Homological Dimension


 Fay Watts
 2 years ago
 Views:
Transcription
1 Homological Dimension David E V Rose April 17, 29 1 Introduction In this note, we explore the notion of homological dimension After introducing the basic concepts, our two main goals are to give a proof of the Hilbert syzygy theorem and to apply the theory of homological dimension to the study of local rings 2 Elementary results from homological algebra Assume throughout that R is a commutative ring with identity We assume some familiarity with the basic concepts from homological algebra, including but not limited to the following: chain complexes and double complexes, projective modules and resolutions, Tor, and Ext We state a number of elementary results without proof, all of which can be found in [4], that will be useful in the duration The first result gives a method for constructing projective resolutions: Lemma 21 Let P 1 P 1 P P M i π M M 1
2 be a diagram of Rmodules where the column is exact and the rows are projective resolutions We can complete this diagram to give the commutative diagram P 1 P M i 1 P 1 i P i M π 1 P 1 π P π M where P i = P i P i give a projective resolution of M and the columns are exact with the canonical inclusion and projection maps Our next result, which follows directly from the long exact sequence for Ext i R(M, ) associated to a short exact sequence of Rmodules, characterizes projective Rmodules Proposition 22 Let M be an Rmodule, then the following are equivalent: 1 M is projective 2 Ext i R (M, N) = for all i > and all Rmodules N 3 Ext 1 R(M, N) = for all Rmodules N 3 Basic concepts Let M be an Rmodule The projective dimension of M, denoted pd R (M) is the smallest natural number n so that there exists an Rmodule projective resolution P n P M If M does not admit a projective resolution of finite length, then we set pd R (M) = Our first result gives several useful conditions characterizing projective dimension: Lemma 31 Let M be an Rmodule, then the following are equivalent: 2
3 1 pd R (M) n 2 Ext i R (M, N) = for all i > n and all Rmodules N 3 Ext n+1 R (M, N) = for all Rmodules N 4 Whenever K P n 1 P M is an exact sequence with each P i projective, K is projective Proof Since Ext i R(, N) is the right derived functor of Hom R (, N), we have that item 4 implies item 1 implies item 2 implies item 3 Now, if A P B is an exact sequence with P projective, the long exact sequence for Ext i R(, N) gives that Ext i R (A, N) = Ext i+1 R (B, N) for i 1 By interlacing long exact sequences with short exact sequences, this in turn implies that if K P n 1 P M is exact with each P i projective, then Ext n+1 R (M, N) = Ext 1 R (K, N) Thus, if item 3 holds then Ext 1 R(K, N) = for all Rmodules N Proposition 22 then gives that K is projective In particular, item 2 gives that if pd R (M) < then pd R (M) = min{i : Ext i+1 R (M, N) = for all Rmodules N} Proposition 31 now gives that for a ring R the following numbers are the same 1 sup{i : Ext i R (M, N) for some Rmodules M and N} 2 sup{pd R (M) : M Rmod} Definition 32 This common number is the global dimension of the ring R, denoted gldim(r) 3
4 4 The Hilbert syzygy theorem Our goal in this section is to give a proof of the Hilbert syzygy theorem The strongest form of this statement is the following: Theorem 41 Let k be a field and suppose that M is a finitely generated k[x 1,, x n ]module If K F n 1 F M (41) is an exact sequence with each F i free and finitely generated, then K is free Our approach is as follows: We will first deduce that gldim(k[x 1,, x n ]) = n (42) Since each F i is projective, Lemma 31 implies that K is projective, and K is finitely generated as k[x 1,, x n ] is noetherian The result then follows from the QuillenSuslin theorem [3, Chapter XXI, 4, Theorem 37]: Theorem 42 Any finitely generated projective k[x 1,, x n ]module is free We now aim to show (42) We follow the presentation given in [4, Section 43], elaborating on the proofs given there The result will follow from a sequence of change of rings results In order to prove the first of these results, we need the following preparatory lemma Lemma 43 Let {M i } i I be a collection of Rmodules, then pd R ( i I ) M i = sup {pd R (M i )} i I Proof Since the direct sum of a family of projective modules is projective, given projective resolutions P,i M i for each M i we have a projective resolution P,i M i which gives pd R ( i I ) M i sup {pd R (M i )} i I Next, if pd R ( M i ) = then the result follows, so suppose pd R ( M i ) = n < For each i, consider an exact sequence K i P n 1,i P,i M i with each P m,i projective This gives an exact sequence i I K i i I P n 1,i i I P,i i I M i and Lemma 31 gives that K i is projective Since direct summands of projective modules are projective, it follows that each K i is projective Lemma 31 then gives pd R (M i ) n and the result follows 4
5 Given a morphism of rings R S, the first result relates the projective dimension of an Smodule M over S to its projective dimension over R Theorem 44 If R S is a morphism of rings and M is an Smodule, then pd R (M) pd R (S) + pd S (M) Proof We can assume that pd S (M) = n < and pd R (S) = d < or else there is nothing to prove Let Q n f n f 1 Q f M be an Smodule projective resolution Choose Rmodule projective resolutions of M = imf and kerf and use Lemma 21 to construct an Rmodule projective resolution P, of Q Now construct Rmodule projective resolutions for each Q i via the same procedure, using the resolution of imf i = kerf i 1 used to construct the projective resolution of Q i 1 and choosing a projective resolution of kerf i We can then patch these short exact sequences of projective resolutions together to obtain the commutative diagram: Q n P,n P 1,n Q 1 P,1 P 1,1 Q P, P 1, where the columns are chain complexes and the rows are Rmodule projective resolutions Now, since each Q i is a projective Smodule, it follows that for each i there exists an Smodule N i so that Q i N i is a free Smodule F i (S) Lemma 43 then implies that pd R (Q i ) pd R (F i (S)) = pd R (S) = d 5
