	11	ideal	ideal	ADJ
pc289g57c3s	7	12	is	be	AUX
pc289g57c3s	7	13	the	the	DET
pc289g57c3s	7	14	alexander	alexander	NOUN
pc289g57c3s	7	15	dual	dual	NOUN
pc289g57c3s	7	16	of	of	ADP
pc289g57c3s	7	17	the	the	DET
pc289g57c3s	7	18	edge	edge	NOUN
pc289g57c3s	7	19	ideal	ideal	NOUN
pc289g57c3s	7	20	of	of	ADP
pc289g57c3s	7	21	the	the	DET
pc289g57c3s	7	22	complement	complement	NOUN
pc289g57c3s	7	23	of	of	ADP
pc289g57c3s	7	24	the	the	DET
pc289g57c3s	7	25	ferrers	ferrer	NOUN
pc289g57c3s	7	26	graph	graph	NOUN
pc289g57c3s	7	27	.	.	PUNCT
pc289g57c3s	8	1	the	the	DET
pc289g57c3s	8	2	second	second	ADJ
pc289g57c3s	8	3	class	class	NOUN
pc289g57c3s	8	4	of	of	ADP
pc289g57c3s	8	5	ideals	ideal	NOUN
pc289g57c3s	8	6	we	we	PRON
pc289g57c3s	8	7	consider	consider	VERB
pc289g57c3s	8	8	are	be	AUX
pc289g57c3s	8	9	specializations	specialization	NOUN
pc289g57c3s	8	10	of	of	ADP
pc289g57c3s	8	11	ferrers	ferrer	NOUN
pc289g57c3s	8	12	ideals	ideal	NOUN
pc289g57c3s	8	13	,	,	PUNCT
pc289g57c3s	8	14	strongly	strongly	ADV
pc289g57c3s	8	15	stable	stable	ADJ
pc289g57c3s	8	16	ideals	ideal	NOUN
pc289g57c3s	8	17	of	of	ADP
pc289g57c3s	8	18	degree	degree	NOUN
pc289g57c3s	8	19	two	two	NUM
pc289g57c3s	8	20	.	.	PUNCT
pc289g57c3s	9	1	we	we	PRON
pc289g57c3s	9	2	find	find	VERB
pc289g57c3s	9	3	a	a	DET
pc289g57c3s	9	4	cellular	cellular	ADJ
pc289g57c3s	9	5	complex	complex	NOUN
pc289g57c3s	9	6	which	which	PRON
pc289g57c3s	9	7	supports	support	VERB
pc289g57c3s	9	8	the	the	DET
pc289g57c3s	9	9	minimal	minimal	ADJ
pc289g57c3s	9	10	free	free	ADJ
pc289g57c3s	9	11	resolution	resolution	NOUN
pc289g57c3s	9	12	for	for	ADP
pc289g57c3s	9	13	the	the	DET
pc289g57c3s	9	14	lcm	lcm	NOUN
pc289g57c3s	9	15	-	-	NOUN
pc289g57c3s	9	16	duals	dual	NOUN
pc289g57c3s	9	17	of	of	ADP
pc289g57c3s	9	18	these	these	DET
pc289g57c3s	9	19	strongly	strongly	ADV
pc289g57c3s	9	20	stable	stable	ADJ
pc289g57c3s	9	21	ideals	ideal	NOUN
pc289g57c3s	9	22	.	.	PUNCT
pc289g57c3s	10	1	we	we	PRON
pc289g57c3s	10	2	describe	describe	VERB
pc289g57c3s	10	3	minimal	minimal	ADJ
pc289g57c3s	10	4	free	free	ADJ
pc289g57c3s	10	5	resolutions	resolution	NOUN
pc289g57c3s	10	6	of	of	ADP
pc289g57c3s	10	7	lcm	lcm	NOUN
pc289g57c3s	10	8	-	-	NOUN
pc289g57c3s	10	9	duals	dual	NOUN
pc289g57c3s	10	10	of	of	ADP
pc289g57c3s	10	11	strongly	strongly	ADV
pc289g57c3s	10	12	stable	stable	ADJ
pc289g57c3s	10	13	ideals	ideal	NOUN
pc289g57c3s	10	14	generated	generate	VERB
pc289g57c3s	10	15	in	in	ADP
pc289g57c3s	10	16	degree	degree	NOUN
pc289g57c3s	10	17	two	two	NUM
pc289g57c3s	10	18	.	.	PUNCT
pc289g57c3s	11	1	we	we	PRON
pc289g57c3s	11	2	describe	describe	VERB
pc289g57c3s	11	3	the	the	DET
pc289g57c3s	11	4	special	special	ADJ
pc289g57c3s	11	5	fiber	fiber	NOUN
pc289g57c3s	11	6	ring	ring	NOUN
pc289g57c3s	11	7	in	in	ADP
pc289g57c3s	11	8	this	this	DET
pc289g57c3s	11	9	case	case	NOUN
pc289g57c3s	11	10	.	.	PUNCT
pc289g57c3s	12	1	using	use	VERB