6 We hence may construct the commutative diagram (43) Q n P,n P d,n Q P, P d, where the columns are chain complexes and the rows are Rmodule projective resolutions by taking P m,i = P m,i for m d and P d,i = P d,i / ker(p d,i P d 1,i ) and using Lemma 31 A standard argument now gives that the complexes Q and Tot(P, ), the total complex associated the double complex P, obtained from truncating the Q i s in the rows of (43), are quasiisomorphic (ie have the same homology) Since Tot(P, ) is a complex of projective Rmodules of length at most n + d, the result follows The proofs of the next two change of rings theorems proceed via induction on pd R (M), so the following lemma aids in these considerations Lemma 45 If the sequence of Rmodules A B C is exact, then pd R (B) max{pd R (A), pd R (C)} Moreover, if we have strict inequality, then pd R (C) = pd R (A) + 1 Proof The first statement follows from Lemma 21 The second follows considering the characterization provided in Lemma 31 Indeed suppose that pd B (B) = n < max{pd R (A), pd R (C)} The long exact sequence for Ext i R(, N) gives that Ext i R(A, N) = Ext i+1 R (C, N) for i n + 1 and all Rmodules N This gives the result provided pd R (A) n + 1 or pd R (C) n + 2 The remaining case is when pd R (C) = n + 1 and pd R (A) < n + 1 If pd R (A) = n then the result holds and otherwise the Ext i R (, N) long exact sequence implies that Extn+1 R (C, N) = for all R modules N, contradicting the fact that pd R (C) = n + 1 6
7 Given a ring R and an element r R that is not a zero divisor, consider an R/rRmodule M Such a module is an Rmodule that is annihilated by r The next result gives the projective dimension of M over R/rR in terms of the projective dimension of M over R This is particularly relevant to our present consideration, since taking R = S[x] for some ring S and r = x, we will be able to deduce information about gldim(s[x]) from gldim(s) Theorem 46 Let r R be a nonzerodivisor, M be a nonzero R/rRmodule, and suppose that pd R/rR (M) <, then pd R (M) = 1 + pd R/rR (M) Proof First, note that pd R (M) Indeed, this would mean that M is a projective Rmodule, implying that M is a direct summand of a free Rmodule and contradicting the fact that M is annihilated by r We hence have that pd R (M) 1 Next, suppose that M is a projective R/rRmodule, then since R r R R/rR (44) gives a projective resolution of R/rR of minimal length, Theorem 44 gives pd R (M) pd R (R/rR) + pd R/rR (M) = 1 so the result holds It now suffices to consider the case when pd R (M), pd R/rR (M) 1 and we proceed via induction on n = pd R/rR (M) < Consider an R/rRmodule projective resolution P n f n f 1 P f A of minimal length Taking K = kerf, we have that K P M is exact and pd R/rR (M) = 1 + pd R/rR (K) Lemma 45 implies that either or 1 = pd R (P ) = max{pd R (K), pd R (M)} (45) Since the induction hypothesis gives that pd R (M) = 1 + pd R (K) (46) pd R (K) = 1 + pd R/rR (K) provided pd R (K) 1, we are done if (46) holds or if (45) holds and pd R (K) =, since pd R (M) 1 7
8 It remains to consider the case when pd R (M) = 1 = pd R/rR (M) and we shall now see that this cannot happen Let P 1 P M be an Rmodule projective resolution of M Applying the functor R R/rR gives the exact sequence Tor R 1 (M, R/rR) P 1 R R/rR P R R/rR M of R/rRmodules Since P 1 R R/rR and P R R/rR are projective R/rRmodules, Lemma 31 implies that Tor1 R (M, R/rR) is projective Computing Tor1 R (M, R/rR) by applying M R to the Rmodule projective resolution (44), we find that Tor R 1 (M, R/rR) = M which implies pd R/rR (M) =, a contradiction The next result examines the case where r R is neither a zero divisor on R nor on M Theorem 47 Let M be an Rmodule and r R be neither a zero divisor on R nor on M, then pd R/rR (M/rM) pd R (M) Proof Since the result is trivial if pd R (M) =, we proceed via induction on pd R (M) < If pd R (M) = then M is projective, hence M/rM = R/rR R M is projective, ie pd R/rR (M/rM) = and the result holds Now suppose n = pd R (M) 1 and let K F M (47) be an exact sequence of Rmodules with F free Lemma 31 gives that pd R (K) = n 1 and the induction hypothesis gives Applying R R/rR to (47) gives pd R/rR (K/rK) n 1 Tor R 1 (M, R/rR) K/rK F/rF M/rM and a calculation gives that Tor R 1 (M, R/rR) =, so K/rK F/rF M/rM is exact Lemma 45 now gives that either = pd R/rR (F/rF) = max{pd R/rR (K/rK), pd R/rR (M/rM)} 8
9 so pd R/rR (M/rM) = or that which gives the result pd R/rR (M/rM) = pd R/rR (K/rK) + 1 Corollary 48 Let M be an Rmodule, then (n 1) + 1 = n pd R (M) = pd R[x] (R[x] R M) Proof Since x is a nonzerodivisor on R[x] and R[x] R M, Theorem 47 gives pd R (M) pd R[x] (R[x] R M) Now, if P M is an Rmodule projective resolution, then applying R[x] R gives an R[x]module projective resolution R[x] R P R[x] R M since R[x] is a flat Rmodule This gives pd R (M) pd R[x] (R[x] R M) We now arrive at the main theorem in this section Theorem 49 Let R be a ring, then gldim(r[x]) = gldim(r) + 1 Proof If gldim(r) = then Corollary 48 implies that gldim(r[x]) = and the result holds We now assume that gldim(r) = n < If N is an Rmodule, then N is an R[x]module (let x act trivially) and Theorem 46 gives pd R[x] (N) = 1 + pd R (N) and hence gldim(r[x]) n + 1 Now, let M be an R[x]module A computation shows that the sequence of R[x]modules with R[x] R M ϕ R[x] R M ψ M ϕ(f m) = xf m f xm ψ(f m) = fm 9
10 is exact Lemma 45 then gives that either pd R[x] (M) = 1 + pd R[x] (R[x] R M) or pd R[x] (M) pd R[x] (R[x] R M) This gives pd R[x] (M) 1 + pd R[x] (R[x] R M) = 1 + pd R (M) 1 + n where we have used Corollary 48 This implies gldim(r[x]) n + 1 which completes the proof This result immediately implies the following corollary Corollary 41 Let R be a ring, then gldim(r[x 1,, x n ]) = gldim(r) + n Our proof of the Hilbert syzygy theorem follows directly from this result, noting that if k is a field, then gldim(k) = since all kmodules are free 5 Local Rings For the duration, we consider a noetherian local ring R with maximal ideal m Recall that such a ring is called a regular local ring provided dim(r) = dim k (m/m 2 ) where k = R/m is the residue field, dim( ) denotes Krull dimension, and dim k ( ) denotes vector space dimension Such rings arise naturally in algebraic geometry, where they characterize nonsingularity of varieties It is a standard fact that if R is a noetherian local ring then dim(r) is finite and in fact dim(r) dim k (m/m 2 ) [1, Chapter 11] In this section we aim to prove the following result [4, Corollary 4418]: Theorem 51 If R is a regular local ring and p R is a prime ideal, then R p is a regular local ring It is interesting to note that from the statement of this theorem, one would not suspect that homological algebra would play any role in its proof As in the last section, we will prove this result using a sequence of preliminary results We again follow the presentation given in [4, Section 44] We begin with an additional change of rings theorem, a strengthening of Theorem 47 in the case that M is finitely generated Theorem 52 Let R a noetherian local ring with maximal ideal m If M is a finitely generated Rmodule and r m is neither a zero divisor on R nor on M, then pd R/rR (M/rM) = pd R (M) 1
11 Proof We have pd R/rR (M/rM) pd R (M) by Theorem 47 so we are done if pd R/rR (M/rM) = Otherwise, we may proceed via induction on n = pd R/rR (M/rM) If n = then M/rM is projective and it follows that M/rM is free since R/rR is local and projective modules over local rings are free We now claim that this implies that M is free, ie pd R (M) = Indeed, let x 1,, x m M be elements mapping to a basis for M/rM and let (x 1,,x m ) denote the Rsubmodule they generate We have M = (x 1,, x m ) + rm = (x 1,, x m ) + mm so Nakayama s lemma implies (x 1,, x m ) = M To see that these elements are independent, suppose there exist s 1,, s m R so that s i x i = This gives that s i x i = in M/rM where x i denotes the image of x i under the canonical quotient map For each i we have, s i = r s i, so r ( s i x i) = and since r is not a zero divisor on M this implies s i x i = We may iterate this procedure and obtain an increasing sequence of ideals where s (j) i s (l) i R = s (l+1) i s i R s i R s(j) i R = rs (j+1) i, which must stabilize since R is noetherian This gives that R for some l and hence there exists a R so that ars (l+1) i = s (l+1) i Rearranging, this gives (1 ar)s (l+1) i is a unit [1, Proposition 19] This implies s (l+1) = and since r m we have that 1 ar = so s i = r l+1 s (l+1) i = for all i Having handled the n = case, we now easily finish the proof Let K F M be an exact sequence of Rmodules with F free, so pd R (K) = pd R (M) 1 Applying the functor R R/R gives the exact sequence K/rK F/rF M/rM since Tor R 1 (M, R/rR) = if r is not a zero divisor on M Since F/rF is free, pd R/rR (K/rK) = n 1, and since M is finitely generated, we can suppose that F is as well As R is noetherian, this implies that K is finitely generated The induction hypothesis then gives that pd R (K) = n 1 and so pd R (M) = pd R (K) + 1 = n We now need to introduce further machinery for analyzing local rings Let M be an Rmodule, then a regular sequence on M is a sequence r 1, r d m so that r 1 is not a zero divisor on M and r i is not a zero divisor on M/(r 1,, r i 1 )M We let G(M) denote the length of the longest regular sequence on M Now recall the following result from the dimension theory of noetherian local rings [1, Corollary 1118]: i 11
12 Proposition 53 Let R be a noetherian local ring and r m not a zero divisor, then dim R/rR = dimr 1 This shows that G(R) dim(r) and R is called CohenMacaulay if we have G(R) = dim(r) Proposition 54 A regular local ring R is CohenMacaulay Moreover, any set r 1,,r d m mapping to a basis of m/m 2 is a regular sequence on R Proof It suffices to show that dim(r) = dim k (m/m 2 ) G(R), and we proceed via induction on d = dim(r) The case d = is trivial, so assume d 1 and let r 1,, r d m map to a basis of m/m 2 Since regular local rings are integral domains [1, Lemma 1123], r 1 is not a zero divisor and hence, by Proposition 53, dim(r/r 1 R) = d 1 Let r 2,, r d denote the images in R = R/r 1 R of r 2,, r d and note that r 2,, r d map to a basis of m/ m 2 where m = m R, hence R is regular The induction hypothesis implies that r 2,, r d is a regular sequence on R, and thus r 1,,r d is a regular sequence on R We now consider a result describing noetherian local rings R with G(R) = Note that this condition is equivalent to all elements of m being zerodivisors Lemma 55 Let R be a noetherian local ring with G(R) =, then if M is a finitely generated Rmodule either pd R (M) = or pd R (M) = Proof Suppose G(R) = and that pd R (M) = n, There then exists an exact sequence P n f n P f M with each P i projective and finitely generated Let K = imf n 1, which is a finitely generated syzygy of M with pd R (K) = 1 Let m 1,, m t M be elements which map to a basis for M/mM and hence generate M We now have an exact sequence P R t K where R t K is the obvious map, and Lemma 31 gives that P is projective, hence free Since mr t is the kernel of the map R t K K/mK, it follows that P mr t Now, since m consists only of zerodivisors, it is contained in the union of the associated primes of R and as it is maximal, it must be one of the associated primes (by prime avoidance) By definition, this gives that there exists s R so that m = {r R rs = } This gives that sp =, contradicting the fact that P is free We state the next two results, Theorem 422 and Corollary 4412 from [4] without proof, as the proofs would necessitate the introduction of a number of additional concepts (in particular injective dimension, flat dimension, and Tordimension [4, Section 41]) 12