pc289g57c3s	12	2	the	the	DET
pc289g57c3s	12	3	same	same	ADJ
pc289g57c3s	12	4	technique	technique	NOUN
pc289g57c3s	12	5	,	,	PUNCT
pc289g57c3s	12	6	we	we	PRON
pc289g57c3s	12	7	consider	consider	VERB
pc289g57c3s	12	8	a	a	DET
pc289g57c3s	12	9	larger	large	ADJ
pc289g57c3s	12	10	class	class	NOUN
pc289g57c3s	12	11	of	of	ADP
pc289g57c3s	12	12	ideals	ideal	NOUN
pc289g57c3s	12	13	generated	generate	VERB
pc289g57c3s	12	14	in	in	ADP
pc289g57c3s	12	15	degree	degree	NOUN
pc289g57c3s	12	16	two	two	NUM
pc289g57c3s	12	17	,	,	PUNCT
pc289g57c3s	12	18	and	and	CCONJ
pc289g57c3s	12	19	find	find	VERB
pc289g57c3s	12	20	their	their	PRON
pc289g57c3s	12	21	minimal	minimal	ADJ
pc289g57c3s	12	22	free	free	ADJ
pc289g57c3s	12	23	resolutions	resolution	NOUN
pc289g57c3s	12	24	.	.	PUNCT
pc289g57c3s	13	1	we	we	PRON
pc289g57c3s	13	2	also	also	ADV
pc289g57c3s	13	3	show	show	VERB
pc289g57c3s	13	4	that	that	SCONJ
pc289g57c3s	13	5	when	when	SCONJ
pc289g57c3s	13	6	the	the	DET
pc289g57c3s	13	7	height	height	NOUN
pc289g57c3s	13	8	of	of	ADP
pc289g57c3s	13	9	a	a	DET
pc289g57c3s	13	10	monomial	monomial	ADJ
pc289g57c3s	13	11	ideal	ideal	NOUN
pc289g57c3s	13	12	i	i	PRON
pc289g57c3s	13	13	is	be	AUX
pc289g57c3s	13	14	at	at	ADP
pc289g57c3s	13	15	least	least	ADJ
pc289g57c3s	13	16	2	2	NUM
pc289g57c3s	13	17	,	,	PUNCT
pc289g57c3s	13	18	the	the	DET
pc289g57c3s	13	19	special	special	ADJ
pc289g57c3s	13	20	fiber	fiber	NOUN
pc289g57c3s	13	21	ring	ring	NOUN
pc289g57c3s	13	22	of	of	ADP
pc289g57c3s	13	23	i	i	PRON
pc289g57c3s	13	24	is	be	AUX
pc289g57c3s	13	25	isomorphic	isomorphic	ADJ
pc289g57c3s	13	26	to	to	ADP
pc289g57c3s	13	27	the	the	DET
pc289g57c3s	13	28	special	special	ADJ
pc289g57c3s	13	29	fiber	fiber	NOUN
pc289g57c3s	13	30	ring	ring	NOUN
pc289g57c3s	13	31	of	of	ADP
pc289g57c3s	13	32	the	the	DET
pc289g57c3s	13	33	lcm	lcm	NOUN
pc289g57c3s	13	34	-	-	NOUN
pc289g57c3s	13	35	dual	dual	NOUN
pc289g57c3s	13	36	of	of	ADP
pc289g57c3s	13	37	i.	i.	PROPN
pc289g57c3s	13	38	we	we	PRON
pc289g57c3s	13	39	use	use	VERB
pc289g57c3s	13	40	this	this	PRON
pc289g57c3s	13	41	to	to	PART
pc289g57c3s	13	42	describe	describe	VERB
pc289g57c3s	13	43	the	the	DET
pc289g57c3s	13	44	special	special	ADJ
pc289g57c3s	13	45	fiber	fiber	NOUN
pc289g57c3s	13	46	rings	ring	NOUN
pc289g57c3s	13	47	of	of	ADP
pc289g57c3s	13	48	lcm	lcm	NOUN
pc289g57c3s	13	49	-	-	NOUN
pc289g57c3s	13	50	duals	dual	NOUN
pc289g57c3s	13	51	of	of	ADP
pc289g57c3s	13	52	strongly	strongly	ADV
pc289g57c3s	13	53	stable	stable	ADJ
pc289g57c3s	13	54	ideals	ideal	NOUN
pc289g57c3s	13	55	in	in	ADP
pc289g57c3s	13	56	degree	degree	NOUN
pc289g57c3s	13	57	two	two	NUM
pc289g57c3s	13	58	.	.	PUNCT
pc289g57c3s	14	1	the	the	DET
pc289g57c3s	14	2	third	third	ADJ
pc289g57c3s	14	3	problem	problem	NOUN
pc289g57c3s	14	4	we	we	PRON
pc289g57c3s	14	5	consider	consider	VERB
pc289g57c3s	14	6	deals	deal	NOUN
pc289g57c3s	14	7	with	with	ADP