13 Proposition 56 A ring R is semisimple (ie every ideal is a direct summand) iff gldim(r) = Proposition 57 Let R be a noetherian local ring with residue field k, then gldim(r) = pd R (k) We now arrive at our main theorem, which characterizes regular local rings via their global dimension Theorem 51 will follow as an easy corollary Theorem 58 A noetherian local ring R is regular iff gldim(r) Moreover, if R is regular then dim(r) = gldim(r) Proof We first assume that R is regular and show gldim(r) <, proceeding via induction on d = dim(r) If d = then m = m 2 so Nakayama s lemma implies that R is a field and the result holds Now suppose d >, then Proposition 54 gives G(R) = d so there exists r m so that r is not a zero divisor on R Since R = R/rR is regular and dim( R) = d 1, we have that gldim( R) = d 1 by the induction hypothesis Now, we compute gldim(r) = pd R (R/m) = pd R ( R) + pd R(R/m) = 1 + pd R( R/ m) = 1 + gldim( R) = d where m = m R is the maximal ideal of R and we have used Proposition 57 and Theorem 44 We now assume that gldim(r) < and deduce that R is regular; again, we proceed via induction Suppose that gldim(r) =, then R is local and semisimple (by Proposition 56) so R is a field, hence regular Now, let gldim(r), so by Lemma 55 there exists r m which is not a zerodivisor Infact, we can even choose r m m 2 ; indeed, if m m 2 consists only of zero divisors, then m m 2 p 1 p m where p 1,, p m are the associated primes of R (it is a standard fact that the union of these ideals precisely equals the zero divisors in R) This gives m m 2 p 1 p m and the strong form of prime avoidance [2, Lemma 33] shows that either m p j for some j or m = This implies that all elements of m are zero divisors Hence let r m m 2 be a nonzerodivisor Consider R = R/rR, which we now aim to show is regular We have exact sequences m R R k (51) and rr/rm m/rm α m R (52) 13
14 The Tor long exact sequence associated to (51) gives that rr/rm = Tor R 1 (R/rR, k), and a standard computation shows that Tor R 1 (R/rR, k) = k Now let r 2,, r n m be such that the images of r, r 2,,r n in m/m 2 form a basis (this is possible since r / m 2 ) Let n = (r 2,, r n )R+rm and note that n/rm surjects onto m R via α Since kerα = rr/rm is isomorphic to a field and contains r + rm (which is not in n/rm) we have that (rr/rm) (n/rm) = which implies n/rm = m R It then follows that m/rm = k m R as Rmodules We now compute gldim( R) = pd R(k) pd R(m/rm) = pd R (m) = pd R (k) 1 = gldim(r) 1 where we have used Proposition 57, Lemma 43, and Theorem 52 The induction hypothesis now implies that R is regular This in turn implies that R is regular Indeed, let d = dim(r) and note that dim( R) = d 1 Since R is regular, Proposition 54 gives that there exist elements r 2,, r d m whose images r 2,, r d m R generate m R and give an Rregular sequence It follows that r, r 2,,r d give an Rregular sequence and generate m This shows dim k (m/m 2 ) G(R) and since we always have G(R) dim(r) dim k (m/m 2 ), this implies R is regular We now explain how this result implies Theorem 51 Suppose R is a regular local ring and p R is a prime ideal Let M be an R p module, then viewing M as an Rmodule, there is a projective resolution P n P M where n gldim(r) < Applying the functor R p R then gives an exact sequence R p R P n R p R P M of R p modules, since R p is a flat Rmodule and R p R M = M Each R p R P i is a projective R p module, so this implies gldim(r p ) gldim(r) < By Theorem 58, R p is regular 14
15 References [1] M F Atiyah and I G MacDonald, Introduction to commutative algebra, AddisonWesley Series in Mathematics, Westview Press, Boulder, Colorado, 1969 [2] David Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, Springer, New York, 24 [3] Serge Lang, Algebra, revised third ed, Graduate Texts in Mathematics, Springer, New York, 22 [4] Charles A Weibel, An introduction to homological algebra, Cambridge studies in advanced mathematics, Cambridge University Press, Cambridge, UK,
COHENMACAULAY RINGS SELECTED EXERCISES. 1. Problem 1.1.9
COHENMACAULAY RINGS SELECTED EXERCISES KELLER VANDEBOGERT 1. Problem 1.1.9 Proceed by induction, and suppose x R is a U and Nregular element for the base case. Suppose now that xm = 0 for some m M. We
More informationINJECTIVE MODULES: PREPARATORY MATERIAL FOR THE SNOWBIRD SUMMER SCHOOL ON COMMUTATIVE ALGEBRA
INJECTIVE MODULES: PREPARATORY MATERIAL FOR THE SNOWBIRD SUMMER SCHOOL ON COMMUTATIVE ALGEBRA These notes are intended to give the reader an idea what injective modules are, where they show up, and, to
More informationHilbert function, Betti numbers. Daniel Gromada
Hilbert function, Betti numbers 1 Daniel Gromada References 2 David Eisenbud: Commutative Algebra with a View Toward Algebraic Geometry 19, 110 David Eisenbud: The Geometry of Syzygies 1A, 1B My own notes
More informationNotes for Boot Camp II
Notes for Boot Camp II Mengyuan Zhang Last updated on September 7, 2016 1 The following are notes for Boot Camp II of the Commutative Algebra Student Seminar. No originality is claimed anywhere. The main
More informationLIVIA HUMMEL AND THOMAS MARLEY
THE AUSLANDERBRIDGER FORMULA AND THE GORENSTEIN PROPERTY FOR COHERENT RINGS LIVIA HUMMEL AND THOMAS MARLEY Abstract. The concept of Gorenstein dimension, defined by Auslander and Bridger for finitely
More informationHomological Methods in Commutative Algebra
Homological Methods in Commutative Algebra Olivier Haution LudwigMaximiliansUniversität München Sommersemester 2017 1 Contents Chapter 1. Associated primes 3 1. Support of a module 3 2. Associated primes
More informationCohenMacaulay Dimension for Coherent Rings
University of Nebraska  Lincoln DigitalCommons@University of Nebraska  Lincoln Dissertations, Theses, and Student Research Papers in Mathematics Mathematics, Department of 52016 CohenMacaulay Dimension
More informationCohomology and Base Change
Cohomology and Base Change Let A and B be abelian categories and T : A B and additive functor. We say T is halfexact if whenever 0 M M M 0 is an exact sequence of Amodules, the sequence T (M ) T (M)
More informationA NEW PROOF OF SERRE S HOMOLOGICAL CHARACTERIZATION OF REGULAR LOCAL RINGS
A NEW PROOF OF SERRE S HOMOLOGICAL CHARACTERIZATION OF REGULAR LOCAL RINGS RAVI JAGADEESAN AND AARON LANDESMAN Abstract. We give a new proof of Serre s result that a Noetherian local ring is regular if
More informationFormal power series rings, inverse limits, and Iadic completions of rings
Formal power series rings, inverse limits, and Iadic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely
More informationInjective Modules and Matlis Duality
Appendix A Injective Modules and Matlis Duality Notes on 24 Hours of Local Cohomology William D. Taylor We take R to be a commutative ring, and will discuss the theory of injective Rmodules. The following
More informationCRing Project, Chapter 16
Contents 16 Homological theory of local rings 3 1 Depth........................................... 3 1.1 Depth over local rings.............................. 3 1.2 Regular sequences................................
More informationCOURSE SUMMARY FOR MATH 508, WINTER QUARTER 2017: ADVANCED COMMUTATIVE ALGEBRA
COURSE SUMMARY FOR MATH 508, WINTER QUARTER 2017: ADVANCED COMMUTATIVE ALGEBRA JAROD ALPER WEEK 1, JAN 4, 6: DIMENSION Lecture 1: Introduction to dimension. Define Krull dimension of a ring A. Discuss
More informationFROBENIUS AND HOMOLOGICAL DIMENSIONS OF COMPLEXES
FROBENIUS AND HOMOLOGICAL DIMENSIONS OF COMPLEXES TARAN FUNK AND THOMAS MARLEY Abstract. It is proved that a module M over a Noetherian local ring R of prime characteristic and positive dimension has finite
More information11. Finitelygenerated modules
11. Finitelygenerated modules 11.1 Free modules 11.2 Finitelygenerated modules over domains 11.3 PIDs are UFDs 11.4 Structure theorem, again 11.5 Recovering the earlier structure theorem 11.6 Submodules
More informationREPRESENTATION THEORY WEEK 9
REPRESENTATION THEORY WEEK 9 1. JordanHölder theorem and indecomposable modules Let M be a module satisfying ascending and descending chain conditions (ACC and DCC). In other words every increasing sequence
More informationFILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.
FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0
More informationarxiv: v1 [math.ac] 16 Nov 2018
On Hilbert Functions of Points in Projective Space and Structure of Graded Modules Damas Karmel Mgani arxiv:1811.06790v1 [math.ac] 16 Nov 2018 Department of Mathematics, College of Natural and Mathematical
More informationBUILDING MODULES FROM THE SINGULAR LOCUS
BUILDING MODULES FROM THE SINGULAR LOCUS JESSE BURKE, LARS WINTHER CHRISTENSEN, AND RYO TAKAHASHI Abstract. A finitely generated module over a commutative noetherian ring of finite Krull dimension can
More informationLecture 2. (1) Every P L A (M) has a maximal element, (2) Every ascending chain of submodules stabilizes (ACC).
Lecture 2 1. Noetherian and Artinian rings and modules Let A be a commutative ring with identity, A M a module, and φ : M N an Alinear map. Then ker φ = {m M : φ(m) = 0} is a submodule of M and im φ is
More informationOn Extensions of Modules
On Extensions of Modules Janet Striuli Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA Abstract In this paper we study closely Yoneda s correspondence between short exact sequences
More informationMath 210B. Artin Rees and completions
Math 210B. Artin Rees and completions 1. Definitions and an example Let A be a ring, I an ideal, and M an Amodule. In class we defined the Iadic completion of M to be M = lim M/I n M. We will soon show
More informationOn the vanishing of Tor of the absolute integral closure
On the vanishing of Tor of the absolute integral closure Hans Schoutens Department of Mathematics NYC College of Technology City University of New York NY, NY 11201 (USA) Abstract Let R be an excellent
More informationTHE AUSLANDER BUCHSBAUM FORMULA. 0. Overview. This talk is about the AuslanderBuchsbaum formula:
THE AUSLANDER BUCHSBAUM FORMULA HANNO BECKER Abstract. This is the script for my talk about the AuslanderBuchsbaum formula [AB57, Theorem 3.7] at the Auslander Memorial Workshop, 15 th 18 th of November
More informationNotes on pdivisible Groups
Notes on pdivisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationALGEBRA HW 4. M 0 is an exact sequence of Rmodules, then M is Noetherian if and only if M and M are.
ALGEBRA HW 4 CLAY SHONKWILER (a): Show that if 0 M f M g M 0 is an exact sequence of Rmodules, then M is Noetherian if and only if M and M are. Proof. ( ) Suppose M is Noetherian. Then M injects into
More information12. Projective modules The blanket assumptions about the base ring k, the kalgebra A, and Amodules enumerated at the start of 11 continue to hold.