pc289g57c3s	14	8	hilbert	hilbert	NOUN
pc289g57c3s	14	9	functions	function	NOUN
pc289g57c3s	14	10	of	of	ADP
pc289g57c3s	14	11	cohen	cohen	NOUN
pc289g57c3s	14	12	-	-	PUNCT
pc289g57c3s	14	13	macaulay	macaulay	NOUN
pc289g57c3s	14	14	local	local	ADJ
pc289g57c3s	14	15	rings	ring	NOUN
pc289g57c3s	14	16	.	.	PUNCT
pc289g57c3s	15	1	given	give	VERB
pc289g57c3s	15	2	a	a	DET
pc289g57c3s	15	3	cohen	cohen	ADJ
pc289g57c3s	15	4	-	-	PUNCT
pc289g57c3s	15	5	macaulay	macaulay	NOUN
pc289g57c3s	15	6	local	local	ADJ
pc289g57c3s	15	7	ring	ring	NOUN
pc289g57c3s	15	8	(	(	PUNCT
pc289g57c3s	15	9	r	r	NOUN
pc289g57c3s	15	10	,	,	PUNCT
pc289g57c3s	15	11	m	m	NOUN
pc289g57c3s	15	12	)	)	PUNCT
pc289g57c3s	15	13	of	of	ADP
pc289g57c3s	15	14	dimension	dimension	NOUN
pc289g57c3s	15	15	d	d	PROPN
pc289g57c3s	15	16	&	&	CCONJ
pc289g57c3s	15	17	gt	gt	PROPN
pc289g57c3s	15	18	;	;	PUNCT
pc289g57c3s	15	19	0	0	NUM
pc289g57c3s	15	20	,	,	PUNCT
pc289g57c3s	15	21	we	we	PRON
pc289g57c3s	15	22	study	study	VERB
pc289g57c3s	15	23	the	the	DET
pc289g57c3s	15	24	depth	depth	NOUN
pc289g57c3s	15	25	of	of	ADP
pc289g57c3s	15	26	the	the	DET
pc289g57c3s	15	27	associated	associate	VERB
pc289g57c3s	15	28	graded	grade	VERB
pc289g57c3s	15	29	ring	ring	NOUN
pc289g57c3s	15	30	of	of	ADP
pc289g57c3s	15	31	r	r	NOUN
pc289g57c3s	15	32	with	with	ADP
pc289g57c3s	15	33	respect	respect	NOUN
pc289g57c3s	15	34	to	to	ADP
pc289g57c3s	15	35	the	the	DET
pc289g57c3s	15	36	maximal	maximal	ADJ
pc289g57c3s	15	37	ideal	ideal	NOUN
pc289g57c3s	15	38	m.	m.	NOUN
pc289g57c3s	15	39	we	we	PRON
pc289g57c3s	15	40	analyze	analyze	VERB
pc289g57c3s	15	41	the	the	DET
pc289g57c3s	15	42	case	case	NOUN
pc289g57c3s	15	43	where	where	SCONJ
pc289g57c3s	15	44	the	the	DET
pc289g57c3s	15	45	multiplicity	multiplicity	NOUN
pc289g57c3s	15	46	e	e	NOUN
pc289g57c3s	15	47	is	be	AUX
pc289g57c3s	15	48	h+3	h+3	SPACE
pc289g57c3s	15	49	,	,	PUNCT
pc289g57c3s	15	50	where	where	SCONJ
pc289g57c3s	15	51	h	h	NOUN
pc289g57c3s	15	52	is	be	AUX
pc289g57c3s	15	53	the	the	DET
pc289g57c3s	15	54	embedding	embed	VERB
pc289g57c3s	15	55	codimension	codimension	NOUN
pc289g57c3s	15	56	.	.	PUNCT
pc289g57c3s	16	1	the	the	DET
pc289g57c3s	16	2	tool	tool	NOUN
pc289g57c3s	16	3	we	we	PRON
pc289g57c3s	16	4	use	use	VERB
pc289g57c3s	16	5	is	be	AUX
pc289g57c3s	16	6	the	the	DET
pc289g57c3s	16	7	bigraded	bigrade	VERB
pc289g57c3s	16	8	sally	sally	ADJ
pc289g57c3s	16	9	module	module	NOUN
pc289g57c3s	16	10	s	s	NOUN
pc289g57c3s	16	11	of	of	ADP
pc289g57c3s	16	12	m	m	PROPN
pc289g57c3s	16	13	with	with	ADP
pc289g57c3s	16	14	respect	respect	NOUN
pc289g57c3s	16	15	to	to	ADP
pc289g57c3s	16	16	j	j	PROPN
pc289g57c3s	16	17	,	,	PUNCT
pc289g57c3s	16	18	where	where	SCONJ
pc289g57c3s	16	19	j	j	PROPN
pc289g57c3s	16	20	is	be	AUX
pc289g57c3s	16	21	a	a	DET
pc289g57c3s	16	22	minimal	minimal	ADJ
pc289g57c3s	16	23	reduction	reduction	NOUN
pc289g57c3s	16	24	of	of	ADP
pc289g57c3s	16	25	m.	m.	NOUN