12. Projective modules The blanket assumptions about the base ring k, the kalgebra A, and Amodules enumerated at the start of 11 continue to hold. 12.1. Indecomposability of M and the localness of End
More informationLECTURE NOTES DAVID WHITE
LECTURE NOTES DAVID WHITE 1. Motivation for Spectra 1 It is VERY hard to compute homotopy groups. We want to put as much algebraic structure as possible in order to make computation easier. You can t add
More informationVANISHING THEOREMS FOR COMPLETE INTERSECTIONS. Craig Huneke, David A. Jorgensen and Roger Wiegand. August 15, 2000
VANISHING THEOREMS FOR COMPLETE INTERSECTIONS Craig Huneke, David A. Jorgensen and Roger Wiegand August 15, 2000 The starting point for this work is Auslander s seminal paper [A]. The main focus of his
More informationHILBERT FUNCTIONS. 1. Introduction
HILBERT FUCTIOS JORDA SCHETTLER 1. Introduction A Hilbert function (so far as we will discuss) is a map from the nonnegative integers to themselves which records the lengths of composition series of each
More informationHomework 2  Math 603 Fall 05 Solutions
Homework 2  Math 603 Fall 05 Solutions 1. (a): In the notation of AtiyahMacdonald, Prop. 5.17, we have B n j=1 Av j. Since A is Noetherian, this implies that B is f.g. as an Amodule. (b): By Noether
More information38 Irreducibility criteria in rings of polynomials
38 Irreducibility criteria in rings of polynomials 38.1 Theorem. Let p(x), q(x) R[x] be polynomials such that p(x) = a 0 + a 1 x +... + a n x n, q(x) = b 0 + b 1 x +... + b m x m and a n, b m 0. If b m
More informationMATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA
MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to
More informationRing Theory Problems. A σ
Ring Theory Problems 1. Given the commutative diagram α A σ B β A σ B show that α: ker σ ker σ and that β : coker σ coker σ. Here coker σ = B/σ(A). 2. Let K be a field, let V be an infinite dimensional
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ANDREW SALCH 1. Hilbert s Nullstellensatz. The last lecture left off with the claim that, if J k[x 1,..., x n ] is an ideal, then
More informationGorenstein homological dimensions
Journal of Pure and Applied Algebra 189 (24) 167 193 www.elsevier.com/locate/jpaa Gorenstein homological dimensions Henrik Holm Matematisk Afdeling, Universitetsparken 5, Copenhagen DK21, Denmark Received
More informationNOTES ON BASIC HOMOLOGICAL ALGEBRA 0 L M N 0
NOTES ON BASIC HOMOLOGICAL ALGEBRA ANDREW BAKER 1. Chain complexes and their homology Let R be a ring and Mod R the category of right Rmodules; a very similar discussion can be had for the category of
More informationA generalized Koszul theory and its applications in representation theory
A generalized Koszul theory and its applications in representation theory A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Liping Li IN PARTIAL FULFILLMENT
More informationABSTRACT NONSINGULAR CURVES
ABSTRACT NONSINGULAR CURVES Affine Varieties Notation. Let k be a field, such as the rational numbers Q or the complex numbers C. We call affine nspace the collection A n k of points P = a 1, a,..., a
More informationRees Algebras of Modules
Rees Algebras of Modules ARON SIMIS Departamento de Matemática Universidade Federal de Pernambuco 50740540 Recife, PE, Brazil email: aron@dmat.ufpe.br BERND ULRICH Department of Mathematics Michigan
More informationTensor Product of modules. MA499 Project II
Tensor Product of modules A Project Report Submitted for the Course MA499 Project II by Subhash Atal (Roll No. 07012321) to the DEPARTMENT OF MATHEMATICS INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI GUWAHATI
More informationPRIMARY DECOMPOSITION OF MODULES
PRIMARY DECOMPOSITION OF MODULES SUDHIR R. GHORPADE AND JUGAL K. VERMA Department of Mathematics, Indian Institute of Technology, Mumbai 400 076, India EMail: {srg, jkv}@math.iitb.ac.in Abstract. Basics
More informationTHE RADIUS OF A SUBCATEGORY OF MODULES
THE RADIUS OF A SUBCATEGORY OF MODULES HAILONG DAO AND RYO TAKAHASHI Dedicated to Professor Craig Huneke on the occasion of his sixtieth birthday Abstract. We introduce a new invariant for subcategories
More informationCOMPRESSIBLE MODULES. Abhay K. Singh Department of Applied Mathematics, Indian School of Mines Dhanbad India. Abstract
COMPRESSIBLE MODULES Abhay K. Singh Department of Applied Mathematics, Indian School of Mines Dhanbad826004 India Abstract The main purpose of this paper is to study under what condition compressible
More informationIntegral Extensions. Chapter Integral Elements Definitions and Comments Lemma
Chapter 2 Integral Extensions 2.1 Integral Elements 2.1.1 Definitions and Comments Let R be a subring of the ring S, and let α S. We say that α is integral over R if α isarootofamonic polynomial with coefficients
More informationAzumaya Algebras. Dennis Presotto. November 4, Introduction: Central Simple Algebras
Azumaya Algebras Dennis Presotto November 4, 2015 1 Introduction: Central Simple Algebras Azumaya algebras are introduced as generalized or global versions of central simple algebras. So the first part
More informationSEQUENCES FOR COMPLEXES II
SEQUENCES FOR COMPLEXES II LARS WINTHER CHRISTENSEN 1. Introduction and Notation This short paper elaborates on an example given in [4] to illustrate an application of sequences for complexes: Let R be
More informationALMOST GORENSTEIN RINGS TOWARDS A THEORY OF HIGHER DIMENSION
ALMOST GORENSTEIN RINGS TOWARDS A THEORY OF HIGHER DIMENSION SHIRO GOTO, RYO TAKAHASHI, AND NAOKI TANIGUCHI Abstract. The notion of almost Gorenstein local ring introduced by V. Barucci and R. Fröberg
More informationThe Proj Construction
The Proj Construction Daniel Murfet May 16, 2006 Contents 1 Basic Properties 1 2 Functorial Properties 2 3 Products 6 4 Linear Morphisms 9 5 Projective Morphisms 9 6 Dimensions of Schemes 11 7 Points of
More informationLINKAGE CLASSES OF GRADE 3 PERFECT IDEALS
LINKAGE CLASSES OF GRADE 3 PERFECT IDEALS LARS WINTHER CHRISTENSEN, OANA VELICHE, AND JERZY WEYMAN Abstract. While every grade 2 perfect ideal in a regular local ring is linked to a complete intersection
More informationTHE GHOST DIMENSION OF A RING
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 00029939(XX)00000 THE GHOST DIMENSION OF A RING MARK HOVEY AND KEIR LOCKRIDGE (Communicated by Birge HuisgenZimmerman)
More informationA TALE OF TWO FUNCTORS. Marc Culler. 1. Hom and Tensor
A TALE OF TWO FUNCTORS Marc Culler 1. Hom and Tensor It was the best of times, it was the worst of times, it was the age of covariance, it was the age of contravariance, it was the epoch of homology, it
More informationMATH 233B, FLATNESS AND SMOOTHNESS.
MATH 233B, FLATNESS AND SMOOTHNESS. The discussion of smooth morphisms is one place were Hartshorne doesn t do a very good job. Here s a summary of this week s material. I ll also insert some (optional)
More informationDESCENT OF THE CANONICAL MODULE IN RINGS WITH THE APPROXIMATION PROPERTY
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 6, June 1996 DESCENT OF THE CANONICAL MODULE IN RINGS WITH THE APPROXIMATION PROPERTY CHRISTEL ROTTHAUS (Communicated by Wolmer V. Vasconcelos)
More informationEXT, TOR AND THE UCT
EXT, TOR AND THE UCT CHRIS KOTTKE Contents 1. Left/right exact functors 1 2. Projective resolutions 2 3. Two useful lemmas 3 4. Ext 6 5. Ext as a covariant derived functor 8 6. Universal Coefficient Theorem
More informationCOURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA
COURSE SUMMARY FOR MATH 504, FALL QUARTER 20178: MODERN ALGEBRA JAROD ALPER Week 1, Sept 27, 29: Introduction to Groups Lecture 1: Introduction to groups. Defined a group and discussed basic properties
More informationON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF
ON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF MATTHEW H. BAKER AND JÁNOS A. CSIRIK Abstract. We give a new proof of the isomorphism between the dualizing sheaf and the canonical
More informationCHAPTER 0 PRELIMINARY MATERIAL. Paul Vojta. University of California, Berkeley. 18 February 1998
CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University of California, Berkeley 18 February 1998 This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic
More informationSubrings of Finite Commutative Rings
Subrings of Finite Commutative Rings Francisco Franco Munoz arxiv:1712.02025v1 [math.ac] 6 Dec 2017 Abstract We record some basic results on subrings of finite commutative rings. Among them we establish
More informationPostgraduate Diploma Dissertation
The Abdus Salam International Centre for Theoretical Physics Trieste, Italy Postgraduate Diploma Dissertation Title Syzygies Author Yairon Cid Ruiz Supervisor Prof. Lothar Göttsche  August, 2016  ii
More informationAN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES
AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES MATTHEW H. BAKER AND JÁNOS A. CSIRIK This paper was written in conjunction with R. Hartshorne s Spring 1996 Algebraic Geometry course at
More informationMath 711: Lecture of September 7, Symbolic powers
Math 711: Lecture of September 7, 2007 Symbolic powers We want to make a number of comments about the behavior of symbolic powers of prime ideals in Noetherian rings, and to give at least one example of
More informationINJECTIVE MODULES AND THE INJECTIVE HULL OF A MODULE, November 27, 2009
INJECTIVE ODULES AND THE INJECTIVE HULL OF A ODULE, November 27, 2009 ICHIEL KOSTERS Abstract. In the first section we will define injective modules and we will prove some theorems. In the second section,
More informationOn the Existence of Gorenstein Projective Precovers
Rend. Sem. Mat. Univ. Padova 1xx (201x) Rendiconti del Seminario Matematico della Università di Padova c European Mathematical Society On the Existence of Gorenstein Projective Precovers Javad Asadollahi
More informationCatenary rings. Anna Bot. Bachelor thesis. supervised by Prof. Dr. Richard Pink
Catenary rings Anna Bot Bachelor thesis supervised by Prof. Dr. Richard Pink 2.10.2017 Contents 1 Introduction 1 2 Prerequisites 2 3 Regular sequences 3 4 Depth 5 5 Depth in light of height, localisation
More informationCategory O and its basic properties
Category O and its basic properties Daniil Kalinov 1 Definitions Let g denote a semisimple Lie algebra over C with fixed Cartan and Borel subalgebras h b g. Define n = α>0 g α, n = α
More informationFORMAL GLUEING OF MODULE CATEGORIES
FORMAL GLUEING OF MODULE CATEGORIES BHARGAV BHATT Fix a noetherian scheme X, and a closed subscheme Z with complement U. Our goal is to explain a result of Artin that describes how coherent sheaves on
More informationx y B =. v u Note that the determinant of B is xu + yv = 1. Thus B is invertible, with inverse u y v x On the other hand, d BA = va + ub 2
5. Finitely Generated Modules over a PID We want to give a complete classification of finitely generated modules over a PID. ecall that a finitely generated module is a quotient of n, a free module. Let
More informationINTRO TO TENSOR PRODUCTS MATH 250B
INTRO TO TENSOR PRODUCTS MATH 250B ADAM TOPAZ 1. Definition of the Tensor Product Throughout this note, A will denote a commutative ring. Let M, N be two Amodules. For a third Amodule Z, consider the
More informationCRing Project, Chapter 5
Contents 5 Noetherian rings and modules 3 1 Basics........................................... 3 1.1 The noetherian condition............................ 3 1.2 Stability properties................................
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationHOMOLOGICAL PROPERTIES OF QUANTIZED COORDINATE RINGS OF SEMISIMPLE GROUPS
HOMOLOGICAL PROPERTIES OF QUANTIZED COORDINATE RINGS OF SEMISIMPLE GROUPS K.R. GOODEARL AND J.J. ZHANG Abstract. We prove that the generic quantized coordinate ring O q(g) is Auslanderregular, CohenMacaulay,
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationKrull Dimension and GoingDown in Fixed Rings
David Dobbs Jay Shapiro April 19, 2006 Basics R will always be a commutative ring and G a group of (ring) automorphisms of R. We let R G denote the fixed ring, that is, Thus R G is a subring of R R G =
More informationarxiv:math/ v1 [math.ac] 4 Oct 2006
ON THE VNISHING OF (CO)HOMOLOGY OVER LOCL RINGS PETTER NDRES BERGH arxiv:math/0610150v1 [mathc] 4 Oct 2006 bstract Considering modules of finite complete intersection dimension over commutative Noetherian
More information11. Dimension. 96 Andreas Gathmann
96 Andreas Gathmann 11. Dimension We have already met several situations in this course in which it seemed to be desirable to have a notion of dimension (of a variety, or more generally of a ring): for
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics GROUP ACTIONS ON POLYNOMIAL AND POWER SERIES RINGS Peter Symonds Volume 195 No. 1 September 2000 PACIFIC JOURNAL OF MATHEMATICS Vol. 195, No. 1, 2000 GROUP ACTIONS ON POLYNOMIAL
More informationNOTES ON LINEAR ALGEBRA OVER INTEGRAL DOMAINS. Contents. 1. Introduction 1 2. Rank and basis 1 3. The set of linear maps 4. 1.
NOTES ON LINEAR ALGEBRA OVER INTEGRAL DOMAINS Contents 1. Introduction 1 2. Rank and basis 1 3. The set of linear maps 4 1. Introduction These notes establish some basic results about linear algebra over
More informationThe most important result in this section is undoubtedly the following theorem.
28 COMMUTATIVE ALGEBRA 6.4. Examples of Noetherian rings. So far the only rings we can easily prove are Noetherian are principal ideal domains, like Z and k[x], or finite. Our goal now is to develop theorems
More informationGOLDIE S THEOREM RICHARD G. SWAN
GOLDIE S THEOREM RICHARD G. SWAN Abstract. This is an exposition of Goldie s theorem with a section on Ore localization and with an application to defining ranks for finitely generated modules over non
More informationGorenstein modules, finite index, and finite Cohen Macaulay type
J. Pure and Appl. Algebra, (), 1 14 (2001) Gorenstein modules, finite index, and finite Cohen Macaulay type Graham J. Leuschke Department of Mathematics, University of Kansas, Lawrence, KS 66045 gleuschke@math.ukans.edu
More informationSTABLE MODULE THEORY WITH KERNELS
Math. J. Okayama Univ. 43(21), 31 41 STABLE MODULE THEORY WITH KERNELS Kiriko KATO 1. Introduction Auslander and Bridger introduced the notion of projective stabilization mod R of a category of finite
More informationSemidualizing Modules. Sean SatherWagstaff
Semidualizing Modules Sean SatherWagstaff Department of Mathematics, North Dakota State University Department # 2750, PO Box 6050, Fargo, ND 581086050, USA Email address: sean.satherwagstaff@ndsu.edu
More informationREMARKS ON REFLEXIVE MODULES, COVERS, AND ENVELOPES
REMARKS ON REFLEXIVE MODULES, COVERS, AND ENVELOPES RICHARD BELSHOFF Abstract. We present results on reflexive modules over Gorenstein rings which generalize results of Serre and Samuel on reflexive modules
More informationLie Algebra Cohomology
Lie Algebra Cohomology Carsten Liese 1 Chain Complexes Definition 1.1. A chain complex (C, d) of Rmodules is a family {C n } n Z of Rmodules, together with Rmodul maps d n : C n C n 1 such that d d
More informationGENERALIZED MORPHIC RINGS AND THEIR APPLICATIONS. Haiyan Zhu and Nanqing Ding Department of Mathematics, Nanjing University, Nanjing, China
Communications in Algebra, 35: 2820 2837, 2007 Copyright Taylor & Francis Group, LLC ISSN: 00927872 print/15324125 online DOI: 10.1080/00927870701354017 GENERALIZED MORPHIC RINGS AND THEIR APPLICATIONS
More informationRelative Left Derived Functors of Tensor Product Functors. Junfu Wang and Zhaoyong Huang
Relative Left Derived Functors of Tensor Product Functors Junfu Wang and Zhaoyong Huang Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, China Abstract We introduce and
More informationSolutions to Homework 1. All rings are commutative with identity!
Solutions to Homework 1. All rings are commutative with identity! (1) [4pts] Let R be a finite ring. Show that R = NZD(R). Proof. Let a NZD(R) and t a : R R the map defined by t a (r) = ar for all r R.
More informationALMOST GORENSTEIN RINGS
ALMOST GORENSTEIN RINGS SHIRO GOTO, NAOYUKI MATSUOKA, AND TRAN THI PHUONG Abstract. The notion of almost Gorenstein ring given by Barucci and Fröberg [2] in the case where the local rings are analytically
More informationF SINGULARITIES AND FROBENIUS SPLITTING NOTES 9/
F SINGULARITIES AND FROBENIUS SPLITTING NOTES 9/212010 KARL SCHWEDE 1. F rationality Definition 1.1. Given (M, φ) as above, the module τ(m, φ) is called the test submodule of (M, φ). With Ψ R : F ω
More informationINFINITE RINGS WITH PLANAR ZERODIVISOR GRAPHS
INFINITE RINGS WITH PLANAR ZERODIVISOR GRAPHS YONGWEI YAO Abstract. For any commutative ring R that is not a domain, there is a zerodivisor graph, denoted Γ(R), in which the vertices are the nonzero zerodivisors
More informationNOTES ON CHAIN COMPLEXES
NOTES ON CHAIN COMPLEXES ANDEW BAKE These notes are intended as a very basic introduction to (co)chain complexes and their algebra, the intention being to point the beginner at some of the main ideas which
More informationA NOTE ON BASIC FREE MODULES AND THE S n CONDITION. 1 Introduction
PORTUGALIAE MATHEMATICA Vol. 63 Fasc.4 2006 Nova Série A NOTE ON BASIC FREE MODULES AND THE S n CONDITION Agustín Marcelo, Félix Marcelo and César Rodríguez Abstract: A new definition of basic free submodule
More informationCotorsion pairs generated by modules of bounded projective dimension
Cotorsion pairs generated by modules of bounded projective dimension Silvana Bazzoni Dipartimento di Matematica Pura e Applicata, Università di Padova Via Trieste 63, 35121 Padova, Italy email: bazzoni@math.unipd.it
More informationNOTES ON SPLITTING FIELDS
NOTES ON SPLITTING FIELDS CİHAN BAHRAN I will try to define the notion of a splitting field of an algebra over a field using my words, to understand it better. The sources I use are Peter Webb s and T.Y
More information1. Algebraic vector bundles. Affine Varieties
0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasiprojective varieties over a field k Affine Varieties 1.
More informationA MODELTHEORETIC PROOF OF HILBERT S NULLSTELLENSATZ
A MODELTHEORETIC PROOF OF HILBERT S NULLSTELLENSATZ NICOLAS FORD Abstract. The goal of this paper is to present a proof of the Nullstellensatz using tools from a branch of logic called model theory. In
More informationModules Over Principal Ideal Domains
Modules Over Principal Ideal Domains Brian Whetter April 24, 2014 This work is licensed under the Creative Commons AttributionNonCommercialShareAlike 4.0 International License. To view a copy of this
More informationCHAPTER 1. AFFINE ALGEBRAIC VARIETIES
CHAPTER 1. AFFINE ALGEBRAIC VARIETIES During this first part of the course, we will establish a correspondence between various geometric notions and algebraic ones. Some references for this part of the
More information